WEBVTT 00:00:00.000 --> 00:00:02.410 align:middle line:90% [SQUEAKING] 00:00:02.410 --> 00:00:04.338 align:middle line:90% [RUSTLING] 00:00:04.338 --> 00:00:07.230 align:middle line:90% [CLICKING] 00:00:07.230 --> 00:00:10.620 align:middle line:90% 00:00:10.620 --> 00:00:13.030 align:middle line:84% GIAN PAOLO BERETTA: OK, here we are. 00:00:13.030 --> 00:00:14.230 align:middle line:90% Good afternoon. 00:00:14.230 --> 00:00:17.460 align:middle line:90% 00:00:17.460 --> 00:00:20.183 align:middle line:84% So we have done chemical kinetics. 00:00:20.183 --> 00:00:24.580 align:middle line:90% 00:00:24.580 --> 00:00:30.360 align:middle line:84% And today, again, with chemical kinetics, 00:00:30.360 --> 00:00:42.050 align:middle line:84% we are going to approach the truly non-equilibrium 00:00:42.050 --> 00:00:47.810 align:middle line:84% thermodynamics topics, including Onsager relations, 00:00:47.810 --> 00:00:54.650 align:middle line:84% that we will introduce today from the chemical kinetics 00:00:54.650 --> 00:00:56.210 align:middle line:90% point of view. 00:00:56.210 --> 00:01:01.860 align:middle line:84% So let me remind you of where we were. 00:01:01.860 --> 00:01:04.709 align:middle line:90% 00:01:04.709 --> 00:01:12.290 align:middle line:84% And also, it will allow us to introduce some new ideas 00:01:12.290 --> 00:01:15.040 align:middle line:90% on things that we already know. 00:01:15.040 --> 00:01:19.720 align:middle line:84% So consider a single chemical reaction mechanism, 00:01:19.720 --> 00:01:25.290 align:middle line:84% which we have learned to represent by this equation 00:01:25.290 --> 00:01:27.990 align:middle line:84% here, stoichiometric coefficients and the symbols 00:01:27.990 --> 00:01:30.750 align:middle line:90% for the chemical compounds. 00:01:30.750 --> 00:01:36.780 align:middle line:84% And this is the way we have written the reaction 00:01:36.780 --> 00:01:42.520 align:middle line:84% rates or more precisely, the rates of change of the reaction 00:01:42.520 --> 00:01:43.306 align:middle line:90% coordinates. 00:01:43.306 --> 00:01:46.180 align:middle line:90% 00:01:46.180 --> 00:01:52.720 align:middle line:84% And just as a reminder, these are the concentrations, 00:01:52.720 --> 00:01:56.440 align:middle line:84% stoichiometric coefficients of the forward and backward half 00:01:56.440 --> 00:01:58.890 align:middle line:90% reactions. 00:01:58.890 --> 00:02:01.020 align:middle line:84% These are the reaction constants, 00:02:01.020 --> 00:02:03.750 align:middle line:84% and for the reaction constants, we 00:02:03.750 --> 00:02:11.490 align:middle line:84% have developed two equivalent ways of writing them, 00:02:11.490 --> 00:02:18.630 align:middle line:84% one in terms of the Arrhenius activation energy 00:02:18.630 --> 00:02:26.480 align:middle line:84% and the other in terms of these equilibrium 00:02:26.480 --> 00:02:32.800 align:middle line:84% constants of the forward and backward reactions, half 00:02:32.800 --> 00:02:33.650 align:middle line:90% reactions. 00:02:33.650 --> 00:02:37.821 align:middle line:90% 00:02:37.821 --> 00:02:40.440 align:middle line:84% And of course, the net reaction rate 00:02:40.440 --> 00:02:43.380 align:middle line:84% is the difference between the forward and the backward. 00:02:43.380 --> 00:02:48.410 align:middle line:84% We define this ratio, and the logarithm of this ratio 00:02:48.410 --> 00:02:51.030 align:middle line:84% we call the degree of this equilibrium. 00:02:51.030 --> 00:02:53.760 align:middle line:84% And we have seen examples in which 00:02:53.760 --> 00:02:56.820 align:middle line:84% this degree of disequilibrium builds up 00:02:56.820 --> 00:03:04.590 align:middle line:84% as the chemical dynamics isn't fast enough 00:03:04.590 --> 00:03:07.510 align:middle line:84% to catch up and maintain equilibrium, for example, 00:03:07.510 --> 00:03:09.410 align:middle line:84% in the nozzle expansions that we have seen. 00:03:09.410 --> 00:03:12.240 align:middle line:90% 00:03:12.240 --> 00:03:15.240 align:middle line:84% Now, we can write this difference, also, 00:03:15.240 --> 00:03:17.730 align:middle line:90% by using this ratio. 00:03:17.730 --> 00:03:26.070 align:middle line:84% In this way, we factor epsilon plus out, and inside remains 00:03:26.070 --> 00:03:28.740 align:middle line:84% epsilon minus over epsilon plus, which 00:03:28.740 --> 00:03:32.030 align:middle line:90% is e to the minus the affinity. 00:03:32.030 --> 00:03:35.940 align:middle line:84% Or we can factor, also, epsilon dot minus, 00:03:35.940 --> 00:03:40.020 align:middle line:84% and this is the equivalent expression. 00:03:40.020 --> 00:03:47.730 align:middle line:84% Clearly, this shows that the reaction rates are functions 00:03:47.730 --> 00:04:01.440 align:middle line:90% of the affinities of-- 00:04:01.440 --> 00:04:05.860 align:middle line:84% each reaction is a function of its own affinity 00:04:05.860 --> 00:04:09.330 align:middle line:84% through this exponential, but it is also 00:04:09.330 --> 00:04:13.610 align:middle line:84% a function of a bunch of other stuff, which 00:04:13.610 --> 00:04:17.730 align:middle line:84% is included in here and in these concentrations. 00:04:17.730 --> 00:04:22.770 align:middle line:84% So it is not correct to say that the reaction rates are functions 00:04:22.770 --> 00:04:25.710 align:middle line:90% of the affinity, period. 00:04:25.710 --> 00:04:28.920 align:middle line:84% They are also function of the affinities. 00:04:28.920 --> 00:04:31.980 align:middle line:84% This is important because in a moment, 00:04:31.980 --> 00:04:39.300 align:middle line:84% we will see that you may want to construct other relations 00:04:39.300 --> 00:04:41.170 align:middle line:90% between rates and affinities. 00:04:41.170 --> 00:04:45.060 align:middle line:90% 00:04:45.060 --> 00:04:47.920 align:middle line:90% You can also do the following. 00:04:47.920 --> 00:04:54.040 align:middle line:84% You can divide the reaction rate by the affinity 00:04:54.040 --> 00:04:54.765 align:middle line:90% of that reaction. 00:04:54.765 --> 00:04:58.940 align:middle line:90% 00:04:58.940 --> 00:05:07.460 align:middle line:84% And therefore, this would look like-- 00:05:07.460 --> 00:05:19.050 align:middle line:84% and if I write epsilon dot j equals Ljj Yj, 00:05:19.050 --> 00:05:25.100 align:middle line:90% it looks like a linear relation. 00:05:25.100 --> 00:05:30.220 align:middle line:84% But it is not linear because the Ljj's depend on the affinities 00:05:30.220 --> 00:05:32.370 align:middle line:84% and depend on a bunch of other stuff. 00:05:32.370 --> 00:05:36.160 align:middle line:90% 00:05:36.160 --> 00:05:43.350 align:middle line:84% Nevertheless, this ratio, this L proportionality 00:05:43.350 --> 00:05:46.840 align:middle line:90% constant between-- 00:05:46.840 --> 00:05:50.740 align:middle line:84% which constant is not-- between the rate and affinity, 00:05:50.740 --> 00:05:52.480 align:middle line:90% can also be viewed as-- 00:05:52.480 --> 00:05:56.740 align:middle line:84% if you do a substitution, except for the gas 00:05:56.740 --> 00:05:58.750 align:middle line:84% constant at the denominator-- this 00:05:58.750 --> 00:06:04.000 align:middle line:84% is the logarithmic mean of the forward and the backward 00:06:04.000 --> 00:06:04.800 align:middle line:90% reactions. 00:06:04.800 --> 00:06:08.735 align:middle line:90% 00:06:08.735 --> 00:06:11.570 align:middle line:84% Now we are interested in seeing what happens 00:06:11.570 --> 00:06:14.420 align:middle line:90% if we are near equilibrium. 00:06:14.420 --> 00:06:17.570 align:middle line:84% We know that, at equilibrium, the affinity is equal to 0. 00:06:17.570 --> 00:06:24.600 align:middle line:84% So if we approach equilibrium, so the affinity 00:06:24.600 --> 00:06:29.030 align:middle line:84% is small, positive or negative, but small, 00:06:29.030 --> 00:06:35.820 align:middle line:84% then we can linearize these expressions here, e to the minus 00:06:35.820 --> 00:06:38.240 align:middle line:90% x. 00:06:38.240 --> 00:06:42.440 align:middle line:84% The first linear approximation is 1 minus x, 00:06:42.440 --> 00:06:47.370 align:middle line:90% or e to the x is 1 plus x. 00:06:47.370 --> 00:06:51.540 align:middle line:84% In either case, these linearizes-- 00:06:51.540 --> 00:06:53.380 align:middle line:90% becomes just the affinity. 00:06:53.380 --> 00:06:56.670 align:middle line:90% 00:06:56.670 --> 00:07:02.610 align:middle line:84% And so we get an expression like this one, 00:07:02.610 --> 00:07:11.310 align:middle line:84% but with the coefficient now being the value at Y equals 0. 00:07:11.310 --> 00:07:13.710 align:middle line:84% In the Taylor series expansion, you 00:07:13.710 --> 00:07:19.320 align:middle line:84% have to take the constant terms and evaluate them where 00:07:19.320 --> 00:07:20.970 align:middle line:90% you do the expansion about. 00:07:20.970 --> 00:07:25.860 align:middle line:84% In this case, it's when the affinity is equal to 0. 00:07:25.860 --> 00:07:29.790 align:middle line:84% And when the affinity is equal to 0, the epsilon plus and minus 00:07:29.790 --> 00:07:34.530 align:middle line:84% tend to the same value, which is epsilon 0, the equilibrium 00:07:34.530 --> 00:07:37.280 align:middle line:90% value, which is also-- 00:07:37.280 --> 00:07:39.590 align:middle line:84% so since they are equal, also their mean, 00:07:39.590 --> 00:07:42.625 align:middle line:84% even the logarithmic mean, is equal. 00:07:42.625 --> 00:07:45.570 align:middle line:90% 00:07:45.570 --> 00:07:51.420 align:middle line:84% And you can also express that if you use this definition down 00:07:51.420 --> 00:07:56.220 align:middle line:84% here and what we have done in the previous lectures. 00:07:56.220 --> 00:08:02.010 align:middle line:84% You can also connect it to the concentration 00:08:02.010 --> 00:08:05.253 align:middle line:84% at equilibrium of the activated complex. 00:08:05.253 --> 00:08:10.666 align:middle line:90% 00:08:10.666 --> 00:08:12.920 align:middle line:90% So we've linearized. 00:08:12.920 --> 00:08:16.900 align:middle line:90% Now we have a linear relation. 00:08:16.900 --> 00:08:19.210 align:middle line:84% And if we substitute this linear relation 00:08:19.210 --> 00:08:21.460 align:middle line:84% in the expression for the entropy, 00:08:21.460 --> 00:08:28.610 align:middle line:84% rate of change of the entropy, which for a closed system, 00:08:28.610 --> 00:08:32.250 align:middle line:84% is also the entropy produced by reversibility. 00:08:32.250 --> 00:08:35.419 align:middle line:90% So it has to be greater than 0. 00:08:35.419 --> 00:08:42.789 align:middle line:84% You see that it becomes the square-- 00:08:42.789 --> 00:08:49.880 align:middle line:84% It's a constant, this L, times the affinity-- 00:08:49.880 --> 00:08:56.030 align:middle line:84% I mean, yes, the affinity squared, 00:08:56.030 --> 00:09:01.790 align:middle line:84% or the inverse of L times the rate squared. 00:09:01.790 --> 00:09:09.590 align:middle line:84% This is why we can identify the inverse of this ratio L 00:09:09.590 --> 00:09:17.990 align:middle line:84% as a sort of a chemical resistance to going back 00:09:17.990 --> 00:09:21.040 align:middle line:90% to equilibrium. 00:09:21.040 --> 00:09:28.970 align:middle line:84% The higher this, it means the lower the L. 00:09:28.970 --> 00:09:31.710 align:middle line:84% And therefore, for a given disequilibrium, 00:09:31.710 --> 00:09:34.490 align:middle line:84% because the affinity is disequilibrium, for a given 00:09:34.490 --> 00:09:41.910 align:middle line:84% disequilibrium, you have a lesser rate towards equilibrium. 00:09:41.910 --> 00:09:47.426 align:middle line:90% 00:09:47.426 --> 00:09:52.620 align:middle line:84% Now, in practice, we have seen, like in this example that we've 00:09:52.620 --> 00:10:00.180 align:middle line:84% discussed a few times, hydrogen, the detailed kinetic mechanism 00:10:00.180 --> 00:10:05.460 align:middle line:84% contains a lot of chemical reactions. 00:10:05.460 --> 00:10:10.080 align:middle line:84% And you need them in order to describe the dynamics. 00:10:10.080 --> 00:10:13.440 align:middle line:84% However, you may notice that these reactions are not 00:10:13.440 --> 00:10:16.230 align:middle line:90% all independent of one another. 00:10:16.230 --> 00:10:22.030 align:middle line:84% Just for an example, here, if you 00:10:22.030 --> 00:10:30.380 align:middle line:84% take the 11 and the 14 reaction and subtract them, 00:10:30.380 --> 00:10:32.450 align:middle line:84% it gives you the same result as if you 00:10:32.450 --> 00:10:37.400 align:middle line:84% take the first and the third reaction and you subtract them. 00:10:37.400 --> 00:10:39.900 align:middle line:90% So they are linearly combined. 00:10:39.900 --> 00:10:43.690 align:middle line:90% They are not independent. 00:10:43.690 --> 00:10:50.710 align:middle line:84% And there may be-- there are many other linear dependencies 00:10:50.710 --> 00:10:52.300 align:middle line:90% here. 00:10:52.300 --> 00:10:53.690 align:middle line:90% So let's go back. 00:10:53.690 --> 00:10:55.810 align:middle line:84% So up to here, it's the same slide 00:10:55.810 --> 00:10:59.230 align:middle line:90% that we were on a minute ago. 00:10:59.230 --> 00:11:01.430 align:middle line:90% This is the new part. 00:11:01.430 --> 00:11:09.590 align:middle line:84% So the idea is that now we have a complex chemical detailed 00:11:09.590 --> 00:11:14.630 align:middle line:90% scheme or a reaction network. 00:11:14.630 --> 00:11:25.990 align:middle line:84% And it contains a lot of linearly-dependent reactions. 00:11:25.990 --> 00:11:31.390 align:middle line:84% That means that we can identify a subset of linearly independent 00:11:31.390 --> 00:11:42.270 align:middle line:84% ones, with respect to which you can write all the others. 00:11:42.270 --> 00:11:44.940 align:middle line:90% 00:11:44.940 --> 00:11:49.350 align:middle line:84% So let's suppose that the stoichiometric coefficients-- 00:11:49.350 --> 00:11:51.400 align:middle line:84% now, here, I'm running out of symbols. 00:11:51.400 --> 00:11:58.410 align:middle line:84% So instead of new, I'm using this other letter 00:11:58.410 --> 00:12:02.700 align:middle line:84% that I wish your Greek colleague was here today, because I 00:12:02.700 --> 00:12:05.430 align:middle line:90% forget what the name is. 00:12:05.430 --> 00:12:10.670 align:middle line:84% In any case, this is the stoichiometric coefficients 00:12:10.670 --> 00:12:13.940 align:middle line:84% of the linearly independent mechanism, with respect to which 00:12:13.940 --> 00:12:16.540 align:middle line:90% I can write all the other ones. 00:12:16.540 --> 00:12:22.610 align:middle line:84% So all the other schemes, for example, in the 24 for hydrogen, 00:12:22.610 --> 00:12:23.940 align:middle line:90% are linear combinations. 00:12:23.940 --> 00:12:26.810 align:middle line:84% So these x's are the linear combination coefficients 00:12:26.810 --> 00:12:30.260 align:middle line:90% of the independent ones. 00:12:30.260 --> 00:12:45.070 align:middle line:84% And due to the linearity of the way the affinities are defined, 00:12:45.070 --> 00:12:47.350 align:middle line:84% also, the affinity of the j-th reaction 00:12:47.350 --> 00:12:50.710 align:middle line:84% is the same combination with the same linear coefficients 00:12:50.710 --> 00:12:55.100 align:middle line:84% of the affinities of the independent ones. 00:12:55.100 --> 00:12:58.610 align:middle line:84% And so when you substitute these into the expression 00:12:58.610 --> 00:13:05.520 align:middle line:84% for the rate of entropy change, which for many reactions 00:13:05.520 --> 00:13:09.060 align:middle line:84% is the sum of the products of the rate times the affinity 00:13:09.060 --> 00:13:14.600 align:middle line:84% for each reaction, that substitution plus a change of-- 00:13:14.600 --> 00:13:17.390 align:middle line:90% 00:13:17.390 --> 00:13:31.470 align:middle line:84% a switch of summation symbols and recognizing that this one-- 00:13:31.470 --> 00:13:35.660 align:middle line:84% I'm sorry, that this linear combination of the rates 00:13:35.660 --> 00:13:41.150 align:middle line:84% is also the rate of the independent reaction, 00:13:41.150 --> 00:13:44.090 align:middle line:84% it turns out that this summation, which before 00:13:44.090 --> 00:13:48.590 align:middle line:84% had the 24 terms, now has the minimum number of terms, which 00:13:48.590 --> 00:13:52.220 align:middle line:84% is the number of independent reactions, their rates 00:13:52.220 --> 00:13:53.300 align:middle line:90% and their affinities. 00:13:53.300 --> 00:13:56.430 align:middle line:84% It's the same product of a rate and an affinity. 00:13:56.430 --> 00:14:06.100 align:middle line:90% 00:14:06.100 --> 00:14:08.505 align:middle line:90% That's what we just got. 00:14:08.505 --> 00:14:11.780 align:middle line:90% 00:14:11.780 --> 00:14:13.850 align:middle line:84% And above is just a reminder what 00:14:13.850 --> 00:14:20.010 align:middle line:84% was the essential of the previous slide. 00:14:20.010 --> 00:14:33.200 align:middle line:84% So if we keep going and substitute, again, 00:14:33.200 --> 00:14:43.600 align:middle line:84% the expression for the affinities of the j-th reaction 00:14:43.600 --> 00:14:52.000 align:middle line:84% in terms of this other upsilon affinity 00:14:52.000 --> 00:14:55.515 align:middle line:84% of the independent ones, you come up with this. 00:14:55.515 --> 00:14:58.600 align:middle line:90% 00:14:58.600 --> 00:15:06.700 align:middle line:84% And therefore, you have a quadratic form 00:15:06.700 --> 00:15:15.900 align:middle line:84% in the coefficients of the linear combination that couple 00:15:15.900 --> 00:15:20.250 align:middle line:84% and combine the L's, so the ratio 00:15:20.250 --> 00:15:24.640 align:middle line:84% rate to affinity of the single non-independent reactions. 00:15:24.640 --> 00:15:27.935 align:middle line:84% So this linear combination, if you notice-- 00:15:27.935 --> 00:15:30.800 align:middle line:90% 00:15:30.800 --> 00:15:33.780 align:middle line:84% not linear, this is a quadratic combination of the L's-- 00:15:33.780 --> 00:15:36.740 align:middle line:90% If you notice, is symmetric. 00:15:36.740 --> 00:15:41.290 align:middle line:84% Because if you change k and ell it doesn't change. 00:15:41.290 --> 00:15:45.940 align:middle line:90% 00:15:45.940 --> 00:15:49.390 align:middle line:84% So we come up with the fact that the entropy production 00:15:49.390 --> 00:15:52.420 align:middle line:84% is a quadratic form in the affinities 00:15:52.420 --> 00:15:57.710 align:middle line:90% of the independent reactions. 00:15:57.710 --> 00:16:03.410 align:middle line:84% And the matrix of the quadratic form is non-negative. 00:16:03.410 --> 00:16:05.510 align:middle line:84% That's required by the fact that you 00:16:05.510 --> 00:16:09.950 align:middle line:84% need that to be non-negative to satisfy the principle 00:16:09.950 --> 00:16:12.230 align:middle line:90% of entropy nondecrease. 00:16:12.230 --> 00:16:14.100 align:middle line:90% And it is also symmetrical. 00:16:14.100 --> 00:16:19.380 align:middle line:90% 00:16:19.380 --> 00:16:21.793 align:middle line:84% You take this matrix and take its inverse. 00:16:21.793 --> 00:16:25.200 align:middle line:90% 00:16:25.200 --> 00:16:35.320 align:middle line:84% You can rewrite this double summation-- 00:16:35.320 --> 00:16:37.900 align:middle line:84% instead of in terms of the quadratic form 00:16:37.900 --> 00:16:41.800 align:middle line:84% in the affinities, it's a quadratic form 00:16:41.800 --> 00:16:45.430 align:middle line:84% in the rates of the independent reactions, where 00:16:45.430 --> 00:16:47.140 align:middle line:90% the coefficients are these. 00:16:47.140 --> 00:16:52.656 align:middle line:84% And these R's are some sort of generalized resistance. 00:16:52.656 --> 00:16:59.010 align:middle line:90% 00:16:59.010 --> 00:17:03.030 align:middle line:84% Now, these independent mechanisms 00:17:03.030 --> 00:17:11.750 align:middle line:84% are coupled via these linear combinations. 00:17:11.750 --> 00:17:17.280 align:middle line:84% So if you try to write the rate of each one, 00:17:17.280 --> 00:17:25.690 align:middle line:84% it is not any longer just dependent on its affinity. 00:17:25.690 --> 00:17:28.275 align:middle line:90% It depends also on the others. 00:17:28.275 --> 00:17:31.350 align:middle line:90% 00:17:31.350 --> 00:17:38.560 align:middle line:84% And this coupling is important and is 00:17:38.560 --> 00:17:42.240 align:middle line:84% typical of non-equilibrium behavior. 00:17:42.240 --> 00:17:45.370 align:middle line:90% 00:17:45.370 --> 00:17:48.310 align:middle line:84% And we will see more about this because that's 00:17:48.310 --> 00:17:52.840 align:middle line:84% a good introduction to what happens in many other phenomena 00:17:52.840 --> 00:17:55.970 align:middle line:90% for non-equilibrium. 00:17:55.970 --> 00:17:59.947 align:middle line:84% Of course, I mean, so far, we didn't linearize yet. 00:17:59.947 --> 00:18:01.280 align:middle line:90% But of course, we can linearize. 00:18:01.280 --> 00:18:05.600 align:middle line:84% Again, we can assume that we go near equilibrium, where 00:18:05.600 --> 00:18:08.750 align:middle line:90% all the affinities go to 0. 00:18:08.750 --> 00:18:12.770 align:middle line:84% So that's the linearization that we did before. 00:18:12.770 --> 00:18:18.000 align:middle line:84% And this is the linear combination of the rates. 00:18:18.000 --> 00:18:21.060 align:middle line:84% So the rate of the independent reactions 00:18:21.060 --> 00:18:23.700 align:middle line:90% becomes summation, like this. 00:18:23.700 --> 00:18:25.870 align:middle line:84% And if you keep going, in the end, 00:18:25.870 --> 00:18:37.530 align:middle line:84% you find that it is a summation of those affinities 00:18:37.530 --> 00:18:44.140 align:middle line:84% times these generalized inverse resistance. 00:18:44.140 --> 00:18:51.670 align:middle line:84% So let's call it the generalized conductivity matrix, symmetric. 00:18:51.670 --> 00:18:57.300 align:middle line:84% And again, the rate of entropy production 00:18:57.300 --> 00:19:01.200 align:middle line:84% is a quadratic form either in the affinities or in the rates. 00:19:01.200 --> 00:19:04.770 align:middle line:84% This time, it is a real quadratic form. 00:19:04.770 --> 00:19:11.990 align:middle line:84% Up here, I was a bit sloppy in saying it is a quadratic form. 00:19:11.990 --> 00:19:20.430 align:middle line:84% But these coefficients may also depend on the affinities 00:19:20.430 --> 00:19:22.270 align:middle line:84% when you're far from equilibrium. 00:19:22.270 --> 00:19:25.110 align:middle line:84% So it's not really a quadratic dependence 00:19:25.110 --> 00:19:26.770 align:middle line:90% when you're off equilibrium. 00:19:26.770 --> 00:19:29.550 align:middle line:84% Whereas, when you linearize, then it 00:19:29.550 --> 00:19:33.560 align:middle line:90% becomes a quadratic form. 00:19:33.560 --> 00:19:36.500 align:middle line:90% 00:19:36.500 --> 00:19:39.860 align:middle line:84% And these properties here, since they 00:19:39.860 --> 00:19:44.530 align:middle line:84% are evaluated at equilibrium, they are equilibrium properties. 00:19:44.530 --> 00:19:49.540 align:middle line:84% So if you want to measure them, for example, 00:19:49.540 --> 00:19:52.330 align:middle line:84% in a molecular dynamics simulation, 00:19:52.330 --> 00:20:03.460 align:middle line:84% you will have go and wait for identifying an equilibrium state 00:20:03.460 --> 00:20:06.560 align:middle line:90% in order to evaluate this. 00:20:06.560 --> 00:20:11.810 align:middle line:84% Which makes it easy because equilibrium is reached quickly 00:20:11.810 --> 00:20:14.270 align:middle line:90% because of collisions. 00:20:14.270 --> 00:20:21.620 align:middle line:84% And so these are easily measurable properties. 00:20:21.620 --> 00:20:28.160 align:middle line:84% Or since we know if we have the full reaction mechanism, 00:20:28.160 --> 00:20:31.860 align:middle line:84% like the 25, 24 reactions for hydrogen, 00:20:31.860 --> 00:20:36.520 align:middle line:84% then you can infer these values from those rates. 00:20:36.520 --> 00:20:48.914 align:middle line:90% 00:20:48.914 --> 00:20:52.390 align:middle line:84% So now let's step back one moment 00:20:52.390 --> 00:21:03.690 align:middle line:84% and assume we don't have developed a theory 00:21:03.690 --> 00:21:08.050 align:middle line:90% for the reaction rates. 00:21:08.050 --> 00:21:12.590 align:middle line:84% And all we know is, from the entropy balance, 00:21:12.590 --> 00:21:14.600 align:middle line:84% this expression, which is one of the first 00:21:14.600 --> 00:21:16.890 align:middle line:84% that we derived when we did chemical reactions. 00:21:16.890 --> 00:21:19.760 align:middle line:84% Because remember that the affinity is the derivative 00:21:19.760 --> 00:21:23.160 align:middle line:90% of S with respect to epsilon. 00:21:23.160 --> 00:21:26.280 align:middle line:84% So this is simply the rate of change, the derivative 00:21:26.280 --> 00:21:29.490 align:middle line:84% of S with respect to t, which here 00:21:29.490 --> 00:21:33.090 align:middle line:84% is taken to be the entropy produced 00:21:33.090 --> 00:21:37.810 align:middle line:84% by reversibility because we are considering the isolated system. 00:21:37.810 --> 00:21:42.530 align:middle line:90% So that is our constraint. 00:21:42.530 --> 00:21:44.810 align:middle line:84% It's part of our model with which 00:21:44.810 --> 00:21:51.320 align:middle line:84% we like to describe properties, even far from equilibrium, 00:21:51.320 --> 00:21:56.220 align:middle line:84% from chemical equilibrium, but assuming a pseudo equilibrium. 00:21:56.220 --> 00:22:02.510 align:middle line:90% 00:22:02.510 --> 00:22:07.880 align:middle line:84% We also know that at maximum entropy, which 00:22:07.880 --> 00:22:10.280 align:middle line:84% is stable equilibrium, chemical equilibrium, 00:22:10.280 --> 00:22:12.890 align:middle line:84% the affinities must be equal to 0. 00:22:12.890 --> 00:22:15.800 align:middle line:84% And now suppose that our theory, we 00:22:15.800 --> 00:22:19.490 align:middle line:84% want to develop a theory in which the rates are functions 00:22:19.490 --> 00:22:28.370 align:middle line:84% of the affinities and nothing else 00:22:28.370 --> 00:22:30.200 align:middle line:84% or maybe some other parameters that 00:22:30.200 --> 00:22:33.840 align:middle line:84% can be kept constant while you change the affinities. 00:22:33.840 --> 00:22:36.450 align:middle line:90% 00:22:36.450 --> 00:22:40.210 align:middle line:84% Clearly, when you are at equilibrium, 00:22:40.210 --> 00:22:41.750 align:middle line:90% the net rate must be 0. 00:22:41.750 --> 00:22:46.160 align:middle line:84% So you want this function to be 0 when the argument is 0. 00:22:46.160 --> 00:22:50.070 align:middle line:90% And this argument is a vector. 00:22:50.070 --> 00:22:51.560 align:middle line:90% So that's why it's boldface. 00:22:51.560 --> 00:22:52.740 align:middle line:90% So this is 0. 00:22:52.740 --> 00:23:00.600 align:middle line:84% This is a list of conditions, so that you can substitute that 00:23:00.600 --> 00:23:02.380 align:middle line:84% into the expression for the entropy. 00:23:02.380 --> 00:23:06.020 align:middle line:84% And you get that the entropy, the rate of change 00:23:06.020 --> 00:23:10.530 align:middle line:84% is also a function of the affinity. 00:23:10.530 --> 00:23:17.320 align:middle line:84% Now you linearize, linearize around equilibrium, 00:23:17.320 --> 00:23:26.170 align:middle line:84% these postulated constitutive equation, which presumably, 00:23:26.170 --> 00:23:28.500 align:middle line:90% should represent our rates. 00:23:28.500 --> 00:23:32.590 align:middle line:90% 00:23:32.590 --> 00:23:39.560 align:middle line:84% So we do the linear Taylor series. 00:23:39.560 --> 00:23:42.565 align:middle line:84% And we will stop at the first term that is on 0. 00:23:42.565 --> 00:23:45.640 align:middle line:90% 00:23:45.640 --> 00:23:47.210 align:middle line:90% So the first term is 0. 00:23:47.210 --> 00:23:49.270 align:middle line:90% This should be a boldface 0. 00:23:49.270 --> 00:23:54.100 align:middle line:84% Then we have plus the first derivative evaluated at 0. 00:23:54.100 --> 00:24:00.580 align:middle line:84% This is also another bold and that multiplied by Y. 00:24:00.580 --> 00:24:05.750 align:middle line:84% So this is the linear term plus dot, dot, dots, 00:24:05.750 --> 00:24:12.700 align:middle line:84% because, in fact, this is already different from 0. 00:24:12.700 --> 00:24:17.960 align:middle line:84% I'm going to call these partial derivative 00:24:17.960 --> 00:24:23.710 align:middle line:84% as L, our generalized conductivity, the inverse 00:24:23.710 --> 00:24:25.910 align:middle line:90% of the resistance matrix. 00:24:25.910 --> 00:24:29.060 align:middle line:90% It's a matrix. 00:24:29.060 --> 00:24:30.890 align:middle line:84% And it's a matrix that we can also 00:24:30.890 --> 00:24:38.600 align:middle line:84% write as the Jacobian matrix of the various rates 00:24:38.600 --> 00:24:45.420 align:middle line:84% as functions of the various affinities, big matrix to be 00:24:45.420 --> 00:24:48.070 align:middle line:90% evaluated at Y is equal to 0. 00:24:48.070 --> 00:24:51.610 align:middle line:90% 00:24:51.610 --> 00:24:54.580 align:middle line:84% Then you take this expression, substitute it 00:24:54.580 --> 00:24:57.040 align:middle line:84% into the expression for the entropy production, 00:24:57.040 --> 00:25:01.750 align:middle line:84% and get that the entropy production is a quadratic form 00:25:01.750 --> 00:25:03.970 align:middle line:84% in the affinities, with the matrix 00:25:03.970 --> 00:25:08.200 align:middle line:84% of the generalized conductivity set to serve as coefficients 00:25:08.200 --> 00:25:15.240 align:middle line:84% of that form that requires that, being this, 00:25:15.240 --> 00:25:17.700 align:middle line:84% the dominating term in your expansion, 00:25:17.700 --> 00:25:20.050 align:middle line:84% this matrix must be non-negative in order 00:25:20.050 --> 00:25:22.740 align:middle line:84% to satisfy the principle of entropy nondecrease. 00:25:22.740 --> 00:25:26.730 align:middle line:90% 00:25:26.730 --> 00:25:32.250 align:middle line:84% You may also do the linear expansion of this expression 00:25:32.250 --> 00:25:34.860 align:middle line:84% here for the entropy with respect 00:25:34.860 --> 00:25:41.810 align:middle line:84% to, again, to Y around the equilibrium. 00:25:41.810 --> 00:25:49.310 align:middle line:84% If you do that, so you get the term at equilibrium 00:25:49.310 --> 00:25:51.360 align:middle line:84% is 0 because the affinities are 0. 00:25:51.360 --> 00:25:52.440 align:middle line:90% You put 0 here. 00:25:52.440 --> 00:25:54.200 align:middle line:90% You get 0. 00:25:54.200 --> 00:26:02.610 align:middle line:84% The first derivative of S dot with respect to Y at equilibrium 00:26:02.610 --> 00:26:05.250 align:middle line:90% is also 0. 00:26:05.250 --> 00:26:07.886 align:middle line:84% When you evaluate it, here is the evaluation. 00:26:07.886 --> 00:26:10.610 align:middle line:90% 00:26:10.610 --> 00:26:14.720 align:middle line:84% See-- you have to take the derivative properly 00:26:14.720 --> 00:26:18.380 align:middle line:84% of this expression here with respect to the Y's, and it's 00:26:18.380 --> 00:26:21.180 align:middle line:84% better to do them with the indexes. 00:26:21.180 --> 00:26:24.145 align:middle line:84% Otherwise, you will miss some details. 00:26:24.145 --> 00:26:27.480 align:middle line:90% 00:26:27.480 --> 00:26:31.670 align:middle line:90% So it's d S with respect to Yj. 00:26:31.670 --> 00:26:34.640 align:middle line:90% 00:26:34.640 --> 00:26:38.450 align:middle line:84% it picks up from this summation the epsilon 00:26:38.450 --> 00:26:43.550 align:middle line:84% j plus the summation of the Yi's times 00:26:43.550 --> 00:26:48.220 align:middle line:84% the derivative of epsilon i with respect to Yj. 00:26:48.220 --> 00:26:52.800 align:middle line:84% And at equilibrium, the epsilons are 0. 00:26:52.800 --> 00:26:53.760 align:middle line:90% The Y's are 0. 00:26:53.760 --> 00:26:54.770 align:middle line:90% So this is 0. 00:26:54.770 --> 00:26:59.690 align:middle line:84% That is why this first term, the linear term in the entropy 00:26:59.690 --> 00:27:06.000 align:middle line:84% disappears, showing that indeed, we are 00:27:06.000 --> 00:27:09.790 align:middle line:90% at the maximum or a minimum. 00:27:09.790 --> 00:27:13.190 align:middle line:90% 00:27:13.190 --> 00:27:16.490 align:middle line:90% The derivative is 0. 00:27:16.490 --> 00:27:20.610 align:middle line:84% And then we go to the second term in the Taylor expansion. 00:27:20.610 --> 00:27:23.150 align:middle line:90% 00:27:23.150 --> 00:27:24.570 align:middle line:90% So this is the general one. 00:27:24.570 --> 00:27:26.320 align:middle line:84% But then you remember that the coefficient 00:27:26.320 --> 00:27:28.940 align:middle line:90% has to be evaluated at 0. 00:27:28.940 --> 00:27:32.270 align:middle line:84% So you have to evaluate this Hessian 00:27:32.270 --> 00:27:41.800 align:middle line:84% of the matrix of the second order coefficients, second order 00:27:41.800 --> 00:27:44.990 align:middle line:90% derivatives. 00:27:44.990 --> 00:27:48.590 align:middle line:84% And so you take its derivative with respect 00:27:48.590 --> 00:27:51.720 align:middle line:90% to Yk of this first derivative. 00:27:51.720 --> 00:27:56.480 align:middle line:84% So you get the derivative of epsilon j with respect to Yk. 00:27:56.480 --> 00:28:02.980 align:middle line:84% Then from the summation corresponding to the Yk, 00:28:02.980 --> 00:28:07.960 align:middle line:84% you pick up the coefficient that multiplies Yk 00:28:07.960 --> 00:28:12.830 align:middle line:84% is partial of epsilon k with respect to Yj. 00:28:12.830 --> 00:28:17.190 align:middle line:84% And then the summation, also has some leftovers, 00:28:17.190 --> 00:28:21.690 align:middle line:84% which are the second derivatives of the epsilons. 00:28:21.690 --> 00:28:25.120 align:middle line:84% However, when you evaluate them at equilibrium, these are 0. 00:28:25.120 --> 00:28:27.010 align:middle line:90% So the summation disappears. 00:28:27.010 --> 00:28:30.390 align:middle line:90% But these ones do not. 00:28:30.390 --> 00:28:34.920 align:middle line:84% And notice that this is not equal to that, 00:28:34.920 --> 00:28:37.780 align:middle line:84% because the one is derivative of j with respect to k. 00:28:37.780 --> 00:28:40.760 align:middle line:84% The other is derivative of k with respect to j. 00:28:40.760 --> 00:28:48.070 align:middle line:84% So, essentially, you are left with this matrix here 00:28:48.070 --> 00:28:51.280 align:middle line:90% plus its transpose. 00:28:51.280 --> 00:28:56.210 align:middle line:84% When you switch the j with k, you get the transpose. 00:28:56.210 --> 00:29:01.460 align:middle line:90% 00:29:01.460 --> 00:29:07.370 align:middle line:84% And so clearly, the sum of the matrix plus its transpose 00:29:07.370 --> 00:29:11.950 align:middle line:84% is a symmetric matrix, which satisfies the fact 00:29:11.950 --> 00:29:15.640 align:middle line:84% that we know like the Schwartz theorem of second derivatives, 00:29:15.640 --> 00:29:17.450 align:middle line:90% that they're symmetric. 00:29:17.450 --> 00:29:19.660 align:middle line:90% It doesn't depend on the order. 00:29:19.660 --> 00:29:23.200 align:middle line:90% So that is OK. 00:29:23.200 --> 00:29:25.575 align:middle line:84% So we get the same expression here. 00:29:25.575 --> 00:29:28.180 align:middle line:90% 00:29:28.180 --> 00:29:34.750 align:middle line:84% However, in addition to non-negativity, 00:29:34.750 --> 00:29:39.220 align:middle line:84% there is an additional constraint that was proved. 00:29:39.220 --> 00:29:42.940 align:middle line:84% And we will try to see how we get there 00:29:42.940 --> 00:29:47.000 align:middle line:84% as we go on, was proved by Onsager in 1931, 00:29:47.000 --> 00:29:49.760 align:middle line:84% for which he got the Nobel Prize in '68. 00:29:49.760 --> 00:29:57.100 align:middle line:84% And he used the statistical considerations 00:29:57.100 --> 00:30:02.336 align:middle line:84% to show that this matrix needs to be symmetric. 00:30:02.336 --> 00:30:05.510 align:middle line:90% 00:30:05.510 --> 00:30:10.140 align:middle line:84% And this symmetry is not a trivial result. 00:30:10.140 --> 00:30:15.260 align:middle line:84% It's actually the basis for a number of rationalizations 00:30:15.260 --> 00:30:21.380 align:middle line:84% of coupled non-equilibrium phenomena 00:30:21.380 --> 00:30:26.450 align:middle line:84% that are used in a variety of applications, of which we 00:30:26.450 --> 00:30:33.090 align:middle line:84% will touch a few as we go on and until the end of the course. 00:30:33.090 --> 00:30:35.810 align:middle line:84% A few lectures that we are left are 00:30:35.810 --> 00:30:41.660 align:middle line:84% going to be all around seeing the consequences 00:30:41.660 --> 00:30:45.980 align:middle line:84% of these important result, in addition to seeing 00:30:45.980 --> 00:30:49.850 align:middle line:90% how you could prove it. 00:30:49.850 --> 00:30:53.420 align:middle line:90% 00:30:53.420 --> 00:31:03.220 align:middle line:84% Now, this is just a reminder of the simple mathematics 00:31:03.220 --> 00:31:12.490 align:middle line:84% of when you have a system, like for a 2 by 2 matrix L, what 00:31:12.490 --> 00:31:14.860 align:middle line:84% is the consequence if the matrix is 00:31:14.860 --> 00:31:18.050 align:middle line:84% symmetric and positive and positive 00:31:18.050 --> 00:31:20.090 align:middle line:90% semi-definite or non-negative. 00:31:20.090 --> 00:31:24.350 align:middle line:90% 00:31:24.350 --> 00:31:34.800 align:middle line:84% And also, by seeing these more explicitly, 00:31:34.800 --> 00:31:37.020 align:middle line:84% we will return to these relations 00:31:37.020 --> 00:31:39.280 align:middle line:84% not only for chemistry, like here, 00:31:39.280 --> 00:31:42.630 align:middle line:84% but later on, also, we will have coupled effects. 00:31:42.630 --> 00:31:45.270 align:middle line:84% And we will typically take them 2 by 2. 00:31:45.270 --> 00:31:48.690 align:middle line:84% Because it's already complicated with two. 00:31:48.690 --> 00:31:54.330 align:middle line:84% And going with more, you have to first have understood the two, 00:31:54.330 --> 00:31:56.290 align:middle line:84% and then you go to more complicated. 00:31:56.290 --> 00:32:02.750 align:middle line:90% 00:32:02.750 --> 00:32:06.710 align:middle line:90% So that's the linear relation. 00:32:06.710 --> 00:32:12.410 align:middle line:84% So it says that the rate of reaction 1 00:32:12.410 --> 00:32:16.670 align:middle line:90% is proportional to the-- 00:32:16.670 --> 00:32:20.150 align:middle line:84% is partly due to the fact that the reaction one 00:32:20.150 --> 00:32:23.320 align:middle line:84% is in disequilibrium, epsilon 1 (I meant, Y 1). 00:32:23.320 --> 00:32:30.300 align:middle line:84% But also, it's coupled with the disequilibrium of reaction 2. 00:32:30.300 --> 00:32:38.560 align:middle line:84% So in some extreme cases, you might have a reaction 1 00:32:38.560 --> 00:32:45.440 align:middle line:84% that advances not due to its own degree of disequilibrium, 00:32:45.440 --> 00:32:47.830 align:middle line:84% but due to the degree of disequilibrium 00:32:47.830 --> 00:32:48.675 align:middle line:90% of another reaction. 00:32:48.675 --> 00:32:53.770 align:middle line:90% 00:32:53.770 --> 00:32:57.470 align:middle line:84% And you may rewrite this system of equations in various ways. 00:32:57.470 --> 00:32:58.430 align:middle line:90% You can solve it. 00:32:58.430 --> 00:33:01.550 align:middle line:84% So you can write the affinities as functions of the rates. 00:33:01.550 --> 00:33:03.110 align:middle line:84% Or you can do something in between, 00:33:03.110 --> 00:33:10.500 align:middle line:84% in which, say, the rate of 1 is given by its own-- 00:33:10.500 --> 00:33:14.230 align:middle line:84% a coefficient times its own affinity, 00:33:14.230 --> 00:33:19.020 align:middle line:84% plus something that relates it to the rate 00:33:19.020 --> 00:33:22.190 align:middle line:84% of the other equation, of the other reaction. 00:33:22.190 --> 00:33:24.870 align:middle line:90% 00:33:24.870 --> 00:33:31.410 align:middle line:84% Now, when a matrix is positive semi-definite, 00:33:31.410 --> 00:33:41.150 align:middle line:84% it means that the elements on the diagonal are non-negative. 00:33:41.150 --> 00:33:46.030 align:middle line:84% It means that the eigenvalues are non-negative. 00:33:46.030 --> 00:33:48.530 align:middle line:84% Therefore, also, the product of the eigenvalues, 00:33:48.530 --> 00:33:51.620 align:middle line:84% which is the determinant, is non-negative. 00:33:51.620 --> 00:33:55.170 align:middle line:84% The sum, which is the trace of the matrix, 00:33:55.170 --> 00:33:57.030 align:middle line:90% is non-negative, and so on. 00:33:57.030 --> 00:33:59.780 align:middle line:90% 00:33:59.780 --> 00:34:02.450 align:middle line:84% And the rate of entropy production 00:34:02.450 --> 00:34:05.650 align:middle line:84% is a quadratic form of this matrix. 00:34:05.650 --> 00:34:10.380 align:middle line:84% And the quadratic form is written in general, this way. 00:34:10.380 --> 00:34:13.460 align:middle line:90% 00:34:13.460 --> 00:34:16.050 align:middle line:84% However, if you write it in this way, 00:34:16.050 --> 00:34:22.010 align:middle line:84% you notice that this double product in the middle 00:34:22.010 --> 00:34:26.000 align:middle line:84% may not have a definite sign, because the affinities can 00:34:26.000 --> 00:34:27.380 align:middle line:90% be positive and negative. 00:34:27.380 --> 00:34:30.630 align:middle line:84% So suppose Y1 is positive and Y2 is negative. 00:34:30.630 --> 00:34:33.100 align:middle line:84% This is negative, and so you have a negative term. 00:34:33.100 --> 00:34:36.510 align:middle line:90% 00:34:36.510 --> 00:34:42.030 align:middle line:84% So there are ways, like we did, also, for the LeChatelier Brown 00:34:42.030 --> 00:34:49.159 align:middle line:84% theorem, to write a quadratic form in canonical forms 00:34:49.159 --> 00:34:51.110 align:middle line:84% either in terms of the affinities 00:34:51.110 --> 00:34:54.659 align:middle line:84% or in terms of the rates, so that it's, essentially, 00:34:54.659 --> 00:35:01.390 align:middle line:84% a sum of squares multiplied by positive coefficients. 00:35:01.390 --> 00:35:05.530 align:middle line:90% And so you make sure that you-- 00:35:05.530 --> 00:35:08.330 align:middle line:84% I mean, if you can manage to write it this way, 00:35:08.330 --> 00:35:16.540 align:middle line:84% you have proved that you are in shape in terms of non-negativity 00:35:16.540 --> 00:35:18.380 align:middle line:90% of the entropy production. 00:35:18.380 --> 00:35:26.000 align:middle line:90% 00:35:26.000 --> 00:35:26.510 align:middle line:90% Sorry. 00:35:26.510 --> 00:35:32.740 align:middle line:84% So the proof that Onsager gave requires 00:35:32.740 --> 00:35:34.660 align:middle line:90% some statistical mechanics. 00:35:34.660 --> 00:35:40.870 align:middle line:84% And I might be able to give some hints about it later on. 00:35:40.870 --> 00:35:50.470 align:middle line:84% But hopefully, it will be given in some later versions 00:35:50.470 --> 00:35:55.770 align:middle line:84% of this course, when Professor Hadjiconstantinou will join in. 00:35:55.770 --> 00:35:58.340 align:middle line:90% 00:35:58.340 --> 00:36:02.780 align:middle line:84% But for the moment, let us prove it 00:36:02.780 --> 00:36:09.420 align:middle line:84% from this principle of maximum entropy production 00:36:09.420 --> 00:36:15.450 align:middle line:84% rate, proposed by Ziegler around 1957, 00:36:15.450 --> 00:36:28.410 align:middle line:84% '58, which is simple enough and doesn't contain 00:36:28.410 --> 00:36:34.320 align:middle line:84% a lot of physics, except the idea 00:36:34.320 --> 00:36:45.130 align:middle line:84% that when a system is in a non-equilibrium state, 00:36:45.130 --> 00:36:49.510 align:middle line:84% we already know that it tends to reach equilibrium, developing 00:36:49.510 --> 00:36:55.360 align:middle line:84% entropy, until this entropy gets to the maximum allowed 00:36:55.360 --> 00:36:59.830 align:middle line:84% by the constraints, for example, the energy, 00:36:59.830 --> 00:37:01.610 align:middle line:90% if the system is isolated. 00:37:01.610 --> 00:37:04.870 align:middle line:90% 00:37:04.870 --> 00:37:08.930 align:middle line:84% But this is not about the fact that the entropy increases. 00:37:08.930 --> 00:37:15.340 align:middle line:84% It's the fact that the system, when it is in non-equilibrium, 00:37:15.340 --> 00:37:22.420 align:middle line:84% searches for the path of steepest ascent 00:37:22.420 --> 00:37:26.310 align:middle line:84% in the entropy allowed by the constraints. 00:37:26.310 --> 00:37:29.190 align:middle line:84% So it's like you need to climb a mountain 00:37:29.190 --> 00:37:32.600 align:middle line:84% and go to the maximum of the entropy. 00:37:32.600 --> 00:37:36.140 align:middle line:84% And there are many paths that you can choose in order 00:37:36.140 --> 00:37:39.450 align:middle line:90% to climb that mountain. 00:37:39.450 --> 00:37:43.580 align:middle line:84% It turns out the systems like to go there 00:37:43.580 --> 00:37:46.640 align:middle line:84% in the most direct way as possible. 00:37:46.640 --> 00:37:48.590 align:middle line:84% Of course, there are constraints, 00:37:48.590 --> 00:37:51.710 align:middle line:84% like the internal structure of your system 00:37:51.710 --> 00:37:53.875 align:middle line:84% or the molecules, the collisions, and all that. 00:37:53.875 --> 00:37:59.340 align:middle line:90% 00:37:59.340 --> 00:38:03.850 align:middle line:84% So here, the idea is the following. 00:38:03.850 --> 00:38:09.540 align:middle line:84% What are the constraints for a chemically reacting system? 00:38:09.540 --> 00:38:12.400 align:middle line:84% We'll take as a constraint this equation, 00:38:12.400 --> 00:38:17.760 align:middle line:84% which comes from the entropy balance, plus our model 00:38:17.760 --> 00:38:19.030 align:middle line:90% for the properties. 00:38:19.030 --> 00:38:22.850 align:middle line:90% 00:38:22.850 --> 00:38:30.470 align:middle line:84% And then we make an assumption, like we did before, where OK, 00:38:30.470 --> 00:38:37.600 align:middle line:84% we want a theory in which the rates are linearly related 00:38:37.600 --> 00:38:39.232 align:middle line:90% to the affinities. 00:38:39.232 --> 00:38:44.290 align:middle line:90% 00:38:44.290 --> 00:38:47.880 align:middle line:90% So you want a linear theory. 00:38:47.880 --> 00:38:50.790 align:middle line:84% That means that the entropy production is 00:38:50.790 --> 00:38:55.590 align:middle line:90% given by this quadratic form. 00:38:55.590 --> 00:38:59.200 align:middle line:84% And when we take the derivative with respect to the affinity, 00:38:59.200 --> 00:39:01.250 align:middle line:90% we have this expression here. 00:39:01.250 --> 00:39:04.130 align:middle line:90% 00:39:04.130 --> 00:39:16.150 align:middle line:84% Notice Lij and Lji, I do not know if they're equal or not. 00:39:16.150 --> 00:39:17.480 align:middle line:90% I'm not assuming. 00:39:17.480 --> 00:39:20.930 align:middle line:84% I want to prove that if you assume 00:39:20.930 --> 00:39:23.840 align:middle line:84% that this rate of entropy production 00:39:23.840 --> 00:39:28.650 align:middle line:84% is maximal, subject to that constraint, 00:39:28.650 --> 00:39:31.445 align:middle line:90% then we prove the symmetry. 00:39:31.445 --> 00:39:40.864 align:middle line:90% 00:39:40.864 --> 00:39:44.000 align:middle line:84% Now, the method for maximizing is simple. 00:39:44.000 --> 00:39:44.500 align:middle line:90% We know it. 00:39:44.500 --> 00:39:47.790 align:middle line:84% It's the Lagrange multipliers method. 00:39:47.790 --> 00:39:53.050 align:middle line:84% We have only one constraint, which is this one. 00:39:53.050 --> 00:39:56.870 align:middle line:84% So in order to take the constrained maximum, 00:39:56.870 --> 00:39:59.410 align:middle line:84% you take the unconstrained maximum 00:39:59.410 --> 00:40:04.120 align:middle line:84% of your function minus the undetermined Lagrange 00:40:04.120 --> 00:40:11.110 align:middle line:84% multiplier multiplied by the constraint, written in the form, 00:40:11.110 --> 00:40:13.340 align:middle line:90% something is equal to 0. 00:40:13.340 --> 00:40:17.070 align:middle line:84% So this is equivalent to that equation set equal to 0. 00:40:17.070 --> 00:40:20.520 align:middle line:84% So it's the left-hand side of that equation. 00:40:20.520 --> 00:40:25.010 align:middle line:84% And then all you do is to take the derivative of this 00:40:25.010 --> 00:40:28.300 align:middle line:84% with respect to the affinities and set it equal to 0. 00:40:28.300 --> 00:40:31.780 align:middle line:90% 00:40:31.780 --> 00:40:38.240 align:middle line:84% And so the affinities are in here, in here, and in here. 00:40:38.240 --> 00:40:40.690 align:middle line:90% So you get three terms. 00:40:40.690 --> 00:40:44.390 align:middle line:84% One is multiplied by 1 and one by minus lambda. 00:40:44.390 --> 00:40:46.120 align:middle line:84% So 1 minus lambda, the derivative 00:40:46.120 --> 00:40:48.940 align:middle line:84% of S with respect to Y, S dot with respect to Y, 00:40:48.940 --> 00:40:57.840 align:middle line:84% plus this last term, which contributes an epsilon dot times 00:40:57.840 --> 00:40:58.340 align:middle line:90% lambda. 00:40:58.340 --> 00:41:01.846 align:middle line:90% 00:41:01.846 --> 00:41:03.440 align:middle line:90% So you set that equal to 0. 00:41:03.440 --> 00:41:06.260 align:middle line:84% You get this relation, which says, 00:41:06.260 --> 00:41:10.730 align:middle line:84% OK, then if you want to do that maximization, 00:41:10.730 --> 00:41:15.160 align:middle line:84% your rates must be equal to lambda minus 1 00:41:15.160 --> 00:41:17.770 align:middle line:84% over lambda times this derivative. 00:41:17.770 --> 00:41:20.660 align:middle line:84% This is called Ziegler orthogonality principle. 00:41:20.660 --> 00:41:29.110 align:middle line:84% And we'll try to interpret that in a while. 00:41:29.110 --> 00:41:32.360 align:middle line:84% For the moment, this is the result 00:41:32.360 --> 00:41:34.610 align:middle line:84% in terms of the Lagrange multiplier. 00:41:34.610 --> 00:41:38.450 align:middle line:84% Then, as typical in the method, you substitute this back 00:41:38.450 --> 00:41:43.370 align:middle line:84% into the constraints in order to find the value for the Lagrange 00:41:43.370 --> 00:41:44.100 align:middle line:90% multiplier. 00:41:44.100 --> 00:41:46.580 align:middle line:84% So you take this one, substitute it back 00:41:46.580 --> 00:41:51.370 align:middle line:84% into the constraint, which is here, 00:41:51.370 --> 00:41:55.040 align:middle line:90% and do the things carefully. 00:41:55.040 --> 00:42:00.280 align:middle line:84% So when this derivative is written up here, 00:42:00.280 --> 00:42:07.500 align:middle line:84% and it has the L's plus the transpose of L-- 00:42:07.500 --> 00:42:13.800 align:middle line:84% but now these are multiplied by the Yi and Yj 00:42:13.800 --> 00:42:17.030 align:middle line:90% under a double summation. 00:42:17.030 --> 00:42:21.640 align:middle line:84% If you switch the i's and j's here, 00:42:21.640 --> 00:42:26.840 align:middle line:84% things don't change, because of the double summation. 00:42:26.840 --> 00:42:34.630 align:middle line:84% Therefore, this is twice, this double summations here with just 00:42:34.630 --> 00:42:42.370 align:middle line:84% one L. But this double summation written this way 00:42:42.370 --> 00:42:44.655 align:middle line:90% is the entropy production rate. 00:42:44.655 --> 00:42:47.300 align:middle line:90% 00:42:47.300 --> 00:42:50.320 align:middle line:84% So we've got entropy production rate is equal to lambda minus 1 00:42:50.320 --> 00:42:53.450 align:middle line:84% over lambda times twice the rate. 00:42:53.450 --> 00:42:58.780 align:middle line:84% So that imposes that this combination must be equal to 1. 00:42:58.780 --> 00:43:02.270 align:middle line:84% And therefore, lambda, you get it when lambda is equal to 2. 00:43:02.270 --> 00:43:03.890 align:middle line:84% So we found the Lagrange multiplier, 00:43:03.890 --> 00:43:07.350 align:middle line:90% lambda is equal to 2. 00:43:07.350 --> 00:43:10.840 align:middle line:84% And we can substitute it into the result. 00:43:10.840 --> 00:43:13.860 align:middle line:84% And so the result of the maximization 00:43:13.860 --> 00:43:16.440 align:middle line:90% is this expression here. 00:43:16.440 --> 00:43:20.760 align:middle line:84% The orthogonality coefficient is 1/2. 00:43:20.760 --> 00:43:23.480 align:middle line:90% 00:43:23.480 --> 00:43:36.560 align:middle line:84% Finally, we can substitute into here our assumption 00:43:36.560 --> 00:43:38.480 align:middle line:90% for epsilon dot. 00:43:38.480 --> 00:43:42.420 align:middle line:84% So this is for the left-hand side, 00:43:42.420 --> 00:43:46.090 align:middle line:84% and this is for the right-hand side. 00:43:46.090 --> 00:43:47.350 align:middle line:90% And you get these expression. 00:43:47.350 --> 00:43:50.730 align:middle line:90% 00:43:50.730 --> 00:43:54.010 align:middle line:84% Then you move things left and right, 00:43:54.010 --> 00:44:01.890 align:middle line:84% so to keep the summation of the Lij's together and leave 00:44:01.890 --> 00:44:04.560 align:middle line:90% the Lji's on the other side. 00:44:04.560 --> 00:44:09.270 align:middle line:84% And here is the final result, that this summation 00:44:09.270 --> 00:44:10.175 align:middle line:90% must be equal to 0. 00:44:10.175 --> 00:44:12.820 align:middle line:90% 00:44:12.820 --> 00:44:17.097 align:middle line:84% And it has to be equal to 0 wherever your theory applies. 00:44:17.097 --> 00:44:19.720 align:middle line:90% 00:44:19.720 --> 00:44:26.510 align:middle line:84% That means in any of the non equilibrium states. 00:44:26.510 --> 00:44:30.047 align:middle line:84% So for arbitrary choices of the affinities-- 00:44:30.047 --> 00:44:31.630 align:middle line:84% and remember that here, the affinities 00:44:31.630 --> 00:44:36.470 align:middle line:84% are many because you have several reactions that 00:44:36.470 --> 00:44:41.160 align:middle line:84% are coupled, like we've seen in the example. 00:44:41.160 --> 00:44:44.480 align:middle line:84% So among the various choices, you 00:44:44.480 --> 00:44:49.550 align:middle line:84% can select one, for example, in which only one of the reactions 00:44:49.550 --> 00:44:55.700 align:middle line:84% is off equilibrium, and the others are at equilibrium. 00:44:55.700 --> 00:44:59.600 align:middle line:84% That, therefore, picks up only one term in this summation 00:44:59.600 --> 00:45:05.570 align:middle line:84% and imposes that the coefficients must be 0. 00:45:05.570 --> 00:45:10.035 align:middle line:84% And so that proves the symmetry of the matrix. 00:45:10.035 --> 00:45:18.990 align:middle line:90% 00:45:18.990 --> 00:45:26.140 align:middle line:84% So we proved that, at least, you get Onsager symmetry, 00:45:26.140 --> 00:45:30.437 align:middle line:84% if you accept the principle of maximum entropy production rate. 00:45:30.437 --> 00:45:37.600 align:middle line:90% 00:45:37.600 --> 00:45:43.410 align:middle line:84% It's-- I'm not sure you can do it also in reverse. 00:45:43.410 --> 00:45:48.210 align:middle line:84% In other words, is this an "if and only if" theorem? 00:45:48.210 --> 00:45:49.690 align:middle line:90% I'm not so sure. 00:45:49.690 --> 00:45:56.860 align:middle line:84% In other words, does Onsager reciprocity 00:45:56.860 --> 00:46:01.240 align:middle line:84% prove that the principle of maximum entropy production rate? 00:46:01.240 --> 00:46:04.650 align:middle line:84% Well, I think you'd probably need a little more. 00:46:04.650 --> 00:46:08.650 align:middle line:90% 00:46:08.650 --> 00:46:11.320 align:middle line:90% This is Ziegler. 00:46:11.320 --> 00:46:15.400 align:middle line:84% And this is a pictorial representation 00:46:15.400 --> 00:46:19.320 align:middle line:90% of his orthogonality relation. 00:46:19.320 --> 00:46:29.160 align:middle line:90% 00:46:29.160 --> 00:46:33.350 align:middle line:84% So suppose we have two reactions. 00:46:33.350 --> 00:46:47.680 align:middle line:84% And this plot wants to represent on a plane the affinities. 00:46:47.680 --> 00:46:52.360 align:middle line:84% So for example, suppose you are in this point here. 00:46:52.360 --> 00:46:56.080 align:middle line:84% It represents a non-equilibrium state 00:46:56.080 --> 00:47:02.280 align:middle line:84% in which the affinity of reaction 1 is this much. 00:47:02.280 --> 00:47:05.520 align:middle line:84% The center is 0, and that's equilibrium. 00:47:05.520 --> 00:47:08.375 align:middle line:84% And the affinity of the second reaction is this much. 00:47:08.375 --> 00:47:11.630 align:middle line:90% 00:47:11.630 --> 00:47:16.530 align:middle line:84% Here, the rate of entropy production, 00:47:16.530 --> 00:47:21.370 align:middle line:84% so the quadratic form, has a certain value 00:47:21.370 --> 00:47:23.620 align:middle line:90% different than zero. 00:47:23.620 --> 00:47:26.410 align:middle line:84% And these contour plots represent 00:47:26.410 --> 00:47:29.410 align:middle line:84% all other non-equilibrium states that 00:47:29.410 --> 00:47:33.910 align:middle line:84% have that same value of the entropy (production). 00:47:33.910 --> 00:47:35.840 align:middle line:84% As you move further from equilibrium, 00:47:35.840 --> 00:47:38.360 align:middle line:84% the entropy (production) goes up. 00:47:38.360 --> 00:47:41.640 align:middle line:84% As you move towards equilibrium, the entropy (production) 00:47:41.640 --> 00:47:42.190 align:middle line:90% goes down. 00:47:42.190 --> 00:47:46.850 align:middle line:84% So you may think of those contour lines 00:47:46.850 --> 00:47:52.780 align:middle line:84% as those of an ellipsoid that is touching 0 at equilibrium. 00:47:52.780 --> 00:47:56.060 align:middle line:90% 00:47:56.060 --> 00:48:04.150 align:middle line:84% The red curves represent the gradients of these contours. 00:48:04.150 --> 00:48:09.700 align:middle line:84% So these curves are everywhere orthogonal 00:48:09.700 --> 00:48:17.145 align:middle line:84% to the constant entropy (production) contour lines. 00:48:17.145 --> 00:48:26.450 align:middle line:90% 00:48:26.450 --> 00:48:34.400 align:middle line:84% The expression for the gradient, the gradient 00:48:34.400 --> 00:48:36.210 align:middle line:84% is like a generalized derivative. 00:48:36.210 --> 00:48:42.440 align:middle line:84% So the derivative of S dot with respect to Y vector, 00:48:42.440 --> 00:48:44.710 align:middle line:90% this is the gradient. 00:48:44.710 --> 00:48:48.270 align:middle line:90% 00:48:48.270 --> 00:49:00.810 align:middle line:84% And if I plot the gradient, it is a vector which is orthogonal. 00:49:00.810 --> 00:49:02.520 align:middle line:90% And I plot it here. 00:49:02.520 --> 00:49:05.237 align:middle line:84% So it's orthogonal to the (contour) curve 00:49:05.237 --> 00:49:06.445 align:middle line:90% for which it is the gradient. 00:49:06.445 --> 00:49:11.020 align:middle line:90% 00:49:11.020 --> 00:49:16.780 align:middle line:84% Now, what the Ziegler principle orthogonality relation says 00:49:16.780 --> 00:49:19.660 align:middle line:84% is that you take this vector, which is the gradient, 00:49:19.660 --> 00:49:30.840 align:middle line:84% take half of it, and that gives me the rate vector. 00:49:30.840 --> 00:49:33.870 align:middle line:84% The epsilon dot is the rate vector, is just a half 00:49:33.870 --> 00:49:38.700 align:middle line:84% of-- the red is a half of the black. 00:49:38.700 --> 00:49:43.740 align:middle line:84% And this vector here, let me now, the last step, 00:49:43.740 --> 00:49:50.370 align:middle line:90% move it back to here, to the 0. 00:49:50.370 --> 00:49:52.920 align:middle line:84% So then on the same plot, I am now 00:49:52.920 --> 00:49:58.810 align:middle line:84% reading on this axis, the one component. 00:49:58.810 --> 00:50:02.500 align:middle line:84% So that is the rate of the first reaction 00:50:02.500 --> 00:50:05.230 align:middle line:84% and on this other axis, the second component. 00:50:05.230 --> 00:50:07.100 align:middle line:84% So that's the rate of the second reaction. 00:50:07.100 --> 00:50:12.270 align:middle line:90% 00:50:12.270 --> 00:50:18.900 align:middle line:84% And what this shows is that if you 00:50:18.900 --> 00:50:23.640 align:middle line:84% had only one reaction and one affinity, 00:50:23.640 --> 00:50:30.810 align:middle line:84% the proportionality is typically that the rate and the affinity 00:50:30.810 --> 00:50:32.170 align:middle line:90% have the same sign. 00:50:32.170 --> 00:50:42.120 align:middle line:84% So if Y is positive, epsilon (dot) 1 is positive. 00:50:42.120 --> 00:50:47.850 align:middle line:84% And the relation is, like this system here, 00:50:47.850 --> 00:50:50.700 align:middle line:84% but instead of having 2 by 2, it's just 1 by 1. 00:50:50.700 --> 00:50:54.090 align:middle line:84% It's simply the proportionality L11 00:50:54.090 --> 00:50:55.720 align:middle line:90% is the direct proportionality. 00:50:55.720 --> 00:50:58.620 align:middle line:84% And it says that the rate of 1 is 00:50:58.620 --> 00:51:00.420 align:middle line:84% proportional to the affinity of 1. 00:51:00.420 --> 00:51:04.140 align:middle line:84% And also, if you had only 2, reaction 2, 00:51:04.140 --> 00:51:09.030 align:middle line:84% the rate of reaction 2 is proportional to its degree 00:51:09.030 --> 00:51:11.710 align:middle line:90% of disequilibrium, same sign. 00:51:11.710 --> 00:51:15.350 align:middle line:84% Because we said that these L's are positive. 00:51:15.350 --> 00:51:18.030 align:middle line:90% 00:51:18.030 --> 00:51:23.890 align:middle line:84% But here, you see, the coupling makes it so-- 00:51:23.890 --> 00:51:27.310 align:middle line:84% so the cross-- off diagonal terms-- 00:51:27.310 --> 00:51:36.190 align:middle line:84% makes it so, in this particular region of states, that-- 00:51:36.190 --> 00:51:39.310 align:middle line:90% look at reaction 2. 00:51:39.310 --> 00:51:43.090 align:middle line:84% It has a positive affinity, but as a result of the coupling, 00:51:43.090 --> 00:51:45.430 align:middle line:84% it gets a negative reaction rate. 00:51:45.430 --> 00:51:54.430 align:middle line:84% So these off-diagonal terms may actually drive the system 00:51:54.430 --> 00:52:01.180 align:middle line:84% in a somewhat unexpected direction, something 00:52:01.180 --> 00:52:05.180 align:middle line:84% that if it were by itself, would violate 00:52:05.180 --> 00:52:12.280 align:middle line:84% the entropy non-decrease principle. 00:52:12.280 --> 00:52:17.380 align:middle line:84% Because if I didn't have, also the other terms 00:52:17.380 --> 00:52:22.760 align:middle line:84% and only this one, so if I have only one reaction, 00:52:22.760 --> 00:52:26.530 align:middle line:84% it is impossible that the rate be in the opposite sign 00:52:26.530 --> 00:52:29.750 align:middle line:90% of the affinity. 00:52:29.750 --> 00:52:34.390 align:middle line:84% So that is, at least in the business 00:52:34.390 --> 00:52:40.370 align:middle line:84% of reactions, chemical reactions, 00:52:40.370 --> 00:52:43.225 align:middle line:84% the importance of this coupling terms. 00:52:43.225 --> 00:52:47.760 align:middle line:90% 00:52:47.760 --> 00:52:52.470 align:middle line:84% And it is, actually, for things like these, 00:52:52.470 --> 00:52:56.960 align:middle line:84% for many aspects of this chemistry 00:52:56.960 --> 00:53:05.550 align:middle line:84% off equilibrium, that also Ilya Prigogine has gotten a Nobel 00:53:05.550 --> 00:53:06.050 align:middle line:90% Prize-- 00:53:06.050 --> 00:53:09.350 align:middle line:90% I forget now what year. 00:53:09.350 --> 00:53:11.790 align:middle line:84% But I want to-- just that I mentioned him, 00:53:11.790 --> 00:53:15.470 align:middle line:84% I want to also mention that if you're doing your PhD 00:53:15.470 --> 00:53:18.810 align:middle line:84% thesis in the field of thermodynamics, 00:53:18.810 --> 00:53:23.050 align:middle line:84% whatever field of thermodynamics, 00:53:23.050 --> 00:53:26.050 align:middle line:84% make sure you apply for the Ilya Prigogine 00:53:26.050 --> 00:53:32.300 align:middle line:84% prize for thermodynamics, which is held every two years. 00:53:32.300 --> 00:53:36.960 align:middle line:90% And it's a prestigious thing. 00:53:36.960 --> 00:53:41.070 align:middle line:84% It doesn't bring a lot of money, but it gives you some prestige. 00:53:41.070 --> 00:53:45.620 align:middle line:84% And you may look up the website of the prize 00:53:45.620 --> 00:53:49.010 align:middle line:84% and see what the previous winners are, 00:53:49.010 --> 00:53:57.140 align:middle line:84% very diverse and very high level set of guys, most of which 00:53:57.140 --> 00:54:02.750 align:middle line:84% are now renowned professors worldwide. 00:54:02.750 --> 00:54:05.155 align:middle line:84% Because it's an international prize. 00:54:05.155 --> 00:54:11.814 align:middle line:90% 00:54:11.814 --> 00:54:14.940 align:middle line:90% So let's do a little switch. 00:54:14.940 --> 00:54:18.590 align:middle line:90% 00:54:18.590 --> 00:54:22.190 align:middle line:84% First of all, it sounds like a switch back, 00:54:22.190 --> 00:54:25.880 align:middle line:84% and then it will be a switch forward. 00:54:25.880 --> 00:54:29.220 align:middle line:84% So let's go back to heat transfer. 00:54:29.220 --> 00:54:33.610 align:middle line:84% And this is a slide that, more or less, we've already seen, 00:54:33.610 --> 00:54:35.270 align:middle line:84% maybe not exactly in this fashion. 00:54:35.270 --> 00:54:40.850 align:middle line:90% 00:54:40.850 --> 00:54:45.200 align:middle line:84% You take a one-dimensional fluid element 00:54:45.200 --> 00:54:49.200 align:middle line:84% subject to a gradient of temperature. 00:54:49.200 --> 00:54:51.860 align:middle line:90% So it's a fluid element. 00:54:51.860 --> 00:54:54.000 align:middle line:84% On one side, there's one temperature. 00:54:54.000 --> 00:54:56.450 align:middle line:84% So there's a heat interaction at one temperature 00:54:56.450 --> 00:54:59.330 align:middle line:84% with the fluid element on its left 00:54:59.330 --> 00:55:04.910 align:middle line:84% and another heat interaction at a different temperature, 00:55:04.910 --> 00:55:10.170 align:middle line:84% slightly different temperature, with the element on the right. 00:55:10.170 --> 00:55:15.600 align:middle line:84% And for the moment, let's assume that this is in a steady state. 00:55:15.600 --> 00:55:17.060 align:middle line:90% So let's do the energy balance. 00:55:17.060 --> 00:55:20.330 align:middle line:90% 00:55:20.330 --> 00:55:23.000 align:middle line:84% And we do the energy balance per unit 00:55:23.000 --> 00:55:29.330 align:middle line:84% of area of the surface of interaction 00:55:29.330 --> 00:55:31.450 align:middle line:90% between these adjacent elements. 00:55:31.450 --> 00:55:35.310 align:middle line:90% 00:55:35.310 --> 00:55:39.310 align:middle line:84% So 0, because we are at steady state, 00:55:39.310 --> 00:55:43.470 align:middle line:84% so there is no accumulation of energy. 00:55:43.470 --> 00:55:46.020 align:middle line:84% Otherwise, here it would be the derivative 00:55:46.020 --> 00:55:53.680 align:middle line:84% of the energy per unit volume with respect to time. 00:55:53.680 --> 00:55:56.520 align:middle line:84% But it's 0 because we are assuming steady state, 00:55:56.520 --> 00:56:01.260 align:middle line:90% is equal to the heat flux. 00:56:01.260 --> 00:56:03.750 align:middle line:84% I'm using here the notation of heat transfer. 00:56:03.750 --> 00:56:07.145 align:middle line:84% So q double prime is the typical notation for heat transfer. 00:56:07.145 --> 00:56:12.360 align:middle line:90% 00:56:12.360 --> 00:56:17.970 align:middle line:84% And then we have heat flux at x plus dx. 00:56:17.970 --> 00:56:20.826 align:middle line:84% So you may recognize this is the incremental-- 00:56:20.826 --> 00:56:25.510 align:middle line:90% 00:56:25.510 --> 00:56:29.620 align:middle line:84% the increment of the function q double prime as a function of x. 00:56:29.620 --> 00:56:33.220 align:middle line:84% So this is the partial of q double prime with respect 00:56:33.220 --> 00:56:34.810 align:middle line:90% to x times dx. 00:56:34.810 --> 00:56:36.475 align:middle line:90% And it is equal to 0. 00:56:36.475 --> 00:56:40.440 align:middle line:90% 00:56:40.440 --> 00:56:43.983 align:middle line:84% And then we do the entropy balance. 00:56:43.983 --> 00:56:45.400 align:middle line:84% Remember, we are doing the balance 00:56:45.400 --> 00:56:47.130 align:middle line:90% for this element in the center. 00:56:47.130 --> 00:56:50.280 align:middle line:90% 00:56:50.280 --> 00:56:52.860 align:middle line:84% So here, we have to account for-- 00:56:52.860 --> 00:56:56.100 align:middle line:84% here, I've multiplied by A. So we 00:56:56.100 --> 00:57:01.450 align:middle line:84% have to account for the entropy coming in from the left, which 00:57:01.450 --> 00:57:12.760 align:middle line:84% is q dot over T. And q double prime is q dot over A, 00:57:12.760 --> 00:57:18.980 align:middle line:84% is the flux, so is the heat rate per unit area, 00:57:18.980 --> 00:57:25.710 align:middle line:84% minus the entropy that goes out on the other side. 00:57:25.710 --> 00:57:28.140 align:middle line:84% And this is q double prime divided 00:57:28.140 --> 00:57:30.180 align:middle line:84% by the temperature, which prevails there, which 00:57:30.180 --> 00:57:34.260 align:middle line:84% is different, slightly but different, and then 00:57:34.260 --> 00:57:38.080 align:middle line:84% plus the entropy produced by irreversibility. 00:57:38.080 --> 00:57:42.990 align:middle line:84% So here, I'm denoting this lowercase s with the triple 00:57:42.990 --> 00:57:50.440 align:middle line:84% prime is entropy generated by irreversibility per unit volume. 00:57:50.440 --> 00:57:53.260 align:middle line:90% And the dot is the rate, so-- 00:57:53.260 --> 00:57:56.020 align:middle line:84% Soon we will go, actually, over here. 00:57:56.020 --> 00:58:01.890 align:middle line:84% We'll give a special symbol for that. 00:58:01.890 --> 00:58:07.620 align:middle line:84% So from now on, when you see sigma, 00:58:07.620 --> 00:58:09.960 align:middle line:90% it's not the stress tensor. 00:58:09.960 --> 00:58:12.360 align:middle line:90% It's not the surface tension. 00:58:12.360 --> 00:58:15.720 align:middle line:84% It's the rate of entropy production per unit volume. 00:58:15.720 --> 00:58:20.100 align:middle line:84% Sometimes, different fields use the same symbol 00:58:20.100 --> 00:58:21.250 align:middle line:90% for different purposes. 00:58:21.250 --> 00:58:24.575 align:middle line:84% But there is only one Greek alphabet. 00:58:24.575 --> 00:58:32.150 align:middle line:90% 00:58:32.150 --> 00:58:37.705 align:middle line:84% So from this one, if you do properly this-- 00:58:37.705 --> 00:58:41.650 align:middle line:90% 00:58:41.650 --> 00:58:46.880 align:middle line:84% you recognize the incremental ratio Tx 00:58:46.880 --> 00:58:49.310 align:middle line:84% plus the x minus Tx divided by dx, 00:58:49.310 --> 00:58:52.700 align:middle line:84% that's the derivative of T with respect to x. 00:58:52.700 --> 00:59:02.830 align:middle line:84% In the denominator, here you can neglect the difference of-- 00:59:02.830 --> 00:59:06.410 align:middle line:84% so this denominator, you can simply call it T squared. 00:59:06.410 --> 00:59:08.640 align:middle line:84% That's why the approximate symbol. 00:59:08.640 --> 00:59:12.810 align:middle line:90% 00:59:12.810 --> 00:59:16.820 align:middle line:90% So we like to write this as-- 00:59:16.820 --> 00:59:20.960 align:middle line:84% so the entropy production rate, you 00:59:20.960 --> 00:59:25.760 align:middle line:84% like to write it as the heat flux times this gradient 00:59:25.760 --> 00:59:27.830 align:middle line:90% of the inverse temperature. 00:59:27.830 --> 00:59:33.880 align:middle line:84% This is analogous, and this is why we are doing it now. 00:59:33.880 --> 00:59:38.650 align:middle line:84% It's analogous to the chemical idea 00:59:38.650 --> 00:59:42.580 align:middle line:84% that entropy production rates are 00:59:42.580 --> 00:59:47.220 align:middle line:84% given by a product of a reaction rate times an affinity. 00:59:47.220 --> 00:59:52.510 align:middle line:84% If you interpret this gradient of the inverse temperature 00:59:52.510 --> 01:00:01.020 align:middle line:84% as the affinity or the degree of disequilibrium with respect 01:00:01.020 --> 01:00:09.930 align:middle line:84% to heat transfer, then you have a similar concept. 01:00:09.930 --> 01:00:13.570 align:middle line:84% And we have already discussed this in a previous lecture, 01:00:13.570 --> 01:00:15.310 align:middle line:90% so I will not do it now. 01:00:15.310 --> 01:00:20.910 align:middle line:84% But I simply remind you that this part of the slide was 01:00:20.910 --> 01:00:34.320 align:middle line:84% to show that you cannot be at equilibrium if this is different 01:00:34.320 --> 01:00:35.610 align:middle line:90% from 0. 01:00:35.610 --> 01:00:46.200 align:middle line:84% And therefore, when we actually adopt a local equilibrium model 01:00:46.200 --> 01:00:50.820 align:middle line:84% for our fluid element, we are neglecting this difference. 01:00:50.820 --> 01:00:52.860 align:middle line:90% And you will see-- 01:00:52.860 --> 01:00:59.520 align:middle line:84% hopefully, today, we will see what accounting for that 01:00:59.520 --> 01:01:05.790 align:middle line:84% difference makes and how things change 01:01:05.790 --> 01:01:07.470 align:middle line:90% if you do account for that. 01:01:07.470 --> 01:01:10.770 align:middle line:84% But for the moment, if you want to, 01:01:10.770 --> 01:01:12.750 align:middle line:84% just go back to that previous lecture 01:01:12.750 --> 01:01:19.250 align:middle line:84% to find more discussion about this graph. 01:01:19.250 --> 01:01:26.870 align:middle line:84% What we may also add is the following. 01:01:26.870 --> 01:01:29.450 align:middle line:84% Now, this fluid element in the middle 01:01:29.450 --> 01:01:32.360 align:middle line:84% of two things that are at different temperatures, 01:01:32.360 --> 01:01:38.070 align:middle line:84% although minute differences, could be, in principle-- 01:01:38.070 --> 01:01:45.110 align:middle line:84% you see, that heat transfer, direct, with no useful effect, 01:01:45.110 --> 01:01:46.160 align:middle line:90% is dissipated. 01:01:46.160 --> 01:01:49.015 align:middle line:84% And in fact, you have entropy produced by irreversibility. 01:01:49.015 --> 01:01:50.390 align:middle line:84% And we have learned that whenever 01:01:50.390 --> 01:01:55.520 align:middle line:84% you have entropy produced by irreversibility, if you 01:01:55.520 --> 01:01:57.540 align:middle line:84% can avoid that entropy production, 01:01:57.540 --> 01:02:00.230 align:middle line:84% you can do something useful with it. 01:02:00.230 --> 01:02:04.550 align:middle line:84% So here, what you would need to do 01:02:04.550 --> 01:02:09.230 align:middle line:84% is to replace your fluid element by a small heat 01:02:09.230 --> 01:02:16.310 align:middle line:84% engine, a local something, maybe bacteria-- 01:02:16.310 --> 01:02:20.810 align:middle line:84% because here, it may be a small system-- 01:02:20.810 --> 01:02:28.580 align:middle line:84% in any case, a heat engine that while exploiting 01:02:28.580 --> 01:02:32.870 align:middle line:84% this temperature difference, extracts some work, 01:02:32.870 --> 01:02:33.990 align:middle line:90% like a Carnot engine. 01:02:33.990 --> 01:02:40.300 align:middle line:90% 01:02:40.300 --> 01:02:45.190 align:middle line:84% We have learned that in order to compute the maximum work, 01:02:45.190 --> 01:02:48.760 align:middle line:84% you have to do energy balance and entropy balance. 01:02:48.760 --> 01:02:52.420 align:middle line:84% And you have to assume that the entropy produced 01:02:52.420 --> 01:02:54.490 align:middle line:84% by irreversibility is equal to 0. 01:02:54.490 --> 01:02:55.720 align:middle line:90% If you do that-- 01:02:55.720 --> 01:02:58.060 align:middle line:84% so you start from the entropy balance 01:02:58.060 --> 01:03:05.230 align:middle line:84% and impose that the entropy that goes out here 01:03:05.230 --> 01:03:15.320 align:middle line:84% is equal to the entropy that comes in, 01:03:15.320 --> 01:03:19.130 align:middle line:90% that gives you one relation. 01:03:19.130 --> 01:03:22.180 align:middle line:84% Then if you substitute it back into the energy balance, 01:03:22.180 --> 01:03:27.940 align:middle line:90% it gives you that maximum work. 01:03:27.940 --> 01:03:30.430 align:middle line:84% And of course, the answer, we could 01:03:30.430 --> 01:03:33.460 align:middle line:84% have done it even without redoing the energy and entropy 01:03:33.460 --> 01:03:34.270 align:middle line:90% balance. 01:03:34.270 --> 01:03:36.670 align:middle line:84% The answer is that the maximum work 01:03:36.670 --> 01:03:42.900 align:middle line:84% you extract there is equal to the temperature that prevails 01:03:42.900 --> 01:03:48.060 align:middle line:84% around there, more or less it's the same, so 01:03:48.060 --> 01:03:50.700 align:middle line:90% T, times the entropy-- 01:03:50.700 --> 01:03:53.820 align:middle line:84% times sigma-- times the entropy that you 01:03:53.820 --> 01:03:56.310 align:middle line:84% would have produced by irreversibility 01:03:56.310 --> 01:03:58.115 align:middle line:90% if you didn't do that work. 01:03:58.115 --> 01:04:01.140 align:middle line:90% 01:04:01.140 --> 01:04:02.850 align:middle line:90% So you get these relations here. 01:04:02.850 --> 01:04:07.750 align:middle line:90% 01:04:07.750 --> 01:04:13.140 align:middle line:84% If your fluid element is three dimensional instead of-- 01:04:13.140 --> 01:04:15.820 align:middle line:90% 01:04:15.820 --> 01:04:24.230 align:middle line:84% here, we've discussed a one-dimensional case. 01:04:24.230 --> 01:04:28.500 align:middle line:84% But you can make it also three dimensional. 01:04:28.500 --> 01:04:37.800 align:middle line:84% So you have a nice fluid element of the kind that you consider 01:04:37.800 --> 01:04:40.910 align:middle line:90% in 3D heat transfer. 01:04:40.910 --> 01:04:49.920 align:middle line:84% Then of course, you have to account for exchanges of heat. 01:04:49.920 --> 01:04:56.210 align:middle line:84% The heat flux is a vector that has the three components. 01:04:56.210 --> 01:04:59.690 align:middle line:84% And the equivalent of the energy balance 01:04:59.690 --> 01:05:03.410 align:middle line:84% here, this derivative of q with respect to x, is equal to 0, 01:05:03.410 --> 01:05:09.770 align:middle line:84% becomes the divergence of the vector q 01:05:09.770 --> 01:05:12.140 align:middle line:90% double prime is equal to 0. 01:05:12.140 --> 01:05:14.810 align:middle line:84% Or in this other problem, it becomes 01:05:14.810 --> 01:05:20.300 align:middle line:84% the divergence of q over T, of the vector q over T, which 01:05:20.300 --> 01:05:25.990 align:middle line:84% is the entropy flux, is equal to 0. 01:05:25.990 --> 01:05:33.850 align:middle line:84% And this expression for the entropy production rate 01:05:33.850 --> 01:05:41.020 align:middle line:84% per unit volume, from this which is just 01:05:41.020 --> 01:05:44.860 align:middle line:84% the x component of the heat flux times the x component 01:05:44.860 --> 01:05:50.590 align:middle line:84% of the gradient in 1 over T, it becomes the dot product, 01:05:50.590 --> 01:05:53.320 align:middle line:84% the scalar product of the heat flux 01:05:53.320 --> 01:05:58.970 align:middle line:84% into the gradient of 1 over T, which is a vector. 01:05:58.970 --> 01:06:06.620 align:middle line:84% So this is why I like to put the underline under the nabla symbol 01:06:06.620 --> 01:06:13.560 align:middle line:84% because nabla is a vector operator like q. 01:06:13.560 --> 01:06:16.120 align:middle line:90% 01:06:16.120 --> 01:06:17.955 align:middle line:84% I'm sure you are familiar with that. 01:06:17.955 --> 01:06:21.160 align:middle line:90% 01:06:21.160 --> 01:06:25.065 align:middle line:90% So nabla is this operator here. 01:06:25.065 --> 01:06:29.980 align:middle line:90% 01:06:29.980 --> 01:06:34.590 align:middle line:84% It's a vector which has as components 01:06:34.590 --> 01:06:36.440 align:middle line:90% the partial derivative symbols. 01:06:36.440 --> 01:06:47.040 align:middle line:90% 01:06:47.040 --> 01:06:53.860 align:middle line:84% So we have seen the analogy in the structure 01:06:53.860 --> 01:06:56.620 align:middle line:84% for the rate of entropy production 01:06:56.620 --> 01:07:05.130 align:middle line:84% with the chemical formalism, affinity times rate, 01:07:05.130 --> 01:07:10.480 align:middle line:84% affinity times rate, the rate of heat exchange. 01:07:10.480 --> 01:07:15.300 align:middle line:84% And so the affinity is the gradient in the temperature. 01:07:15.300 --> 01:07:19.990 align:middle line:90% So let's do the same procedure. 01:07:19.990 --> 01:07:25.330 align:middle line:84% Let us assume, like we did, that the epsilon dot is 01:07:25.330 --> 01:07:28.750 align:middle line:84% a function of the affinity, with the function being 01:07:28.750 --> 01:07:31.060 align:middle line:90% 0 at equilibrium. 01:07:31.060 --> 01:07:33.050 align:middle line:90% Let's assume the same here. 01:07:33.050 --> 01:07:37.270 align:middle line:84% So let's assume that the heat flux 01:07:37.270 --> 01:07:42.160 align:middle line:84% vector is a function of the gradient 01:07:42.160 --> 01:07:44.820 align:middle line:90% of the inverse temperature. 01:07:44.820 --> 01:07:48.460 align:middle line:90% 01:07:48.460 --> 01:07:52.360 align:middle line:84% And it is equal to 0 if there is no gradient. 01:07:52.360 --> 01:07:55.400 align:middle line:84% Because we know that if the temperature profile is flat, 01:07:55.400 --> 01:07:59.500 align:middle line:84% and there's no gradient, there is no heat flux. 01:07:59.500 --> 01:08:07.080 align:middle line:84% And we linearize this expression here, like we did here. 01:08:07.080 --> 01:08:10.260 align:middle line:90% So you take this derivative. 01:08:10.260 --> 01:08:11.850 align:middle line:84% And so the derivative of a vector 01:08:11.850 --> 01:08:16.569 align:middle line:84% with respect to another vector becomes a tensor. 01:08:16.569 --> 01:08:20.740 align:middle line:90% We call it L double underline. 01:08:20.740 --> 01:08:21.689 align:middle line:90% That's for a tensor. 01:08:21.689 --> 01:08:28.340 align:middle line:90% 01:08:28.340 --> 01:08:32.160 align:middle line:84% And so you're linearizing in gradient of 1 over T. 01:08:32.160 --> 01:08:34.670 align:middle line:84% So that multiplies gradient of 1 over T. 01:08:34.670 --> 01:08:38.250 align:middle line:84% And of course, this tensor is to be evaluated at equilibrium. 01:08:38.250 --> 01:08:39.939 align:middle line:84% So this is an equilibrium property. 01:08:39.939 --> 01:08:42.540 align:middle line:90% 01:08:42.540 --> 01:08:45.471 align:middle line:84% You would evaluate this when there is no gradient, 01:08:45.471 --> 01:08:46.804 align:middle line:90% so when the temperature is flat. 01:08:46.804 --> 01:08:51.109 align:middle line:90% 01:08:51.109 --> 01:09:00.130 align:middle line:84% Well, if you call that L k tensor times T squared, 01:09:00.130 --> 01:09:11.850 align:middle line:84% you obtain the Fourier's law of thermal conduction, 01:09:11.850 --> 01:09:15.955 align:middle line:84% on which most of the heat transfer is based. 01:09:15.955 --> 01:09:18.479 align:middle line:90% 01:09:18.479 --> 01:09:21.399 align:middle line:90% So this is a fundamental step. 01:09:21.399 --> 01:09:26.279 align:middle line:84% Then from this, you construct all the applications. 01:09:26.279 --> 01:09:34.310 align:middle line:84% Notice that k is a tensor, first of all. 01:09:34.310 --> 01:09:41.660 align:middle line:84% And only in some special cases, that tensor is isotropic. 01:09:41.660 --> 01:09:48.450 align:middle line:84% So you need only one variable to represent it. 01:09:48.450 --> 01:09:51.040 align:middle line:84% And so for an isotropic material, 01:09:51.040 --> 01:09:57.390 align:middle line:84% then you can write that q double prime-- 01:09:57.390 --> 01:10:06.030 align:middle line:84% I'm sorry, q double prime is equal to minus k gradient of T. 01:10:06.030 --> 01:10:11.400 align:middle line:84% That's because, of course, the gradient of 1 over T 01:10:11.400 --> 01:10:29.360 align:middle line:84% is equal to minus 1 over T squared times the gradient of T. 01:10:29.360 --> 01:10:32.670 align:middle line:84% So that is the standard Fourier law 01:10:32.670 --> 01:10:35.460 align:middle line:84% of proportionality (between) heat and (gradient 01:10:35.460 --> 01:10:38.750 align:middle line:84% of temperature) that says that the heat flux is indeed 01:10:38.750 --> 01:10:42.715 align:middle line:84% in the direction of the vector gradient. 01:10:42.715 --> 01:10:47.016 align:middle line:90% 01:10:47.016 --> 01:10:49.280 align:middle line:84% And that's true for an isotropic material. 01:10:49.280 --> 01:10:53.800 align:middle line:84% But for an anisotropic material, you 01:10:53.800 --> 01:11:01.700 align:middle line:84% may have that the gradient is in one direction, 01:11:01.700 --> 01:11:05.040 align:middle line:84% and the heat flux is in another direction. 01:11:05.040 --> 01:11:11.440 align:middle line:90% 01:11:11.440 --> 01:11:16.450 align:middle line:84% For example, here, suppose you have a two-dimensional problem 01:11:16.450 --> 01:11:19.525 align:middle line:84% in which you have a two-dimensional anisotropic 01:11:19.525 --> 01:11:20.025 align:middle line:90% material. 01:11:20.025 --> 01:11:23.120 align:middle line:90% 01:11:23.120 --> 01:11:28.724 align:middle line:84% So the two dimensional means that whatever-- 01:11:28.724 --> 01:11:32.420 align:middle line:84% that nothing changes with the z direction, 01:11:32.420 --> 01:11:34.430 align:middle line:84% neither the temperature, nor the heat flux, 01:11:34.430 --> 01:11:35.780 align:middle line:90% nor the boundary conditions. 01:11:35.780 --> 01:11:40.070 align:middle line:84% In that case, every partial derivative with respect to z 01:11:40.070 --> 01:11:41.220 align:middle line:90% is equal to 0. 01:11:41.220 --> 01:11:46.460 align:middle line:84% That's condition for 2D symmetry. 01:11:46.460 --> 01:11:49.910 align:middle line:84% So for example, the temperature field depends only on x and y. 01:11:49.910 --> 01:11:52.580 align:middle line:90% That's to simplify. 01:11:52.580 --> 01:11:56.120 align:middle line:84% And here, so this is the expression. 01:11:56.120 --> 01:12:02.000 align:middle line:84% The Fourier law becomes this linear expression here. 01:12:02.000 --> 01:12:06.140 align:middle line:84% And how do you build non-isotropic materials? 01:12:06.140 --> 01:12:12.270 align:middle line:84% For example, by sandwiching layers of different materials. 01:12:12.270 --> 01:12:16.490 align:middle line:84% For example, you could have a highly conductive layer 01:12:16.490 --> 01:12:25.760 align:middle line:84% of a material like carbon sheet sandwiched 01:12:25.760 --> 01:12:32.850 align:middle line:84% between an epoxy or some other material with low conductivity. 01:12:32.850 --> 01:12:37.480 align:middle line:90% 01:12:37.480 --> 01:12:40.100 align:middle line:90% That makes the anisotropy. 01:12:40.100 --> 01:12:44.965 align:middle line:90% 01:12:44.965 --> 01:12:46.090 align:middle line:90% So these are the equations. 01:12:46.090 --> 01:12:47.790 align:middle line:90% Let's see how you would-- 01:12:47.790 --> 01:12:51.270 align:middle line:90% 01:12:51.270 --> 01:12:59.188 align:middle line:84% so in order to characterize this material, 01:12:59.188 --> 01:13:02.490 align:middle line:84% you don't just have one thermal conductivity. 01:13:02.490 --> 01:13:06.930 align:middle line:84% You need to measure three parameters. 01:13:06.930 --> 01:13:11.510 align:middle line:84% But actually, there are four entries in the matrix. 01:13:11.510 --> 01:13:17.770 align:middle line:84% But thanks to Onsager principle, the off-diagonal ones 01:13:17.770 --> 01:13:23.960 align:middle line:84% must be equal because of his reciprocity, general reciprocity 01:13:23.960 --> 01:13:28.000 align:middle line:84% theorem, which applies also in this business. 01:13:28.000 --> 01:13:30.260 align:middle line:84% And therefore, you need only to measure three. 01:13:30.260 --> 01:13:31.810 align:middle line:84% And here are the three measurements 01:13:31.810 --> 01:13:33.980 align:middle line:90% that you could make. 01:13:33.980 --> 01:13:38.000 align:middle line:84% You cut the sample of your material 01:13:38.000 --> 01:13:49.410 align:middle line:84% thin in the x direction, and assume 01:13:49.410 --> 01:13:52.380 align:middle line:84% that things in the y direction do not 01:13:52.380 --> 01:13:58.610 align:middle line:84% change, or set up an experiment in which they don't change. 01:13:58.610 --> 01:14:03.170 align:middle line:84% And then so that you are set up to use the first equation here, 01:14:03.170 --> 01:14:07.055 align:middle line:84% with no component of the gradient in temperature in the y 01:14:07.055 --> 01:14:07.555 align:middle line:90% direction. 01:14:07.555 --> 01:14:11.940 align:middle line:84% This is 0, and you have only this direct relation. 01:14:11.940 --> 01:14:13.820 align:middle line:90% So you measure the heat flux. 01:14:13.820 --> 01:14:16.250 align:middle line:84% You measure the gradient in temperature 01:14:16.250 --> 01:14:20.180 align:middle line:84% across the small thickness of your sample. 01:14:20.180 --> 01:14:22.850 align:middle line:90% And you take that ratio. 01:14:22.850 --> 01:14:31.430 align:middle line:84% And so this ratio gives you the conductivity, the xx component 01:14:31.430 --> 01:14:33.170 align:middle line:90% of the conductivity tensor. 01:14:33.170 --> 01:14:42.380 align:middle line:84% That must be positive because this tensor is an Onsager 01:14:42.380 --> 01:14:42.930 align:middle line:90% matrix. 01:14:42.930 --> 01:14:47.750 align:middle line:84% So it needs to be positive semi-definite. 01:14:47.750 --> 01:14:53.370 align:middle line:84% Do the same for the y direction, and you measure k double y. 01:14:53.370 --> 01:14:59.600 align:middle line:90% 01:14:59.600 --> 01:15:02.090 align:middle line:84% Now, to measure the off-diagonal terms, 01:15:02.090 --> 01:15:06.570 align:middle line:90% one way is to put your sample. 01:15:06.570 --> 01:15:16.380 align:middle line:84% You squeeze it between insulating materials 01:15:16.380 --> 01:15:22.680 align:middle line:84% in the two faces in the y direction, 01:15:22.680 --> 01:15:25.110 align:middle line:84% with the normal in the y direction. 01:15:25.110 --> 01:15:27.190 align:middle line:90% So this is an insulation. 01:15:27.190 --> 01:15:28.420 align:middle line:90% This is an insulation. 01:15:28.420 --> 01:15:31.810 align:middle line:90% 01:15:31.810 --> 01:15:37.090 align:middle line:84% And you subject your sample to a gradient in temperature 01:15:37.090 --> 01:15:40.070 align:middle line:90% in the x direction. 01:15:40.070 --> 01:15:46.948 align:middle line:90% 01:15:46.948 --> 01:15:51.180 align:middle line:84% Now, since the insulating material-- 01:15:51.180 --> 01:15:56.150 align:middle line:84% since this wall is insulated, the heat flux component 01:15:56.150 --> 01:15:59.390 align:middle line:90% in the y direction here is 0. 01:15:59.390 --> 01:16:00.960 align:middle line:90% And also, here is 0. 01:16:00.960 --> 01:16:06.450 align:middle line:84% So you can assume that it is 0, also, not really precise-- 01:16:06.450 --> 01:16:09.950 align:middle line:84% but that it is 0, also, in between because this 01:16:09.950 --> 01:16:14.245 align:middle line:84% is supposed to be a small thickness of your sample. 01:16:14.245 --> 01:16:18.320 align:middle line:90% 01:16:18.320 --> 01:16:21.560 align:middle line:84% However, in spite of that, so there 01:16:21.560 --> 01:16:25.700 align:middle line:84% is no heat flux in the vertical direction. 01:16:25.700 --> 01:16:29.420 align:middle line:84% Yet, you're going to build up a gradient in temperature 01:16:29.420 --> 01:16:32.940 align:middle line:90% in that direction. 01:16:32.940 --> 01:16:42.530 align:middle line:84% Because of this equation here, which is rewritten here, 01:16:42.530 --> 01:16:46.040 align:middle line:84% the insulation sets this equal to 0 01:16:46.040 --> 01:16:51.040 align:middle line:84% and therefore, makes these two terms equal to one another, 01:16:51.040 --> 01:16:53.415 align:middle line:84% I mean, equal and opposite to one another. 01:16:53.415 --> 01:16:56.960 align:middle line:90% 01:16:56.960 --> 01:17:01.790 align:middle line:84% So by measuring the gradient in temperature that builds up 01:17:01.790 --> 01:17:05.360 align:middle line:84% in the y direction and dividing it 01:17:05.360 --> 01:17:09.480 align:middle line:84% by the imposed gradient in temperature in the x direction, 01:17:09.480 --> 01:17:15.080 align:middle line:84% you get this ratio of the off-diagonal coefficient 01:17:15.080 --> 01:17:16.920 align:middle line:90% versus the direct coefficient. 01:17:16.920 --> 01:17:25.560 align:middle line:90% 01:17:25.560 --> 01:17:30.060 align:middle line:84% And in fact, you notice that in this case, 01:17:30.060 --> 01:17:35.376 align:middle line:84% the x direction component of the heat flux 01:17:35.376 --> 01:17:47.290 align:middle line:84% q double prime x is less than it was in this other example, 01:17:47.290 --> 01:17:48.580 align:middle line:90% in these other situation. 01:17:48.580 --> 01:17:55.120 align:middle line:84% Because you see there is still proportionality between the flux 01:17:55.120 --> 01:17:59.750 align:middle line:84% and the imposed gradient in the x direction, 01:17:59.750 --> 01:18:04.155 align:middle line:84% but the coefficient of proportionality is smaller. 01:18:04.155 --> 01:18:09.940 align:middle line:90% 01:18:09.940 --> 01:18:16.210 align:middle line:84% Physically, you can realize that this sandwich makes it 01:18:16.210 --> 01:18:20.860 align:middle line:84% so that the heat flux finds it easier 01:18:20.860 --> 01:18:26.510 align:middle line:84% to run through the high conductivity layers. 01:18:26.510 --> 01:18:29.010 align:middle line:84% And since those layers are inclined, 01:18:29.010 --> 01:18:34.370 align:middle line:84% as hinted by these inclined lines here, 01:18:34.370 --> 01:18:36.000 align:middle line:90% the heat flux enters here. 01:18:36.000 --> 01:18:46.240 align:middle line:84% But it likes to go up, following the high conductivity materials. 01:18:46.240 --> 01:18:50.260 align:middle line:84% That's why you build up temperature gradient 01:18:50.260 --> 01:18:54.720 align:middle line:84% in the direction where there is no net heat flux. 01:18:54.720 --> 01:18:57.980 align:middle line:90% 01:18:57.980 --> 01:19:06.900 align:middle line:84% So this is also called Righi-Leduc effect, 01:19:06.900 --> 01:19:12.570 align:middle line:84% which was discovered before Onsager theorem. 01:19:12.570 --> 01:19:17.220 align:middle line:84% Many of the things that are rationalized by the Onsager 01:19:17.220 --> 01:19:20.370 align:middle line:84% reciprocity theorem were actually discovered 01:19:20.370 --> 01:19:24.270 align:middle line:84% independently-- and sometimes, not even connected to one 01:19:24.270 --> 01:19:25.440 align:middle line:90% another-- 01:19:25.440 --> 01:19:30.580 align:middle line:84% by various researchers in the 19th century. 01:19:30.580 --> 01:19:36.790 align:middle line:84% And it's only later on that you could connect them. 01:19:36.790 --> 01:19:41.580 align:middle line:84% And so somehow, you had four different effects 01:19:41.580 --> 01:19:44.880 align:middle line:84% that, thanks to Onsager, were related and become 01:19:44.880 --> 01:19:46.670 align:middle line:90% only three independent ones. 01:19:46.670 --> 01:19:49.670 align:middle line:90% 01:19:49.670 --> 01:19:55.890 align:middle line:84% And notice also that, for example, using this idea, 01:19:55.890 --> 01:20:09.720 align:middle line:84% you can construct a sensor, a non-disturbing heat flux sensor, 01:20:09.720 --> 01:20:17.770 align:middle line:84% because in here, by putting the thermocouples across the sample, 01:20:17.770 --> 01:20:23.630 align:middle line:84% which are off the way where the heat flux goes, 01:20:23.630 --> 01:20:26.660 align:middle line:84% by measuring this temperature difference, 01:20:26.660 --> 01:20:30.800 align:middle line:90% you measure the heat flux. 01:20:30.800 --> 01:20:36.530 align:middle line:84% And therefore, you don't have to put thermocouples 01:20:36.530 --> 01:20:40.220 align:middle line:84% in the way of a small sample, because typically, 01:20:40.220 --> 01:20:41.750 align:middle line:84% if you put the thermocouple, that 01:20:41.750 --> 01:20:44.780 align:middle line:84% will disturb the measurement because the thermocouple has 01:20:44.780 --> 01:20:46.230 align:middle line:90% its own size. 01:20:46.230 --> 01:20:51.300 align:middle line:84% And therefore, it alters the actual situation. 01:20:51.300 --> 01:21:02.197 align:middle line:90% 01:21:02.197 --> 01:21:14.030 align:middle line:84% Of course, in the Fourier equation of the complete energy 01:21:14.030 --> 01:21:28.580 align:middle line:84% balance for a continuum with no motion, 01:21:28.580 --> 01:21:31.450 align:middle line:84% no velocity, so not a gas or a fluid, but, say, 01:21:31.450 --> 01:21:40.830 align:middle line:84% a solid or maybe a static fluid, this is the equation. 01:21:40.830 --> 01:21:47.310 align:middle line:84% And if you have, for example, a gas-- 01:21:47.310 --> 01:21:51.790 align:middle line:90% OK, this u, rho u is the-- 01:21:51.790 --> 01:22:02.931 align:middle line:84% OK, let's say that rho u is the energy per unit volume. 01:22:02.931 --> 01:22:12.510 align:middle line:84% So you could write it as m over V times U over m. 01:22:12.510 --> 01:22:16.450 align:middle line:84% And so by seeing that, you see this is rho. 01:22:16.450 --> 01:22:19.190 align:middle line:90% 01:22:19.190 --> 01:22:21.050 align:middle line:90% And this is u. 01:22:21.050 --> 01:22:24.830 align:middle line:84% Here, I'm using u without the star, the asterisk. 01:22:24.830 --> 01:22:32.630 align:middle line:84% But that's the energy per unit mass, the specific energy 01:22:32.630 --> 01:22:33.355 align:middle line:90% per unit mass. 01:22:33.355 --> 01:22:35.870 align:middle line:90% 01:22:35.870 --> 01:22:45.140 align:middle line:84% And for example, for a gas, that would be rho cv times T. 01:22:45.140 --> 01:22:54.790 align:middle line:84% Or at least, the derivative with respect to time of rho 01:22:54.790 --> 01:22:59.330 align:middle line:84% u can be written as this derivative here. 01:22:59.330 --> 01:23:02.220 align:middle line:90% 01:23:02.220 --> 01:23:05.190 align:middle line:84% And if the density is constant, and the specific heat 01:23:05.190 --> 01:23:07.530 align:middle line:84% is constant, it goes out of the derivative, 01:23:07.530 --> 01:23:16.950 align:middle line:84% and you get only rho cv dT over dt 01:23:16.950 --> 01:23:21.990 align:middle line:84% minus the gradient (I meant, the divergence) of q-- 01:23:21.990 --> 01:23:30.360 align:middle line:84% q is-- the Fourier equation, if you have an isotropic material, 01:23:30.360 --> 01:23:31.920 align:middle line:90% if it is-- 01:23:31.920 --> 01:23:35.390 align:middle line:84% OK, let's keep it isotropic just for simplicity. 01:23:35.390 --> 01:23:42.420 align:middle line:84% So I have to substitute here, minus k times the gradient of T. 01:23:42.420 --> 01:23:47.010 align:middle line:90% So that makes it, if k-- 01:23:47.010 --> 01:23:51.990 align:middle line:84% this requires that k be independent of space-- 01:23:51.990 --> 01:23:53.430 align:middle line:90% of temperature. 01:23:53.430 --> 01:23:56.430 align:middle line:84% So if k is a constant, it goes out of this. 01:23:56.430 --> 01:23:59.040 align:middle line:84% Otherwise, you have to keep it inside. 01:23:59.040 --> 01:24:02.280 align:middle line:90% So if k is constant-- 01:24:02.280 --> 01:24:05.850 align:middle line:84% because the conductivity of materials does 01:24:05.850 --> 01:24:07.180 align:middle line:90% change with temperature. 01:24:07.180 --> 01:24:10.870 align:middle line:84% So this step may not be always allowed, 01:24:10.870 --> 01:24:16.170 align:middle line:84% but if you do that, this becomes the Laplacian 01:24:16.170 --> 01:24:17.350 align:middle line:90% of the temperature. 01:24:17.350 --> 01:24:22.290 align:middle line:84% Because the gradient dotted into the gradient 01:24:22.290 --> 01:24:25.970 align:middle line:90% gives the Laplacian operator. 01:24:25.970 --> 01:24:31.450 align:middle line:84% If you dot this vector into another vector equal to itself, 01:24:31.450 --> 01:24:34.630 align:middle line:84% the dot product gives you the products 01:24:34.630 --> 01:24:36.650 align:middle line:84% of the components of the vectors. 01:24:36.650 --> 01:24:38.440 align:middle line:84% And so here, you have the partial 01:24:38.440 --> 01:24:40.760 align:middle line:90% of x times another partial of x. 01:24:40.760 --> 01:24:43.710 align:middle line:84% So that's the second derivative with respect to x twice. 01:24:43.710 --> 01:24:49.210 align:middle line:84% And those are the various ingredients, the three 01:24:49.210 --> 01:24:53.600 align:middle line:90% that represent the Laplacian. 01:24:53.600 --> 01:25:06.180 align:middle line:90% 01:25:06.180 --> 01:25:11.490 align:middle line:84% In any case, this is the equation of Fourier, the Fourier 01:25:11.490 --> 01:25:12.200 align:middle line:90% energy balance. 01:25:12.200 --> 01:25:20.230 align:middle line:90% 01:25:20.230 --> 01:25:26.770 align:middle line:84% If you have a non-isotropic material, 01:25:26.770 --> 01:25:30.900 align:middle line:84% there is an interesting change of coordinates. 01:25:30.900 --> 01:25:35.920 align:middle line:84% And we've seen that the heat flux 01:25:35.920 --> 01:25:41.540 align:middle line:84% and the gradient in temperature are not in the same direction. 01:25:41.540 --> 01:25:44.050 align:middle line:90% 01:25:44.050 --> 01:25:54.880 align:middle line:84% And there is a change of coordinates that you can do. 01:25:54.880 --> 01:25:57.550 align:middle line:84% So define non-orthogonal coordinates 01:25:57.550 --> 01:26:02.630 align:middle line:84% instead of the usual orthogonal x and y, the small x and y. 01:26:02.630 --> 01:26:06.070 align:middle line:90% You can define capital X and Y. 01:26:06.070 --> 01:26:10.330 align:middle line:90% Here, actually, this is wrong. 01:26:10.330 --> 01:26:12.070 align:middle line:90% There is no equal sign here. 01:26:12.070 --> 01:26:15.050 align:middle line:90% This is just the multiplication. 01:26:15.050 --> 01:26:17.950 align:middle line:84% So that's the linear combination of the original coordinates 01:26:17.950 --> 01:26:22.570 align:middle line:84% to give these coordinates, which are 01:26:22.570 --> 01:26:25.640 align:middle line:84% at a certain angle between one another. 01:26:25.640 --> 01:26:32.500 align:middle line:84% And if you write it that way, then the equation, 01:26:32.500 --> 01:26:36.880 align:middle line:84% the Laplacian in terms of the new coordinates 01:26:36.880 --> 01:26:44.510 align:middle line:84% becomes like that for a uniform material. 01:26:44.510 --> 01:26:50.130 align:middle line:84% And where k is equal, the effective conductivity, 01:26:50.130 --> 01:26:53.190 align:middle line:84% uniform conductivity, is equal to this linear combination 01:26:53.190 --> 01:26:56.170 align:middle line:90% of the various terms. 01:26:56.170 --> 01:27:02.190 align:middle line:84% So this is the square root of the determinant 01:27:02.190 --> 01:27:08.260 align:middle line:90% of the conductivity tensor. 01:27:08.260 --> 01:27:13.230 align:middle line:84% And also, the heat flux vector in the original coordinates, 01:27:13.230 --> 01:27:15.020 align:middle line:90% it becomes a linear combination. 01:27:15.020 --> 01:27:18.160 align:middle line:84% Also here, I have to correct this. 01:27:18.160 --> 01:27:21.350 align:middle line:84% It shouldn't be an equal sign, just a multiplication sign. 01:27:21.350 --> 01:27:32.320 align:middle line:84% And it gives the flux components in this new non-orthogonal 01:27:32.320 --> 01:27:33.230 align:middle line:90% coordinates. 01:27:33.230 --> 01:27:39.680 align:middle line:84% And they are related through that conductivity directly. 01:27:39.680 --> 01:27:42.670 align:middle line:84% So there are no off-diagonal terms. 01:27:42.670 --> 01:27:46.910 align:middle line:84% So this is just a useful change of coordinates. 01:27:46.910 --> 01:27:50.230 align:middle line:84% But there is one last item that I wanted-- 01:27:50.230 --> 01:27:54.090 align:middle line:84% I promised I would discuss, which 01:27:54.090 --> 01:28:02.070 align:middle line:84% has to do with this problem, that if you are dealing 01:28:02.070 --> 01:28:08.490 align:middle line:84% with a heat transfer, there is this conundrum 01:28:08.490 --> 01:28:17.270 align:middle line:90% of the fact that the entropy-- 01:28:17.270 --> 01:28:22.210 align:middle line:84% that you cannot be at equilibrium. 01:28:22.210 --> 01:28:29.440 align:middle line:84% And in fact, so the entropy of your sample, of your material, 01:28:29.440 --> 01:28:32.930 align:middle line:90% is off the equilibrium. 01:28:32.930 --> 01:28:37.940 align:middle line:84% So assigning it the equilibrium entropy introduces an error. 01:28:37.940 --> 01:28:43.870 align:middle line:84% And it's interesting to see how you now solve that problem 01:28:43.870 --> 01:28:47.350 align:middle line:90% by assuming, instead-- 01:28:47.350 --> 01:28:51.310 align:middle line:84% assigning to the fluid element or continuum 01:28:51.310 --> 01:28:55.660 align:middle line:84% element, not the equilibrium entropy, but an entropy which 01:28:55.660 --> 01:28:58.460 align:middle line:90% is less than equilibrium. 01:28:58.460 --> 01:29:04.390 align:middle line:84% And it departs from equilibrium more 01:29:04.390 --> 01:29:05.950 align:middle line:90% the higher the heat transfer. 01:29:05.950 --> 01:29:08.800 align:middle line:84% So you make it through a proportionality constant, which 01:29:08.800 --> 01:29:11.710 align:middle line:84% I, for convenience, call b over 2, 01:29:11.710 --> 01:29:15.590 align:middle line:84% proportional to the modulus of the square of the heat transfer. 01:29:15.590 --> 01:29:17.990 align:middle line:84% So clearly, if you're at equilibrium, 01:29:17.990 --> 01:29:19.340 align:middle line:90% there is no heat transfer. 01:29:19.340 --> 01:29:24.260 align:middle line:84% And this difference-- so the entropy is that of equilibrium. 01:29:24.260 --> 01:29:26.300 align:middle line:84% But the higher the heat transfer, 01:29:26.300 --> 01:29:29.660 align:middle line:84% the farther you are from equilibrium. 01:29:29.660 --> 01:29:32.600 align:middle line:84% So if I take the differential of this one, 01:29:32.600 --> 01:29:36.320 align:middle line:84% it gives me the differential of the equilibrium state 01:29:36.320 --> 01:29:39.980 align:middle line:84% plus the differential of that, which is-- 01:29:39.980 --> 01:29:41.310 align:middle line:90% so the 2 goes away. 01:29:41.310 --> 01:29:44.810 align:middle line:90% It's b q dot q plus-- 01:29:44.810 --> 01:29:47.540 align:middle line:90% q dot dq plus dq dot q. 01:29:47.540 --> 01:29:49.640 align:middle line:90% So it's twice q dot dq. 01:29:49.640 --> 01:29:53.520 align:middle line:90% 01:29:53.520 --> 01:29:59.850 align:middle line:84% Now, this is important because if we 01:29:59.850 --> 01:30:07.430 align:middle line:84% do the entropy balance, like we wrote the energy balance here, 01:30:07.430 --> 01:30:10.880 align:middle line:84% we should write also the entropy balance. 01:30:10.880 --> 01:30:19.200 align:middle line:84% And that is given by the divergence of the entropy flux, 01:30:19.200 --> 01:30:27.290 align:middle line:84% which is q over T, plus the rate of entropy production per unit 01:30:27.290 --> 01:30:31.400 align:middle line:90% volume, which is sigma. 01:30:31.400 --> 01:30:36.530 align:middle line:84% Now, usually, these two are related 01:30:36.530 --> 01:30:39.290 align:middle line:90% through the Gibbs relation. 01:30:39.290 --> 01:30:42.470 align:middle line:84% The Gibbs relation you can write it 01:30:42.470 --> 01:30:51.050 align:middle line:84% as d rho s equals 1 over T d of rho u. 01:30:51.050 --> 01:30:59.690 align:middle line:84% And the reason I like to write it in terms of entropy per unit 01:30:59.690 --> 01:31:04.710 align:middle line:84% volume is that in this way, from the Gibbs relation-- 01:31:04.710 --> 01:31:07.340 align:middle line:84% we'll see more later about this-- 01:31:07.340 --> 01:31:09.270 align:middle line:90% the pressure disappears. 01:31:09.270 --> 01:31:12.000 align:middle line:84% The pressure is always a bit problematic to treat. 01:31:12.000 --> 01:31:19.652 align:middle line:84% So if you find a way that it disappears, it's better. 01:31:19.652 --> 01:31:23.010 align:middle line:84% So if this is true for the differential, 01:31:23.010 --> 01:31:26.790 align:middle line:84% it means all possible changes in all directions. 01:31:26.790 --> 01:31:30.000 align:middle line:84% It's true, also, if you take the derivative with respect to time. 01:31:30.000 --> 01:31:34.690 align:middle line:84% So just move only with respect to time. 01:31:34.690 --> 01:31:42.330 align:middle line:84% So this can also be written as partial with respect to t, 01:31:42.330 --> 01:31:46.490 align:middle line:84% and it's 1 over T times the partial with respect to t of rho 01:31:46.490 --> 01:31:46.990 align:middle line:90% u. 01:31:46.990 --> 01:31:49.750 align:middle line:90% 01:31:49.750 --> 01:31:54.340 align:middle line:84% And this is shorthand for the partial derivative with respect 01:31:54.340 --> 01:31:55.070 align:middle line:90% to time. 01:31:55.070 --> 01:32:00.070 align:middle line:84% And so the Gibbs relation couples the energy 01:32:00.070 --> 01:32:02.270 align:middle line:90% and the entropy balance. 01:32:02.270 --> 01:32:11.960 align:middle line:84% And so in that way, this is actually what I just wrote. 01:32:11.960 --> 01:32:14.450 align:middle line:90% Yes, I should-- this is true. 01:32:14.450 --> 01:32:17.100 align:middle line:84% Because the Gibbs relation is valid for equilibrium. 01:32:17.100 --> 01:32:24.300 align:middle line:84% So this is true for the e, for the equilibrium entropy s sub e, 01:32:24.300 --> 01:32:29.510 align:middle line:84% or s sub SES stable equilibrium state. 01:32:29.510 --> 01:32:32.900 align:middle line:84% Here, I should just unify the notation. 01:32:32.900 --> 01:32:39.920 align:middle line:84% Anyway, so if you do this substitution, 01:32:39.920 --> 01:32:43.500 align:middle line:84% you can obtain the expression for the entropy production, 01:32:43.500 --> 01:32:45.440 align:middle line:84% which is the product of the heat flux, 01:32:45.440 --> 01:32:48.950 align:middle line:84% times this thing-- so this is not only 01:32:48.950 --> 01:32:59.180 align:middle line:84% the gradient in 1 over T, but there is also 01:32:59.180 --> 01:33:02.450 align:middle line:84% b times the derivative with respect to t 01:33:02.450 --> 01:33:04.330 align:middle line:90% of q, which comes from here. 01:33:04.330 --> 01:33:09.990 align:middle line:90% 01:33:09.990 --> 01:33:15.980 align:middle line:84% So this term in square brackets is now the affinity. 01:33:15.980 --> 01:33:18.500 align:middle line:84% So the affinity changes, the expression for the affinity 01:33:18.500 --> 01:33:20.960 align:middle line:90% has changed. 01:33:20.960 --> 01:33:25.020 align:middle line:90% And if we assume a linear. 01:33:25.020 --> 01:33:30.890 align:middle line:84% relation between flux and affinity 01:33:30.890 --> 01:33:36.920 align:middle line:84% through a positive constant L, what we obtain 01:33:36.920 --> 01:33:42.320 align:middle line:84% is a correction to the Fourier's law of conduction, 01:33:42.320 --> 01:33:48.810 align:middle line:84% which was proposed by Cattaneo, which gives rise to the Cattaneo 01:33:48.810 --> 01:33:51.350 align:middle line:90% heat equation, 1948. 01:33:51.350 --> 01:33:55.800 align:middle line:84% Cattaneo didn't start from these premises. 01:33:55.800 --> 01:34:00.800 align:middle line:84% He wanted to fix another problem that I'm describing in a second. 01:34:00.800 --> 01:34:06.800 align:middle line:84% But so let's take this one and then substitute back 01:34:06.800 --> 01:34:10.920 align:middle line:84% into the energy balance equation. 01:34:10.920 --> 01:34:14.030 align:middle line:84% So we have an expression for q double prime. 01:34:14.030 --> 01:34:20.070 align:middle line:84% You can take the divergence of that, 01:34:20.070 --> 01:34:26.270 align:middle line:84% and that gives you this term, which is like typical Fourier. 01:34:26.270 --> 01:34:31.820 align:middle line:84% But also, it's the gradient, I mean, the divergence 01:34:31.820 --> 01:34:35.146 align:middle line:84% of the partial derivative with respect to t 01:34:35.146 --> 01:34:37.900 align:middle line:84% and the divergence of this contribution. 01:34:37.900 --> 01:34:44.050 align:middle line:84% You can switch the partial derivative with respect 01:34:44.050 --> 01:34:46.900 align:middle line:84% to time with the partial derivatives with respect 01:34:46.900 --> 01:34:49.210 align:middle line:90% to space. 01:34:49.210 --> 01:34:52.090 align:middle line:90% And so this also appears. 01:34:52.090 --> 01:34:54.750 align:middle line:84% And again, using the energy balance, 01:34:54.750 --> 01:34:58.630 align:middle line:84% this is the partial with respect to time of rho u. 01:34:58.630 --> 01:35:00.550 align:middle line:84% So you have the double partial with respect 01:35:00.550 --> 01:35:06.730 align:middle line:84% to time, which if you then substitute 01:35:06.730 --> 01:35:10.780 align:middle line:84% this expression for the specific heats 01:35:10.780 --> 01:35:16.510 align:middle line:84% and the density then can be rewritten in this way. 01:35:16.510 --> 01:35:22.300 align:middle line:84% So in other words, you obtain an equation with double-- 01:35:22.300 --> 01:35:25.360 align:middle line:84% second derivative with respect to time 01:35:25.360 --> 01:35:28.460 align:middle line:84% of the temperature, first derivative with respect to time, 01:35:28.460 --> 01:35:31.210 align:middle line:90% and then the usual Laplacian. 01:35:31.210 --> 01:35:33.740 align:middle line:84% If you didn't have that term, you 01:35:33.740 --> 01:35:40.510 align:middle line:84% would have the standard Fourier equation. 01:35:40.510 --> 01:35:43.210 align:middle line:84% But with that term, things change, 01:35:43.210 --> 01:35:48.390 align:middle line:84% and it becomes a damped wave equation. 01:35:48.390 --> 01:35:53.970 align:middle line:90% And it actually fixes a problem. 01:35:53.970 --> 01:35:58.260 align:middle line:84% I'm sure you're familiar with this problem from heat transfer, 01:35:58.260 --> 01:36:04.500 align:middle line:84% in which you have a semi-infinite solid, which 01:36:04.500 --> 01:36:08.640 align:middle line:84% is initially at a certain temperature T, T0. 01:36:08.640 --> 01:36:12.630 align:middle line:84% And at time equals 0, you suddenly 01:36:12.630 --> 01:36:18.690 align:middle line:84% change and keep, from then on, the temperature here 01:36:18.690 --> 01:36:20.850 align:middle line:90% on the surface. 01:36:20.850 --> 01:36:25.380 align:middle line:84% As a result, you obtain temperature profiles 01:36:25.380 --> 01:36:30.440 align:middle line:84% that look like that, given by the error function distribution. 01:36:30.440 --> 01:36:34.650 align:middle line:90% 01:36:34.650 --> 01:36:37.710 align:middle line:84% And as time goes on, the distribution 01:36:37.710 --> 01:36:41.220 align:middle line:84% goes like that, so as time proceeds. 01:36:41.220 --> 01:36:42.970 align:middle line:90% But what is the problem? 01:36:42.970 --> 01:36:49.920 align:middle line:84% The nonphysical problem is that already starting from time 0 01:36:49.920 --> 01:37:00.000 align:middle line:84% plus, when the error function sets on, it turns out that since 01:37:00.000 --> 01:37:06.000 align:middle line:84% the error function is non-zero everywhere, it turns out that-- 01:37:06.000 --> 01:37:07.390 align:middle line:90% may be minute-- 01:37:07.390 --> 01:37:12.090 align:middle line:84% but you still have an infinite propagation rate 01:37:12.090 --> 01:37:19.140 align:middle line:84% of what you have done here, which is unphysical, 01:37:19.140 --> 01:37:22.770 align:middle line:90% because there is no infinite-- 01:37:22.770 --> 01:37:27.180 align:middle line:84% if you do something here now, it is 01:37:27.180 --> 01:37:31.320 align:middle line:84% unphysical to think that it is felt immediately 01:37:31.320 --> 01:37:32.585 align:middle line:90% at infinite distance. 01:37:32.585 --> 01:37:35.170 align:middle line:90% 01:37:35.170 --> 01:37:40.790 align:middle line:84% So the Cattaneo equation solves this problem. 01:37:40.790 --> 01:37:45.230 align:middle line:84% And in fact, this equation has been 01:37:45.230 --> 01:37:51.380 align:middle line:84% used to represent the early times of this sharp change 01:37:51.380 --> 01:37:55.940 align:middle line:84% of temperature problem or situations 01:37:55.940 --> 01:38:01.570 align:middle line:84% in which you will change the boundary conditions very quickly 01:38:01.570 --> 01:38:11.670 align:middle line:84% and therefore, you need to account for a propagation 01:38:11.670 --> 01:38:17.100 align:middle line:84% speed, which is this coefficient here becomes now the wave 01:38:17.100 --> 01:38:18.930 align:middle line:90% celerity. 01:38:18.930 --> 01:38:23.680 align:middle line:84% And the gamma becomes the damping coefficient. 01:38:23.680 --> 01:38:34.410 align:middle line:90% 01:38:34.410 --> 01:38:40.280 align:middle line:84% So time is up, and I think we should stop here for today. 01:38:40.280 --> 01:38:52.000 align:middle line:90%