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PROFESSOR: Let's start.

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Any questions?

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00:00:24,545 --> 00:00:27,810

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We've been gradually building
up thermodynamics, culminating

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in what I call the most
important fundamental relation,

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which was the E is
T dS plus Ji dxi.

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And essentially
we said that this

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was some statement of
conservation of energy.

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where there's a heat component
that goes into the system,

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and there's a work component.

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And actually, we ended up
breaking the work component

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into a mechanical part
and a chemical part,

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which we wrote as
mu alpha dN alpha.

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I guess if I don't
write the sums,

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the summation
convention is assumed.

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00:01:38,640 --> 00:01:41,710

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Now, this gave us an idea of
what a thermodynamic system is,

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in the sense of how many
coordinates do we need in order

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to place the system
in equilibrium

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somewhere in some
coordinate space.

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And we said that we can just
count the number of variables

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that we have on this side.

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And indeed, for
the case of energy,

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the natural set of variables
would be S, x, and N.

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And given some place in
the coordinate system that

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is S, x, and N, you
could find a point that

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corresponds to that
thermodynamic equilibrium

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state.

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00:02:24,445 --> 00:02:28,580

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00:02:28,580 --> 00:02:31,830
Now, having gotten this
fundamental relation,

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we can manipulate and write
it in different fashion.

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And in particular,
for example, you

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can write the S is dE by
dT, minus Ji dxi over T,

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minus mu alpha dN alpha over
T. So you could in principle

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express everything rather than
in terms of entropy-- which

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in some circumstances may not
be the right variable-- in terms

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of the energy content of
the system and the variables

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that you need in order to
make work type of changes

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on the system.

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And we saw that other
versions of it is possible.

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So for example, we
could rearrange.

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Rather than looking
at E, we could

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look at E minus TS, the
quantity that we call F.

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And then dF would look very
much like the original equation.

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Rather than T dS, we
would get minus S dT.

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And clearly, for
this quantity F,

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the natural variables
are T, x, and N.

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And there were other
types of these functions

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that we construct.

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So basically at this stage, we
have a formula that tells us

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how to navigate our way
in this space of things

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that are in equilibrium.

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And by taking various
derivatives of equilibrium

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functions with
respect to each other,

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we can express other ones.

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And via this methodology, we can
relate to many different things

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to each other and have
mathematical relations

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between physical observables.

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So following up on
that, we then ask

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what are some other mathematical
forms that we can look at.

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And one set of forms
followed from extensivity.

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And we saw that if we were
under circumstances such

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that we were to multiply all
of the extensive variables

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by some factor-- 2, 3,
whatever-- and the energy

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content would increase
by that same factor,

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then we could in fact
integrate the equation

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and drop the d's and
write it in this fashion--

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TS Jx plus mu N, regarding
these as kind of vectors.

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00:05:30,393 --> 00:05:33,710

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And importantly, that told us
that the intensive variables

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were not independent
of each other.

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And you had relationship that
you-- S dT plus Ji xi dJi

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plus N alpha d mu alpha is 0.

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00:05:57,110 --> 00:06:00,160

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And last time, I
did one calculation

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using this Gibbs-Duhem
relation that I repeat.

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And that was for a gas isotherm.

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00:06:11,150 --> 00:06:15,320

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Isotherm corresponds
to dT goes to 0.

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00:06:18,370 --> 00:06:21,420

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00:06:21,420 --> 00:06:26,870
And the analogue of hydrostatic
work for a gas was minus P

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for J. So this becomes
minus V dP plus N d mu.

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If you have one
component, it goes to 0.

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And we said that this
gives us the relationship

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along an isotherm.

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So for a fixed T, d
mu by dP is V over N.

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I'll come back to that shortly.

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00:06:54,970 --> 00:07:01,690

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Towards the end
of last lecture, I

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mentioned that we
can get a number

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of other so-called
Maxwell relations

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by noting that mixed
second derivatives are

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independent of the order
of taking derivatives.

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So that if you have some df
of x and y written as df by dx

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at constant y dx, plus df
by dy at constant x dy,

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then the order of the
second derivatives, which

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00:07:45,330 --> 00:07:48,670
would be obtained
by taking either

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00:07:48,670 --> 00:07:51,410
a derivative of the
first term with respect

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00:07:51,410 --> 00:07:54,440
to y or the second term
with respect to x--

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doesn't matter which order.

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00:07:55,750 --> 00:08:03,790

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00:08:03,790 --> 00:08:10,870
So let's start with
our initial equation.

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And for simplicity,
let's think of cases

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where we have a fixed
number of particles

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so that we can write
dE as T dS plus Ji dxi.

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00:08:25,597 --> 00:08:29,580

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00:08:29,580 --> 00:08:34,190
So this looks like this dF
that I wrote before for you.

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And I can identify temperature
as dE by dS at constant x,

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and each one of the forces Ji
as dE by dxi at constant S.

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And all of the J's that are
not equal to i kept fixed.

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And then, since the
second derivative

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00:09:06,160 --> 00:09:10,880
is irrespective of order, I
can form the second derivative

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00:09:10,880 --> 00:09:17,440
with respect to the combination
S xi into a different fashion.

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00:09:17,440 --> 00:09:21,820
I can either take a derivative
over here-- I already

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have derivative with
respect to S giving me T,

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so I will get dT by dxi,
essentially at constant S--

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00:09:36,440 --> 00:09:40,400
or I can take the
derivative of this object.

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00:09:40,400 --> 00:09:44,940
I already have the x derivative,
so I take the S derivative.

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So I have dJi with respect
to S at constant x.

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00:09:50,800 --> 00:09:55,160

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So this is an example
of the Maxwell relation.

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00:10:02,275 --> 00:10:05,430

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00:10:05,430 --> 00:10:13,130
And just to remind
you, we can always

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invert these derivatives.

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00:10:15,820 --> 00:10:20,440
So I can invert this
and say that actually,

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if I had wondered how
the entropy changes

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00:10:24,130 --> 00:10:29,810
as a function of a force such
as pressure at constant x,

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00:10:29,810 --> 00:10:36,040
it would be related to how
the corresponding displacement

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to the force changes as a
function of temperature.

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00:10:39,930 --> 00:10:44,570
So you would have a relationship
between some observable

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00:10:44,570 --> 00:10:48,100
such as how the length
of a wire changes

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00:10:48,100 --> 00:10:50,880
if you change
temperature rapidly,

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00:10:50,880 --> 00:10:55,560
and how the entropy would change
as a function of pressure,

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00:10:55,560 --> 00:11:00,340
or corresponding force,
whatever it may be.

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00:11:00,340 --> 00:11:07,220
Now, what I would like to
remind you is that the best

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00:11:07,220 --> 00:11:10,570
way to sort of derive
these relationships

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00:11:10,570 --> 00:11:13,680
is to reconstruct
what kind of second

151
00:11:13,680 --> 00:11:15,360
derivative or mixed
derivative you

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00:11:15,360 --> 00:11:19,370
want to have to give
you the right result.

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So for example, say
somebody told me

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that I want to calculate
something similar-- dS by dJ,

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let's say at
constant temperature.

156
00:11:29,920 --> 00:11:33,190

157
00:11:33,190 --> 00:11:34,625
How do I calculate that?

158
00:11:34,625 --> 00:11:37,990

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A nice way to calculate
that is if I somehow

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manipulate so that S appears
as a first derivative.

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Now, S does not appear as
a first derivative here,

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but certainly appears as
a first derivative here.

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So who says I have to take
E as a function of state?

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I can look at the
second derivative of F.

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So if I look at F,
which is E minus TS--

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and actually, I don't
even need to know

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what the name of this entity is,
whether it's F, G, et cetera.

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All I know is that it will
convert this to minus S dT--

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the TS that I don't want,
so that S appears now

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as a first derivative.

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00:12:30,120 --> 00:12:33,130
Now I want to take a
derivative of S with respect

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00:12:33,130 --> 00:12:42,370
to J. If I just stop here, the
next term that I would have

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is Ji dxi.

174
00:12:43,305 --> 00:12:46,015

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And the natural way that I would
construct a second derivative

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00:12:50,820 --> 00:12:54,390
would give me dS
with respect to xi.

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00:12:54,390 --> 00:12:57,185
But I want the S
with respect to Ji.

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So I say, OK, I'll do
this function instead.

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00:13:00,650 --> 00:13:02,350
As I said, I don't
really care what

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00:13:02,350 --> 00:13:04,570
the name of these functions is.

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00:13:04,570 --> 00:13:06,830
This will get
converted to xi dJ.

182
00:13:06,830 --> 00:13:10,710

183
00:13:10,710 --> 00:13:15,200
Now I can say there is
this first derivative-- d o

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00:13:15,200 --> 00:13:21,990
minus S with respect
to Ji at constant T

185
00:13:21,990 --> 00:13:27,990
is the same thing as d
of minus y with respect

186
00:13:27,990 --> 00:13:42,110
to T at constant J. So I can
get rid of the minus sign

187
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if I want, and I have
the answer that I want.

188
00:13:44,700 --> 00:14:03,570
It is dxi by dT at constant
J-- at constant-- yeah, T-- J.

189
00:14:03,570 --> 00:14:06,410
And you can go and construct
anything else that you like.

190
00:14:06,410 --> 00:14:09,110

191
00:14:09,110 --> 00:14:14,170
Let me, for example,
try to construct

192
00:14:14,170 --> 00:14:18,520
this entity we
already calculated.

193
00:14:18,520 --> 00:14:20,870
So let's see.

194
00:14:20,870 --> 00:14:27,350
What can I say about d mu by
dP at constant temperature,

195
00:14:27,350 --> 00:14:35,230
and I want to calculate
that via a Maxwell relation?

196
00:14:35,230 --> 00:14:36,030
So what do I know?

197
00:14:36,030 --> 00:14:43,250
I know that dE is
T dS minus pdV.

198
00:14:43,250 --> 00:14:45,130
Good.

199
00:14:45,130 --> 00:14:47,740
Oh, I want to have
mu in the play,

200
00:14:47,740 --> 00:14:49,520
so now I have to add mu dN.

201
00:14:49,520 --> 00:14:54,110

202
00:14:54,110 --> 00:14:59,070
That's good, because I have
mu as a first derivative.

203
00:14:59,070 --> 00:15:01,600
I have it over here.

204
00:15:01,600 --> 00:15:05,320
But I have P as a
first derivative,

205
00:15:05,320 --> 00:15:09,740
whereas I want P to appear as
the work element, something

206
00:15:09,740 --> 00:15:11,550
like dP.

207
00:15:11,550 --> 00:15:18,530
So what I will do is I will
make this d of E plus pV.

208
00:15:18,530 --> 00:15:28,166
This converts this to
plus V dP plus mu dN.

209
00:15:28,166 --> 00:15:32,840
That's fine, except that
when I do things now,

210
00:15:32,840 --> 00:15:36,440
I would have calculated
things at constant S.

211
00:15:36,440 --> 00:15:39,420
And I want to calculate
things at constant T.

212
00:15:39,420 --> 00:15:40,870
That's not really important.

213
00:15:40,870 --> 00:15:45,660
I do a minus ST. This
becomes minus S dT.

214
00:15:45,660 --> 00:15:49,890
And then I have a
function of state--

215
00:15:49,890 --> 00:15:53,110
I don't care what
it's called-- that has

216
00:15:53,110 --> 00:16:00,910
the right format for me
to identify that d mu

217
00:16:00,910 --> 00:16:08,710
by dP at constant N and T,
along an isotherm of a fixed

218
00:16:08,710 --> 00:16:17,640
number of particles,
is the same thing as dV

219
00:16:17,640 --> 00:16:28,970
by dN at constant PN T. So
that's my Maxwell relation.

220
00:16:28,970 --> 00:16:31,215
So why didn't I get this result?

221
00:16:31,215 --> 00:16:35,576

222
00:16:35,576 --> 00:16:36,806
Did I make a mistake?

223
00:16:36,806 --> 00:16:41,476

224
00:16:41,476 --> 00:16:43,464
AUDIENCE: [INAUDIBLE].

225
00:16:43,464 --> 00:16:44,955
PROFESSOR: Louder.

226
00:16:44,955 --> 00:16:47,950
AUDIENCE: Is this valid
for extensive systems?

227
00:16:47,950 --> 00:16:50,440
PROFESSOR: Yes, this is
valid for extensive system.

228
00:16:50,440 --> 00:16:54,860
I used extensively in
the relation of this.

229
00:16:54,860 --> 00:16:58,540
And here, I never
used extensivity.

230
00:16:58,540 --> 00:17:01,130
But if I have a system that
I tell you its pressure

231
00:17:01,130 --> 00:17:05,270
and temperature, how
is its volume related

232
00:17:05,270 --> 00:17:07,270
to the number of particles?

233
00:17:07,270 --> 00:17:09,280
I have a box in this room.

234
00:17:09,280 --> 00:17:12,950
I tell you pressure and
temperature are fixed.

235
00:17:12,950 --> 00:17:16,750
If I make the box twice as big,
number of particles, volume

236
00:17:16,750 --> 00:17:18,169
will go twice as big.

237
00:17:18,169 --> 00:17:22,740
So if I were to
apply extensivity,

238
00:17:22,740 --> 00:17:25,665
I would have to conclude that
this is the same thing as V

239
00:17:25,665 --> 00:17:28,202
over N, and I would
be in agreement

240
00:17:28,202 --> 00:17:29,160
with what I had before.

241
00:17:29,160 --> 00:17:35,262

242
00:17:35,262 --> 00:17:35,845
Any questions?

243
00:17:35,845 --> 00:17:40,471

244
00:17:40,471 --> 00:17:40,970
Yes.

245
00:17:40,970 --> 00:17:43,136
AUDIENCE: What about the
constraint of constant PN T

246
00:17:43,136 --> 00:17:44,677
[INAUDIBLE]?

247
00:17:44,677 --> 00:17:45,260
PROFESSOR: OK.

248
00:17:45,260 --> 00:17:47,570
Actually, constraint
of constant PN T

249
00:17:47,570 --> 00:17:50,700
is important because
we said that said there

250
00:17:50,700 --> 00:17:56,088
it is a kind of
Gibbs-Duhem relation.

251
00:17:56,088 --> 00:17:59,360
And the Gibbs-Duhem
relation tells me

252
00:17:59,360 --> 00:18:13,252
for the case of the gas that
the S dT minus V dP plus N d mu

253
00:18:13,252 --> 00:18:14,860
is 0.

254
00:18:14,860 --> 00:18:17,650

255
00:18:17,650 --> 00:18:19,920
And that constraint
is the one that

256
00:18:19,920 --> 00:18:25,700
tells me that once you have
set P and T, which are the two

257
00:18:25,700 --> 00:18:30,820
variables that I have specified
here, you also know mu.

258
00:18:30,820 --> 00:18:35,150
That is, you know all
the intensive quantities.

259
00:18:35,150 --> 00:18:39,050
So once you set PN T-- the
constraints over there--

260
00:18:39,050 --> 00:18:40,800
really, the only thing
that you don't know

261
00:18:40,800 --> 00:18:44,070
is how big the system is.

262
00:18:44,070 --> 00:18:46,620
And that's what's the condition
that I used over here.

263
00:18:46,620 --> 00:18:49,350

264
00:18:49,350 --> 00:18:51,820
But it is only-- so
let me re-emphasize

265
00:18:51,820 --> 00:18:56,570
it this way-- I
will write dV by dN

266
00:18:56,570 --> 00:19:01,990
is the same thing as V over
N, only if I have PN T here.

267
00:19:01,990 --> 00:19:06,840
If I had here PN S, for example,
I wouldn't be able to use this.

268
00:19:06,840 --> 00:19:28,250

269
00:19:28,250 --> 00:19:31,625
I was going to
build this equation

270
00:19:31,625 --> 00:19:34,472
to say, look at the
elements of this.

271
00:19:34,472 --> 00:19:38,530
This part we got
from the first law,

272
00:19:38,530 --> 00:19:43,010
the temperature we got
from the zeroth law,

273
00:19:43,010 --> 00:19:50,480
and this we got from the
second law, which is correct,

274
00:19:50,480 --> 00:19:54,040
but it leaves out
something, which

275
00:19:54,040 --> 00:19:58,820
is that what the second
law had was things

276
00:19:58,820 --> 00:20:01,890
going some particular direction.

277
00:20:01,890 --> 00:20:09,490
And in particular, we had
that for the certain changes

278
00:20:09,490 --> 00:20:19,680
that I won't repeat, the change
in entropy to be positive.

279
00:20:19,680 --> 00:20:22,540
So when you are in
equilibrium, you

280
00:20:22,540 --> 00:20:27,750
have this relationship between
equilibrium state functions.

281
00:20:27,750 --> 00:20:30,430
But in fact, the second
law of thermodynamics

282
00:20:30,430 --> 00:20:34,320
tells you a little
bit more than that.

283
00:20:34,320 --> 00:20:37,390
So the next item that
we are going to look at

284
00:20:37,390 --> 00:20:40,610
is, what are the
consequences of having

285
00:20:40,610 --> 00:20:44,240
underlying this equality
between functions

286
00:20:44,240 --> 00:20:48,280
of state some kind of an
inequality about certain things

287
00:20:48,280 --> 00:20:51,690
being possible and things
not being possible?

288
00:20:51,690 --> 00:20:54,119
And that relates to
stability conditions.

289
00:20:54,119 --> 00:21:04,110

290
00:21:04,110 --> 00:21:10,190
And I find it easier to
again introduce to you these

291
00:21:10,190 --> 00:21:14,960
stability conditions by thinking
of a mechanical analogue.

292
00:21:14,960 --> 00:21:26,950
So let's imagine that you
have some kind of a spring,

293
00:21:26,950 --> 00:21:32,240
and that this spring has some
kind of a potential energy

294
00:21:32,240 --> 00:21:37,620
that I will call phi as a
function of the extension x.

295
00:21:37,620 --> 00:21:41,510

296
00:21:41,510 --> 00:21:46,420
And let's say that it's
some non-linear function

297
00:21:46,420 --> 00:21:48,421
such as this.

298
00:21:48,421 --> 00:21:52,290
There is no reason why we
should use a hook and spring,

299
00:21:52,290 --> 00:21:53,460
but something like this.

300
00:21:53,460 --> 00:21:56,320

301
00:21:56,320 --> 00:21:59,650
Now, we also discussed
last time what

302
00:21:59,650 --> 00:22:07,680
happens if I were to pull
on this with some force J.

303
00:22:07,680 --> 00:22:09,640
So I pull on this.

304
00:22:09,640 --> 00:22:12,780
As a result, presumably
it will no longer

305
00:22:12,780 --> 00:22:16,510
be sitting at x, because
what I need to do

306
00:22:16,510 --> 00:22:21,090
is to minimize something else--
the function that we call H--

307
00:22:21,090 --> 00:22:25,630
which was the potential
energy of the spring

308
00:22:25,630 --> 00:22:29,120
plus the entity that
was exerting the force.

309
00:22:29,120 --> 00:22:32,500
So if I were to make it
in the vertical direction,

310
00:22:32,500 --> 00:22:35,680
you can imagine this as
being the potential energy

311
00:22:35,680 --> 00:22:38,720
of the mass that was
pulling on the spring.

312
00:22:38,720 --> 00:22:47,630
So what that means is
that in the presence of J,

313
00:22:47,630 --> 00:22:51,400
what I need to do is
to find the x that

314
00:22:51,400 --> 00:22:52,790
minimizes this expression.

315
00:22:52,790 --> 00:22:55,780

316
00:22:55,780 --> 00:23:05,500
Once essentially the kinetic
energy has disappeared,

317
00:23:05,500 --> 00:23:09,950
my spring stops oscillating
and goes to a particular value.

318
00:23:09,950 --> 00:23:13,700
And mathematically,
that corresponds

319
00:23:13,700 --> 00:23:18,920
to subtracting a linear
function from this.

320
00:23:18,920 --> 00:23:21,270
Actually, what
will it look like?

321
00:23:21,270 --> 00:23:26,680
Let's not make this go flat,
but go up in some linear fashion

322
00:23:26,680 --> 00:23:30,008
so that at least I can
draw something like this.

323
00:23:30,008 --> 00:23:35,040

324
00:23:35,040 --> 00:23:37,670
So what I need to do
is to minimize this.

325
00:23:37,670 --> 00:23:43,140
So I find that for
a given J, I have

326
00:23:43,140 --> 00:23:55,490
to solve d phi dx equals J.
But especially if you have

327
00:23:55,490 --> 00:24:00,306
a spring that has the kind of
potential that I drew for you,

328
00:24:00,306 --> 00:24:05,170
you start increasing
the-- so basically,

329
00:24:05,170 --> 00:24:08,460
what it says is that
you end up at the place

330
00:24:08,460 --> 00:24:13,260
where the slope is
equal to the derivative

331
00:24:13,260 --> 00:24:15,430
of the potential function.

332
00:24:15,430 --> 00:24:18,300
What happens if you start
increasing this slope,

333
00:24:18,300 --> 00:24:21,490
if it is a non-linear type
of thing, at some point

334
00:24:21,490 --> 00:24:24,050
you can see it's impossible.

335
00:24:24,050 --> 00:24:27,470
Essentially, it's like
a very flat noodle.

336
00:24:27,470 --> 00:24:31,950
You pull on it too much and then
it starts to expand forever.

337
00:24:31,950 --> 00:24:37,170
So essentially, this thing that
we are looking for a minimum,

338
00:24:37,170 --> 00:24:39,210
ultimately also
means that you need

339
00:24:39,210 --> 00:24:43,320
to have a condition on
the second derivative.

340
00:24:43,320 --> 00:24:49,210
d2 phi by dx squared better
be positive actually,

341
00:24:49,210 --> 00:24:54,830
I have to take d2
H y dx squared.

342
00:24:54,830 --> 00:24:56,615
I took one derivative
to identify

343
00:24:56,615 --> 00:25:01,600
where the location of
the best solution is.

344
00:25:01,600 --> 00:25:03,590
If I take a second
derivative, this

345
00:25:03,590 --> 00:25:06,880
became essentially the
J. J is a constant.

346
00:25:06,880 --> 00:25:11,080
It only depends on
the potential energy.

347
00:25:11,080 --> 00:25:16,870
So it says independent of
the force that you apply.

348
00:25:16,870 --> 00:25:22,720
You know that if the shape of
the potential energy is given,

349
00:25:22,720 --> 00:25:25,370
the only places
that are accessible

350
00:25:25,370 --> 00:25:27,690
are the kinds of
places where you

351
00:25:27,690 --> 00:25:30,080
take the second
derivative and the second

352
00:25:30,080 --> 00:25:32,390
has a particular sign.

353
00:25:32,390 --> 00:25:36,900
So this portion of the
curve is in principle

354
00:25:36,900 --> 00:25:38,860
physically accessible.

355
00:25:38,860 --> 00:25:42,920
The portion that corresponds
to essentially changing

356
00:25:42,920 --> 00:25:45,900
the curvature and going the
other way, there's no way,

357
00:25:45,900 --> 00:25:48,720
there's no force
that you can put

358
00:25:48,720 --> 00:25:51,815
that would be able to access
those kinds of displacements.

359
00:25:51,815 --> 00:25:54,320

360
00:25:54,320 --> 00:25:59,040
So there is this
convexity condition.

361
00:25:59,040 --> 00:26:02,640
This was for one variable.

362
00:26:02,640 --> 00:26:09,500
Now suppose you have multiple
variables-- x2, x3, et cetera.

363
00:26:09,500 --> 00:26:11,260
So you will have
the potential energy

364
00:26:11,260 --> 00:26:14,500
as a function of many variables.

365
00:26:14,500 --> 00:26:19,330
And the condition that you have
is that the second derivative

366
00:26:19,330 --> 00:26:26,450
of the potential with respect
to all variables, which forms--

367
00:26:26,450 --> 00:26:31,430
and if there are n variables,
this would be an n by n matrix.

368
00:26:31,430 --> 00:26:34,850
This matrix is
positive definite.

369
00:26:34,850 --> 00:26:37,710
What that means is
that if you think

370
00:26:37,710 --> 00:26:42,590
of this as defining for you
the bottom of a potential,

371
00:26:42,590 --> 00:26:46,710
in whatever direction
you go, this

372
00:26:46,710 --> 00:26:50,370
would describe the change
of potential for you.

373
00:26:50,370 --> 00:26:53,110
The first order
change you already

374
00:26:53,110 --> 00:26:58,490
set to 0 because of the force J.
The second order change better

375
00:26:58,490 --> 00:26:59,250
be positive.

376
00:26:59,250 --> 00:27:03,260

377
00:27:03,260 --> 00:27:07,512
So this is the condition that
you would have for stability,

378
00:27:07,512 --> 00:27:08,720
quite generally mechanically.

379
00:27:08,720 --> 00:27:12,100

380
00:27:12,100 --> 00:27:20,590
And again, this
came from the fact

381
00:27:20,590 --> 00:27:24,070
that if you were
to exert a force,

382
00:27:24,070 --> 00:27:27,240
you would change the energy
by something like J delta

383
00:27:27,240 --> 00:27:32,080
x, which is certainly the
same thing that we have here.

384
00:27:32,080 --> 00:27:39,140
This is just one component
that I have been focusing on.

385
00:27:39,140 --> 00:27:41,860
And so ultimately,
I'm going to try

386
00:27:41,860 --> 00:27:46,000
to generalize that to this form.

387
00:27:46,000 --> 00:27:50,510
But one thing that I don't
quite like in this form

388
00:27:50,510 --> 00:27:57,410
is that I'm focusing too
much on the displacements.

389
00:27:57,410 --> 00:28:01,350
This is written appropriately
using this form.

390
00:28:01,350 --> 00:28:04,900
And we have seen over
here that whether or not

391
00:28:04,900 --> 00:28:09,430
I express things in terms
of the displacements

392
00:28:09,430 --> 00:28:14,150
or in terms of entropy,
temperature, conjugate forces,

393
00:28:14,150 --> 00:28:16,890
et cetera, these are
all equivalent ways

394
00:28:16,890 --> 00:28:20,310
of describing the
equilibrium system.

395
00:28:20,310 --> 00:28:23,440
So I'm going to do a
slight manipulation of this

396
00:28:23,440 --> 00:28:27,090
to make that symmetry between
forces and displacements

397
00:28:27,090 --> 00:28:29,020
apparent.

398
00:28:29,020 --> 00:28:31,350
So the generalization
of this expression

399
00:28:31,350 --> 00:28:35,060
that we have over
here is that once I

400
00:28:35,060 --> 00:28:38,010
know the potential
energy, the force

401
00:28:38,010 --> 00:28:41,620
in this multi-component
space is df by dxi.

402
00:28:41,620 --> 00:28:45,160

403
00:28:45,160 --> 00:28:48,870
So in this multi-component
space, I have some J,

404
00:28:48,870 --> 00:28:52,310
and there's a corresponding x.

405
00:28:52,310 --> 00:28:54,950
They're related by this.

406
00:28:54,950 --> 00:29:00,710
Now, if I were to
make a change in J,

407
00:29:00,710 --> 00:29:04,370
there will be a change in the
position of the equilibrium.

408
00:29:04,370 --> 00:29:08,145
And I can get that by
doing a corresponding cange

409
00:29:08,145 --> 00:29:10,290
in the derivative.

410
00:29:10,290 --> 00:29:13,270
The corresponding
change in the derivative

411
00:29:13,270 --> 00:29:17,550
is the second derivative
of phi with respect

412
00:29:17,550 --> 00:29:29,470
to xi xj delta xj, sum
over J. So what I'm saying

413
00:29:29,470 --> 00:29:34,040
is that this is a
condition that relates

414
00:29:34,040 --> 00:29:37,760
the position to the forces.

415
00:29:37,760 --> 00:29:40,370
If I make a slight
change in the force,

416
00:29:40,370 --> 00:29:44,090
there will be a corresponding
slight change in the position.

417
00:29:44,090 --> 00:29:48,650
And the changes in the
position and the force given

418
00:29:48,650 --> 00:29:51,970
by this equilibrium condition
are related by this formula.

419
00:29:51,970 --> 00:29:54,890

420
00:29:54,890 --> 00:30:00,370
Now, you can see that that
sum is already present here.

421
00:30:00,370 --> 00:30:02,650
So I can rewrite
the expression that

422
00:30:02,650 --> 00:30:07,310
is over here using that
formula as delta xi delta

423
00:30:07,310 --> 00:30:10,960
Ji, sum over i--
better be positive.

424
00:30:10,960 --> 00:30:17,730

425
00:30:17,730 --> 00:30:23,260
So this expression is slightly
different and better way

426
00:30:23,260 --> 00:30:27,430
of writing essentially the same
convexity condition, stability

427
00:30:27,430 --> 00:30:29,930
condition, but in a
manner that treats

428
00:30:29,930 --> 00:30:33,710
the displacements and
the forces equivalently.

429
00:30:33,710 --> 00:30:36,620

430
00:30:36,620 --> 00:30:41,230
And I'm going to
apply that to this

431
00:30:41,230 --> 00:30:44,320
and say that in
general, there will

432
00:30:44,320 --> 00:30:47,310
be more ways of
manipulating the energy

433
00:30:47,310 --> 00:30:51,360
of a thermodynamic system, and
that having the system reach

434
00:30:51,360 --> 00:30:59,890
to equilibrium requires
the generalization of this

435
00:30:59,890 --> 00:31:06,540
as a delta T delta S, plus
sum over i delta Ji delta xi,

436
00:31:06,540 --> 00:31:10,260
plus sum over alpha,
delta mu alpha,

437
00:31:10,260 --> 00:31:13,040
delta N alpha-- to be positive.

438
00:31:13,040 --> 00:31:23,500

439
00:31:23,500 --> 00:31:26,620
And you may say I didn't
really derive this for you

440
00:31:26,620 --> 00:31:31,120
because I started from
mechanical equilibrium

441
00:31:31,120 --> 00:31:33,370
and stability condition
and generalized

442
00:31:33,370 --> 00:31:35,680
to the thermodynamic one.

443
00:31:35,680 --> 00:31:43,220
But really, if you go back
and I use this condition

444
00:31:43,220 --> 00:31:48,140
and do manipulations that
are compatible to this,

445
00:31:48,140 --> 00:31:52,800
you will come up with exactly
this stability condition.

446
00:31:52,800 --> 00:31:54,740
And I have that in the notes.

447
00:31:54,740 --> 00:31:58,410
It is somewhat more
mathematically complicated

448
00:31:58,410 --> 00:32:00,590
than this, because the
derivatives involve

449
00:32:00,590 --> 00:32:03,720
additional factors
of 1 over T. And you

450
00:32:03,720 --> 00:32:06,680
have to do slightly more
work, but you will ultimately

451
00:32:06,680 --> 00:32:08,580
come with the same expression.

452
00:32:08,580 --> 00:32:14,995

453
00:32:14,995 --> 00:32:15,495
OK?

454
00:32:15,495 --> 00:32:20,980

455
00:32:20,980 --> 00:32:23,620
You say, why is that of any use?

456
00:32:23,620 --> 00:32:26,290

457
00:32:26,290 --> 00:32:28,320
It is possible
that you've already

458
00:32:28,320 --> 00:32:32,276
seen a particular use
of this expression.

459
00:32:32,276 --> 00:32:37,200

460
00:32:37,200 --> 00:32:44,390
Let's look at the case
of a gas-- one component.

461
00:32:44,390 --> 00:32:46,710
You would say that the
condition that we have

462
00:32:46,710 --> 00:32:54,165
is that delta T delta S minus
delta P delta V plus delta mu

463
00:32:54,165 --> 00:32:57,100
delta N has to be positive.

464
00:32:57,100 --> 00:33:02,030

465
00:33:02,030 --> 00:33:03,710
Let's even be simpler.

466
00:33:03,710 --> 00:33:07,320
Look at the case
where delta T is 0,

467
00:33:07,320 --> 00:33:11,850
delta N equals to 0, which
means that I have a fixed

468
00:33:11,850 --> 00:33:15,740
amount of gas at a
particular temperature.

469
00:33:15,740 --> 00:33:18,430
So I could have,
for example, a box

470
00:33:18,430 --> 00:33:20,670
in this room which
is in equilibrium

471
00:33:20,670 --> 00:33:24,260
at the same temperature as
the temperature of the room.

472
00:33:24,260 --> 00:33:27,260
But its volume I'm
allowed to adjust

473
00:33:27,260 --> 00:33:30,260
so that the pressure changes.

474
00:33:30,260 --> 00:33:34,990
So in that process, at a
particular temperature,

475
00:33:34,990 --> 00:33:42,385
I will be finding a particular
curve in the pressure volume

476
00:33:42,385 --> 00:33:42,885
space.

477
00:33:42,885 --> 00:33:45,500

478
00:33:45,500 --> 00:33:50,710
And the reason that I
drew this particular way

479
00:33:50,710 --> 00:33:55,600
is because I have the condition
that minus dP dV better

480
00:33:55,600 --> 00:34:03,060
be positive, so that if I
have P as a function of V,

481
00:34:03,060 --> 00:34:13,690
then I know that dP
is dP by dV delta V.

482
00:34:13,690 --> 00:34:20,440
And if the product
dP dV is positive,

483
00:34:20,440 --> 00:34:26,620
then this derivative dP
by dV has to be negative.

484
00:34:26,620 --> 00:34:29,469
This has to be a
decreasing function.

485
00:34:29,469 --> 00:34:31,560
And another way
of saying that is

486
00:34:31,560 --> 00:34:41,090
that the compressibility--
kappa T, which is minus 1

487
00:34:41,090 --> 00:34:47,400
over V, the inverse of this
dV by dP at constant T--

488
00:34:47,400 --> 00:34:49,300
has to be positive.

489
00:34:49,300 --> 00:34:56,620

490
00:34:56,620 --> 00:35:00,800
So this is a curve where
dP by dV is negative.

491
00:35:00,800 --> 00:35:04,970

492
00:35:04,970 --> 00:35:06,060
Fine, you say.

493
00:35:06,060 --> 00:35:08,350
Nothing surprising about that.

494
00:35:08,350 --> 00:35:11,530
Well, it turns out that
if you cool the gas,

495
00:35:11,530 --> 00:35:14,070
the shape of this
isotherm can change.

496
00:35:14,070 --> 00:35:17,500
It can get started.

497
00:35:17,500 --> 00:35:20,580
And there is a critical
isotherm at which

498
00:35:20,580 --> 00:35:23,770
you see a condition such
as this, where there's

499
00:35:23,770 --> 00:35:30,000
a point where the curve
comes with 0 slope.

500
00:35:30,000 --> 00:35:33,700
So you say that for that
particular critical isotherm

501
00:35:33,700 --> 00:35:38,620
occurring at TC, if I were
to make this expansion,

502
00:35:38,620 --> 00:35:41,232
the first term in
this expansion is 0.

503
00:35:41,232 --> 00:35:44,850

504
00:35:44,850 --> 00:35:48,090
You say, OK, let's continue and
write down the second term--

505
00:35:48,090 --> 00:35:53,950
1/2 d2 P by dV squared
at constant P times

506
00:35:53,950 --> 00:35:54,700
delta V squared.

507
00:35:54,700 --> 00:35:58,560

508
00:35:58,560 --> 00:36:02,130
Now the problem is
that if I multiply this

509
00:36:02,130 --> 00:36:07,710
by delta V-- which would have
made this into delta V squared

510
00:36:07,710 --> 00:36:10,600
and then this into
delta V cubed--

511
00:36:10,600 --> 00:36:12,770
I have a problem
because delta V can

512
00:36:12,770 --> 00:36:14,600
be both positive or negative.

513
00:36:14,600 --> 00:36:17,610
I can go one direction
or the other direction.

514
00:36:17,610 --> 00:36:21,680
And a term that is a
cubic can change sign--

515
00:36:21,680 --> 00:36:24,500
can sometimes be positive,
can sometimes be negative,

516
00:36:24,500 --> 00:36:27,670
depending on the sign of that.

517
00:36:27,670 --> 00:36:31,750
And that is disallowed
by this condition.

518
00:36:31,750 --> 00:36:36,090
So that would say that if
an analytical expression

519
00:36:36,090 --> 00:36:41,550
for the expansion exists, the
second derivative has to be 0.

520
00:36:41,550 --> 00:36:44,900
And then you would have
to go to the next term-- d

521
00:36:44,900 --> 00:36:48,440
cubed P dV cubed.

522
00:36:48,440 --> 00:36:53,280
And then delta V-- you would
normally write delta V cubed,

523
00:36:53,280 --> 00:36:56,070
but then I've multiplied by
delta V on the other side.

524
00:36:56,070 --> 00:36:58,250
We'd have delta V to the fourth.

525
00:36:58,250 --> 00:37:01,970
Delta V to the fourth you would
say is definitely positive.

526
00:37:01,970 --> 00:37:04,970
Then I can get things done
by having to make sure

527
00:37:04,970 --> 00:37:08,810
that the third
derivative is negative.

528
00:37:08,810 --> 00:37:12,600
And you say, OK, exactly
what have you done here?

529
00:37:12,600 --> 00:37:14,620
What you have done
here is you've

530
00:37:14,620 --> 00:37:18,460
drawn a curve that
locally looks like a cubic

531
00:37:18,460 --> 00:37:20,240
with the appropriate sign.

532
00:37:20,240 --> 00:37:24,280
And so that's certainly
true that if you

533
00:37:24,280 --> 00:37:30,570
have an isotherm there the
compressibility diverges,

534
00:37:30,570 --> 00:37:34,100
you have a condition
on second derivatives.

535
00:37:34,100 --> 00:37:38,880
And you probably have used this
for calculating critical points

536
00:37:38,880 --> 00:37:41,620
of van der Waals
and other gases.

537
00:37:41,620 --> 00:37:44,390
And we will do that ourselves.

538
00:37:44,390 --> 00:37:49,280
But it's actually not the
right way to go about,

539
00:37:49,280 --> 00:37:52,860
because when you go and look
at things experimentally

540
00:37:52,860 --> 00:37:56,850
with sufficient detail, you find
that these curves at this point

541
00:37:56,850 --> 00:37:58,700
are not analytic.

542
00:37:58,700 --> 00:38:00,960
They don't admit a
Taylor expansion.

543
00:38:00,960 --> 00:38:03,660
So while this
condition is correct,

544
00:38:03,660 --> 00:38:06,590
that the shape should
be something like this,

545
00:38:06,590 --> 00:38:09,167
it is neither cubic
nor a fifth power,

546
00:38:09,167 --> 00:38:10,750
but some kind of a
non-analytic curve.

547
00:38:10,750 --> 00:38:15,931

548
00:38:15,931 --> 00:38:18,770
AUDIENCE: Could you
repeat last statement?

549
00:38:18,770 --> 00:38:20,260
PROFESSOR: That
statement, in order

550
00:38:20,260 --> 00:38:22,280
to fully understand
and appreciate,

551
00:38:22,280 --> 00:38:24,220
you have to come
to next term, where

552
00:38:24,220 --> 00:38:26,150
we talk about phase transitions.

553
00:38:26,150 --> 00:38:28,820
But the statement
is that the shape

554
00:38:28,820 --> 00:38:33,000
of the function in the
vicinity of something like this

555
00:38:33,000 --> 00:38:37,430
is delta P is
proportional to delta V

556
00:38:37,430 --> 00:38:42,420
to some other gamma that is
neither 3 nor 5 nor an integer.

557
00:38:42,420 --> 00:38:46,590
It is some number
that experimentally

558
00:38:46,590 --> 00:38:51,660
has been determined to be of
the order of, I don't know, 4.7.

559
00:38:51,660 --> 00:38:56,430
And why it is 4.7 you have to
do a lot of interesting field

560
00:38:56,430 --> 00:38:57,770
theory to understand.

561
00:38:57,770 --> 00:39:01,424
And hopefully, you'll come next
semester and I'll explain it.

562
00:39:01,424 --> 00:39:06,680
AUDIENCE: So your point is
that gamma is not [INAUDIBLE].

563
00:39:06,680 --> 00:39:11,460
PROFESSOR: Gamma is not-- so
the statement is, in order

564
00:39:11,460 --> 00:39:18,470
for me to write this
curve as something delta V

565
00:39:18,470 --> 00:39:21,060
plus some other thing--
delta V Squared--

566
00:39:21,060 --> 00:39:25,320
plus some other
thing-- delta V cubed--

567
00:39:25,320 --> 00:39:29,610
which we do all the time,
there is an inherent assumption

568
00:39:29,610 --> 00:39:34,090
that an analytical expansion
is possible around this point.

569
00:39:34,090 --> 00:39:40,255
Whereas if I gave you a
function y that is x to the 5/3,

570
00:39:40,255 --> 00:39:44,126
around x of 0, you cannot
write a Taylor series for it.

571
00:39:44,126 --> 00:39:51,541

572
00:39:51,541 --> 00:39:52,040
OK?

573
00:39:52,040 --> 00:40:05,900

574
00:40:05,900 --> 00:40:10,470
Essentially, this
stability condition

575
00:40:10,470 --> 00:40:15,820
tells you a lot of things about
signs of response functions.

576
00:40:15,820 --> 00:40:18,590
In the case of
mechanical cases, it

577
00:40:18,590 --> 00:40:21,160
is really that if
you have a spring,

578
00:40:21,160 --> 00:40:27,860
you better have a force
constant that is positive,

579
00:40:27,860 --> 00:40:31,630
so that when you pull on it,
the change in displacement

580
00:40:31,630 --> 00:40:33,840
has the right sign.

581
00:40:33,840 --> 00:40:36,050
It wants to be contracted.

582
00:40:36,050 --> 00:40:39,050
And similarly here,
you are essentially

583
00:40:39,050 --> 00:40:40,820
compressing the gas.

584
00:40:40,820 --> 00:40:46,890
There's a thermal analogue of
that sine which is worth making

585
00:40:46,890 --> 00:40:48,690
sure we know.

586
00:40:48,690 --> 00:40:51,500
Let's look at the
case where we have

587
00:40:51,500 --> 00:40:52,895
a fixed number of particles.

588
00:40:52,895 --> 00:40:56,620

589
00:40:56,620 --> 00:41:02,480
And then what we would
have is that our dE

590
00:41:02,480 --> 00:41:07,940
is T dS plus Ji dxi.

591
00:41:07,940 --> 00:41:11,250

592
00:41:11,250 --> 00:41:17,100
I am free to choose
any set of variables.

593
00:41:17,100 --> 00:41:20,760
The interesting set of
variables-- well, OK.

594
00:41:20,760 --> 00:41:29,550
So then stability implies
that delta T delta

595
00:41:29,550 --> 00:41:36,500
S plus delta Ji delta xi with
the sum over i is positive.

596
00:41:36,500 --> 00:41:39,390

597
00:41:39,390 --> 00:41:41,640
Now, I can take a look
at this expression

598
00:41:41,640 --> 00:41:46,120
and choose any set of variables
to express these changes

599
00:41:46,120 --> 00:41:47,176
in terms of.

600
00:41:47,176 --> 00:41:50,960

601
00:41:50,960 --> 00:41:58,300
So I'm going to choose for my
variables temperature and xi.

602
00:41:58,300 --> 00:42:01,750
You see, the way that
I've written it here,

603
00:42:01,750 --> 00:42:06,810
S, T, J, x appear
completely equivalently.

604
00:42:06,810 --> 00:42:08,720
I could have chosen
to express things

605
00:42:08,720 --> 00:42:15,900
in terms of S and
J, T and J, S and x.

606
00:42:15,900 --> 00:42:18,215
But I choose to use T and x.

607
00:42:18,215 --> 00:42:20,890

608
00:42:20,890 --> 00:42:22,440
All right, so what
does that mean?

609
00:42:22,440 --> 00:42:28,258
It means that I write my
delta S in terms of dS

610
00:42:28,258 --> 00:42:38,180
by dT at constant x dT,
plus dS by dxi at constant T

611
00:42:38,180 --> 00:42:42,910
and other xj not
equal to i, delta xi.

612
00:42:42,910 --> 00:42:46,160

613
00:42:46,160 --> 00:42:51,960
I also need an expression
for delta Ji here.

614
00:42:51,960 --> 00:42:58,620
so my delta Ji
would be dJi by dT

615
00:42:58,620 --> 00:43:05,810
at constant x, delta T,
plus dJi with respect

616
00:43:05,810 --> 00:43:12,100
to xj at constant T, an
appropriate set of J's,

617
00:43:12,100 --> 00:43:18,450
delta xi-- delta xj.

618
00:43:18,450 --> 00:43:24,510

619
00:43:24,510 --> 00:43:26,000
Sum over J implicit here.

620
00:43:26,000 --> 00:43:29,580

621
00:43:29,580 --> 00:43:34,320
I substitute these--
delta S and delta J--

622
00:43:34,320 --> 00:43:38,420
in the general form of
the stability condition

623
00:43:38,420 --> 00:43:43,440
to get constraints
applicable to combinations

624
00:43:43,440 --> 00:43:46,810
involving delta T and delta x.

625
00:43:46,810 --> 00:43:52,786
So I have to multiply
delta S with delta T.

626
00:43:52,786 --> 00:44:00,480
I have to multiply delta
Ji with delta xi here.

627
00:44:00,480 --> 00:44:00,980
Yes?

628
00:44:00,980 --> 00:44:04,330
AUDIENCE: So here,
J is not equal to i.

629
00:44:04,330 --> 00:44:07,940
And then which one is J?

630
00:44:07,940 --> 00:44:09,640
[INAUDIBLE]

631
00:44:09,640 --> 00:44:10,350
PROFESSOR: OK.

632
00:44:10,350 --> 00:44:18,920
So let's say that I have S
that is a function of-- I

633
00:44:18,920 --> 00:44:22,790
have x1 and x2.

634
00:44:22,790 --> 00:44:25,810
So here, I would
have derivatives

635
00:44:25,810 --> 00:44:29,970
dS by dx1 times delta x1.

636
00:44:29,970 --> 00:44:35,160
I would have derivatives
dS by dx2 times delta x2.

637
00:44:35,160 --> 00:44:37,960
In both cases, T is kept fixed.

638
00:44:37,960 --> 00:44:40,470

639
00:44:40,470 --> 00:44:44,230
In this case, additionally,
x2 is kept fixed.

640
00:44:44,230 --> 00:44:47,780
In this case, additionally,
x1 is kept fixed.

641
00:44:47,780 --> 00:44:48,440
That's all.

642
00:44:48,440 --> 00:44:54,490

643
00:44:54,490 --> 00:44:56,600
OK?

644
00:44:56,600 --> 00:44:58,570
So let's do that multiplication.

645
00:44:58,570 --> 00:45:05,450
We multiply a delta T here and
we get dS by dT at constant x,

646
00:45:05,450 --> 00:45:06,475
delta T squared.

647
00:45:06,475 --> 00:45:09,350

648
00:45:09,350 --> 00:45:22,900
We get here a term which is
delta T delta xi, multiplying

649
00:45:22,900 --> 00:45:27,980
dS by dxi at constant
T, and the others

650
00:45:27,980 --> 00:45:31,320
that I don't bother to write.

651
00:45:31,320 --> 00:45:35,310
But I note that I will
get another term that

652
00:45:35,310 --> 00:45:44,160
is delta xi delta t from the
second term, whose coefficient

653
00:45:44,160 --> 00:45:51,080
is dJi by dT at constant x.

654
00:45:51,080 --> 00:45:53,100
And I group those
two terms together.

655
00:45:53,100 --> 00:45:56,670

656
00:45:56,670 --> 00:46:01,610
And finally, we have
the last term here,

657
00:46:01,610 --> 00:46:17,450
which is delta xi delta
xj, dJi by dxj, constant T.

658
00:46:17,450 --> 00:46:23,040
And the constraint for all
of these stability conditions

659
00:46:23,040 --> 00:46:25,460
that I have written,
I have to be

660
00:46:25,460 --> 00:46:27,730
in the minimum of a potential.

661
00:46:27,730 --> 00:46:31,180
Any deviation that
I make better lead

662
00:46:31,180 --> 00:46:35,780
to an increase in this
combination, which

663
00:46:35,780 --> 00:46:39,000
is kind of equivalent
to being at the minimum

664
00:46:39,000 --> 00:46:41,370
of some kind of
potential energy.

665
00:46:41,370 --> 00:46:43,150
If you do some
manipulations, this

666
00:46:43,150 --> 00:46:45,990
is also equivalent to
being at the maximum

667
00:46:45,990 --> 00:46:47,490
of some entropy function.

668
00:46:47,490 --> 00:46:52,390

669
00:46:52,390 --> 00:47:00,950
Now, let's look at that
manipulation a little bit.

670
00:47:00,950 --> 00:47:03,800
Now, I happen to know
that this form will

671
00:47:03,800 --> 00:47:05,970
simplify a little bit.

672
00:47:05,970 --> 00:47:09,520
And that's why I chose this
combination of variables.

673
00:47:09,520 --> 00:47:13,240
And the simplification
that will happen

674
00:47:13,240 --> 00:47:17,390
is that this entity is 0.

675
00:47:17,390 --> 00:47:22,940
So let me show you
why that entity is 0.

676
00:47:22,940 --> 00:47:28,670
And that relies on one of
these Maxwell relationships

677
00:47:28,670 --> 00:47:30,160
that I had.

678
00:47:30,160 --> 00:47:33,170
So let's find the
Maxwell relation that

679
00:47:33,170 --> 00:47:39,710
is applicable to dS by
dxi at constant T, which

680
00:47:39,710 --> 00:47:40,890
is what I have over here.

681
00:47:40,890 --> 00:47:43,430

682
00:47:43,430 --> 00:47:53,580
OK, let's again start with
dE being T dS plus ji dxi.

683
00:47:53,580 --> 00:47:55,220
I'm at fixed number
of particles.

684
00:47:55,220 --> 00:47:57,640
I don't need to write
the other term-- UDN.

685
00:47:57,640 --> 00:48:00,410

686
00:48:00,410 --> 00:48:04,270
Unfortunately here, S is
not the first derivative.

687
00:48:04,270 --> 00:48:09,150
So what I will do is I
will look at E minus TS.

688
00:48:09,150 --> 00:48:12,170
That's going to turn
this into minus S dT.

689
00:48:12,170 --> 00:48:14,890

690
00:48:14,890 --> 00:48:15,860
Good.

691
00:48:15,860 --> 00:48:20,370
Ji dxi is exactly what I want
because this will tell me

692
00:48:20,370 --> 00:48:25,610
something about the
derivative in this fashion.

693
00:48:25,610 --> 00:48:36,230
dS by dxi at constant
T is equal to what?

694
00:48:36,230 --> 00:48:41,880
It's taking the things
in this fashion,

695
00:48:41,880 --> 00:48:48,700
which is dJi by
dT at constant x.

696
00:48:48,700 --> 00:48:51,581

697
00:48:51,581 --> 00:48:52,080
Good.

698
00:48:52,080 --> 00:48:57,710
So I can see that the sum
total of these two terms is 0.

699
00:48:57,710 --> 00:49:00,880
And that's exactly
what I have here.

700
00:49:00,880 --> 00:49:03,670
So this is 0 by
Maxwell relation.

701
00:49:03,670 --> 00:49:35,090

702
00:49:35,090 --> 00:49:35,590
OK?

703
00:49:35,590 --> 00:49:36,810
OK

704
00:49:36,810 --> 00:49:42,310
So having set that to
0, I have two terms.

705
00:49:42,310 --> 00:49:49,360
One of them is this entity
that goes with delta T squared.

706
00:49:49,360 --> 00:49:55,710
And the other is this entity--
sum over i and J, delta xi,

707
00:49:55,710 --> 00:50:03,650
delta xj, dJi, dxj,
constant T. And this--

708
00:50:03,650 --> 00:50:11,210
for any choice of delta x's
and delta T's-- for any choice,

709
00:50:11,210 --> 00:50:12,850
it better be positive.

710
00:50:12,850 --> 00:50:15,470

711
00:50:15,470 --> 00:50:18,840
So in particular,
given this nice form

712
00:50:18,840 --> 00:50:21,570
that I have over
here, I can certainly

713
00:50:21,570 --> 00:50:26,710
choose all of the delta x's
to be 0, because I do things

714
00:50:26,710 --> 00:50:31,710
at constant displacement.

715
00:50:31,710 --> 00:50:35,730
And then what I have
is that the whole thing

716
00:50:35,730 --> 00:50:41,290
is proportional to delta T
squared and some coefficient.

717
00:50:41,290 --> 00:50:43,958
So that coefficient
better be positive.

718
00:50:43,958 --> 00:50:49,220

719
00:50:49,220 --> 00:50:53,500
So entropy should always
be an increasing function

720
00:50:53,500 --> 00:50:58,450
of temperature at
constant displacement.

721
00:50:58,450 --> 00:51:03,380
And the consequence of that is
that heat capacities measure

722
00:51:03,380 --> 00:51:06,290
at constant displacement.

723
00:51:06,290 --> 00:51:14,768
So this is the
heat capacity-- Cx.

724
00:51:14,768 --> 00:51:20,030
The general definition is
you put some amount of heat

725
00:51:20,030 --> 00:51:24,480
into the system at
constant displacement.

726
00:51:24,480 --> 00:51:27,680
So I have to specify
the procedure

727
00:51:27,680 --> 00:51:30,270
by which heat is
applied to the system

728
00:51:30,270 --> 00:51:36,710
and see how much this
temperature changes.

729
00:51:36,710 --> 00:51:39,870
I can in principle do this
sufficiently stability

730
00:51:39,870 --> 00:51:43,210
so I can measure precisely
the changes in temperature

731
00:51:43,210 --> 00:51:44,910
infinitesimally.

732
00:51:44,910 --> 00:51:47,670
And then I can use
the relationship

733
00:51:47,670 --> 00:51:54,310
between dQ and reversible
heat-- so dS at constant x dT.

734
00:51:54,310 --> 00:51:56,830

735
00:51:56,830 --> 00:52:05,500
And dS by dT at constant
x is precisely the entity

736
00:52:05,500 --> 00:52:12,970
that we have looked
at-- to be positive.

737
00:52:12,970 --> 00:52:17,170

738
00:52:17,170 --> 00:52:26,300
So in the same way that a spring
is stable-- if you pull on it,

739
00:52:26,300 --> 00:52:29,490
its extension should
increase a little bit.

740
00:52:29,490 --> 00:52:32,980
And so the coefficient
that relates

741
00:52:32,980 --> 00:52:35,680
the change in
displacement corresponding

742
00:52:35,680 --> 00:52:39,810
to the change in force has
to be positive for stability.

743
00:52:39,810 --> 00:52:44,840
Heat capacities must have
this positivity in the sense

744
00:52:44,840 --> 00:52:48,210
that if you add
heat to a system,

745
00:52:48,210 --> 00:52:52,310
its temperature should go up.

746
00:52:52,310 --> 00:52:56,710
It it went the other way, it
would actually also violate

747
00:52:56,710 --> 00:52:59,450
the second law of
thermodynamics.

748
00:52:59,450 --> 00:52:59,950
Yes?

749
00:52:59,950 --> 00:53:02,360
AUDIENCE: I'm a bit confused
about the J notation

750
00:53:02,360 --> 00:53:04,288
that we used.

751
00:53:04,288 --> 00:53:06,216
The initial equation
that we start

752
00:53:06,216 --> 00:53:13,990
has only-- it deals with i,
but where does J come from?

753
00:53:13,990 --> 00:53:16,860
PROFESSOR: This equation.

754
00:53:16,860 --> 00:53:17,510
Which equation?

755
00:53:17,510 --> 00:53:20,450
Up here?

756
00:53:20,450 --> 00:53:24,290
AUDIENCE: So index J is
just a dummy variable?

757
00:53:24,290 --> 00:53:27,170

758
00:53:27,170 --> 00:53:33,130
[INAUDIBLE] that dE
equals T dS equals Ji dSi.

759
00:53:33,130 --> 00:53:35,450
PROFESSOR: OK.

760
00:53:35,450 --> 00:53:35,950
Fine.

761
00:53:35,950 --> 00:53:40,050
So let's start from here.

762
00:53:40,050 --> 00:53:45,790
And part of the confusion is
that I'm not really consistent.

763
00:53:45,790 --> 00:53:50,700
Whenever I write an
index that is repeated,

764
00:53:50,700 --> 00:53:56,790
there is a so-called
summation convention

765
00:53:56,790 --> 00:53:59,760
that you may have seen
in quantum mechanics

766
00:53:59,760 --> 00:54:04,390
and other courses which
says that the index that

767
00:54:04,390 --> 00:54:07,230
is repeated twice
is summed over.

768
00:54:07,230 --> 00:54:12,600
So when I write ai bi, using
the summation convention, what

769
00:54:12,600 --> 00:54:19,565
I mean is, sum over all
possible values of i, ai, bi.

770
00:54:19,565 --> 00:54:22,490
So what I meant
here is that there

771
00:54:22,490 --> 00:54:26,250
may be multiple ways of
doing work on the system.

772
00:54:26,250 --> 00:54:28,600
So there could be pressure,
there could be spring,

773
00:54:28,600 --> 00:54:30,590
there could be magnetic work.

774
00:54:30,590 --> 00:54:35,070
And if I do mechanical
work on the system,

775
00:54:35,070 --> 00:54:38,910
the change in energy is the
sum total of all different ways

776
00:54:38,910 --> 00:54:41,830
to do mechanical
works, plus the heat

777
00:54:41,830 --> 00:54:43,750
that is supplied to the system.

778
00:54:43,750 --> 00:54:47,030
So that's statement number one.

779
00:54:47,030 --> 00:54:50,210
Now, I did something
else here, which

780
00:54:50,210 --> 00:54:54,050
is that when I take a derivative
of a function that depends

781
00:54:54,050 --> 00:54:59,630
on multiple variables-- so
f depends on x1, x2, x3,

782
00:54:59,630 --> 00:55:10,920
xn-- then if I look
at the change in this,

783
00:55:10,920 --> 00:55:14,050
any one of these
variables can change

784
00:55:14,050 --> 00:55:17,380
by an amount that is dxi.

785
00:55:17,380 --> 00:55:19,640
And if the second
variable changes,

786
00:55:19,640 --> 00:55:21,830
there is a corresponding
contribution,

787
00:55:21,830 --> 00:55:26,020
which is df with respect
to the second variable.

788
00:55:26,020 --> 00:55:30,050
And I have to add all of
these things together.

789
00:55:30,050 --> 00:55:35,410
So if I'm again
inconsistent, this

790
00:55:35,410 --> 00:55:39,200
is really what I
mean, that when you

791
00:55:39,200 --> 00:55:43,180
do a change in a function
of multiple variables,

792
00:55:43,180 --> 00:55:47,410
you would write it
as dxi, df by dxi.

793
00:55:47,410 --> 00:55:50,120
If you use the
summation convention,

794
00:55:50,120 --> 00:55:54,180
I have to really sum
over all possibilities.

795
00:55:54,180 --> 00:55:59,650
Now, suppose I took one of
these things, like df by dx3.

796
00:55:59,650 --> 00:56:02,340

797
00:56:02,340 --> 00:56:05,290
So it's one of the
derivatives here.

798
00:56:05,290 --> 00:56:08,910
And I took another derivative,
or I looked at the change

799
00:56:08,910 --> 00:56:10,670
in this function.

800
00:56:10,670 --> 00:56:15,470
This function depends also
on variables x1 through xn.

801
00:56:15,470 --> 00:56:22,700
So I have to look at the sum
over all changes with respect

802
00:56:22,700 --> 00:56:28,350
to xj of df by dx3 delta xj.

803
00:56:28,350 --> 00:56:30,960

804
00:56:30,960 --> 00:56:36,230
So you can see that
automatically, I

805
00:56:36,230 --> 00:56:38,620
have kind of assumed
some of these things

806
00:56:38,620 --> 00:56:42,320
as I write things rapidly.

807
00:56:42,320 --> 00:56:49,200
And if it is not clear, you have
to spend a little bit of time

808
00:56:49,200 --> 00:56:53,850
writing things more in
detail, because it's

809
00:56:53,850 --> 00:56:56,550
after all kind of
simple algebra.

810
00:56:56,550 --> 00:56:58,507
Yes, did I make another mistake?

811
00:56:58,507 --> 00:56:59,006
Yes?

812
00:56:59,006 --> 00:57:00,880
AUDIENCE: Oh, not
a mistake, just--

813
00:57:00,880 --> 00:57:01,587
PROFESSOR: Yes?

814
00:57:01,587 --> 00:57:03,670
AUDIENCE: When you were
talking about mechanicals,

815
00:57:03,670 --> 00:57:05,005
you had a non-linear spring.

816
00:57:05,005 --> 00:57:06,954
You start pulling on it.

817
00:57:06,954 --> 00:57:09,120
At some point, it
just increases.

818
00:57:09,120 --> 00:57:12,188
PROFESSOR: If it is a non-linear
spring, yes, that's right.

819
00:57:12,188 --> 00:57:15,082
AUDIENCE: Can you provide
example of thermodynamic system

820
00:57:15,082 --> 00:57:21,262
which, upon some
threshold action upon it,

821
00:57:21,262 --> 00:57:21,970
becomes unstable?

822
00:57:21,970 --> 00:57:38,910

823
00:57:38,910 --> 00:57:41,550
PROFESSOR: OK.

824
00:57:41,550 --> 00:57:44,540
I think I more or
less had that example.

825
00:57:44,540 --> 00:57:47,800
Let's try to develop it
a little bit further.

826
00:57:47,800 --> 00:57:54,665
So I said that typically, if
I look at the pressure volume

827
00:57:54,665 --> 00:57:58,705
isotherms of a gas, it
looks something like this.

828
00:57:58,705 --> 00:58:01,480

829
00:58:01,480 --> 00:58:04,870
And I told you that
there is actually

830
00:58:04,870 --> 00:58:11,140
some kind of a critical isotherm
that looks something like this.

831
00:58:11,140 --> 00:58:12,950
And you may ask,
well, what do things

832
00:58:12,950 --> 00:58:16,900
look below that for
lower temperatures?

833
00:58:16,900 --> 00:58:19,180
And again, you
probably know this,

834
00:58:19,180 --> 00:58:21,310
is that this is the
temperature below which

835
00:58:21,310 --> 00:58:27,600
you have a transition to
liquid gas coexistence.

836
00:58:27,600 --> 00:58:31,090
And so the isotherms
would look like that.

837
00:58:31,090 --> 00:58:35,850
Now, it turns out that while
these are true equilibrium

838
00:58:35,850 --> 00:58:39,830
things-- you would have
to wait sufficiently long

839
00:58:39,830 --> 00:58:44,140
so that the system
explores these isotherms--

840
00:58:44,140 --> 00:58:47,590
it is also possible
in certain systems

841
00:58:47,590 --> 00:58:50,340
to do things a little
bit more rapidly.

842
00:58:50,340 --> 00:58:53,810
And then when you
go below, you will

843
00:58:53,810 --> 00:58:59,010
see a kind of remnant of that
isotherm which kind of looks

844
00:58:59,010 --> 00:58:59,620
like this.

845
00:58:59,620 --> 00:59:02,700

846
00:59:02,700 --> 00:59:09,450
So that isotherm would
have portions of it

847
00:59:09,450 --> 00:59:17,600
that don't satisfy these
conditions of stability.

848
00:59:17,600 --> 00:59:21,170
And the reason that you still
get some remnant of it--

849
00:59:21,170 --> 00:59:22,430
not quite the entire thing.

850
00:59:22,430 --> 00:59:24,150
You get actually
some portion here,

851
00:59:24,150 --> 00:59:28,590
some portion here-- is because
you do things very rapidly

852
00:59:28,590 --> 00:59:31,110
and the system
does not have time

853
00:59:31,110 --> 00:59:33,910
to explore the
entire equilibrium.

854
00:59:33,910 --> 00:59:37,910
So that's one example
that I can think

855
00:59:37,910 --> 00:59:40,920
of where some remnants
of something that

856
00:59:40,920 --> 00:59:47,060
is like that non-linear
noodle takes place.

857
00:59:47,060 --> 00:59:49,665
But even the non-linear
noodle, you cannot really see

858
00:59:49,665 --> 00:59:50,570
in equilibrium.

859
00:59:50,570 --> 00:59:53,140
You can only see it
when you're pulling it.

860
00:59:53,140 --> 00:59:55,710
And you can see that
it is being extended.

861
00:59:55,710 --> 00:59:59,430
There is no equilibrium that
you can see that has that.

862
00:59:59,430 --> 01:00:04,070
So if I want to find an analogue
of that non-linear noodle

863
01:00:04,070 --> 01:00:09,250
showing instability, I'd better
find some near-equilibrium

864
01:00:09,250 --> 01:00:10,145
condition.

865
01:00:10,145 --> 01:00:12,770
And this is the near-equilibrium
condition that I can think of.

866
01:00:12,770 --> 01:00:19,620

867
01:00:19,620 --> 01:00:22,920
There are a number of cases.

868
01:00:22,920 --> 01:00:29,010
And essentially, you've
probably heard hysteresis

869
01:00:29,010 --> 01:00:32,230
in magnets and things like that.

870
01:00:32,230 --> 01:00:35,250
And by definition,
hysteresis means

871
01:00:35,250 --> 01:00:38,970
that for the same
set of conditions,

872
01:00:38,970 --> 01:00:42,630
you see multiple
different states.

873
01:00:42,630 --> 01:00:46,200
So not both of them can
be or multiple of them

874
01:00:46,200 --> 01:00:48,020
can be in equilibrium.

875
01:00:48,020 --> 01:00:50,270
And over some
sufficient time scale,

876
01:00:50,270 --> 01:00:53,520
you would go from one
behavior to another behavior.

877
01:00:53,520 --> 01:00:58,310
But if you explore the vicinity
of the curves that correspond

878
01:00:58,310 --> 01:01:01,460
to these hysteretic
behavior, you

879
01:01:01,460 --> 01:01:05,160
would see the signatures
of these kinds

880
01:01:05,160 --> 01:01:07,120
of thermodynamic instabilities.

881
01:01:07,120 --> 01:01:22,790

882
01:01:22,790 --> 01:01:25,530
There was one other part
of this that maybe I

883
01:01:25,530 --> 01:01:29,730
will write a little bit
in more detail, given

884
01:01:29,730 --> 01:01:30,750
the questions that were.

885
01:01:30,750 --> 01:01:40,260
Asked It is here I looked
at this general condition

886
01:01:40,260 --> 01:01:43,190
for the case where
the delta x's were 0.

887
01:01:43,190 --> 01:01:45,810

888
01:01:45,810 --> 01:01:48,180
Let's look at the same
thing for the case

889
01:01:48,180 --> 01:01:52,490
where delta T equals
to 0, because then I

890
01:01:52,490 --> 01:02:02,470
have the condition that
the Ji by the xj, delta xi,

891
01:02:02,470 --> 01:02:05,240
delta xj should be positive.

892
01:02:05,240 --> 01:02:08,150

893
01:02:08,150 --> 01:02:12,170
And this was done at constant T.

894
01:02:12,170 --> 01:02:20,380
And to be more explicit, there
is a sum over i and J in here.

895
01:02:20,380 --> 01:02:24,190

896
01:02:24,190 --> 01:02:29,210
So if I'm doing things
at constant temperature,

897
01:02:29,210 --> 01:02:35,780
I really have the analogue
of the instability condition.

898
01:02:35,780 --> 01:02:42,370
What I have here, it is
the same thing as this one.

899
01:02:42,370 --> 01:02:48,780
And if you think about this
in a little bit more detail--

900
01:02:48,780 --> 01:02:52,250
let's say I have
something like a gas.

901
01:02:52,250 --> 01:02:55,190

902
01:02:55,190 --> 01:02:58,730
We have two ways of
doing work on the system.

903
01:02:58,730 --> 01:03:01,460
I have dJ1 by dx1.

904
01:03:01,460 --> 01:03:04,245
I have dJ2 by dx2.

905
01:03:04,245 --> 01:03:07,610
I have dJ1 by dx2.

906
01:03:07,610 --> 01:03:10,820
I have dj2 by dx1.

907
01:03:10,820 --> 01:03:14,680
Essentially, this object,
there are four of them.

908
01:03:14,680 --> 01:03:20,830
And those four I can put
into something like this.

909
01:03:20,830 --> 01:03:24,210
And the statement
that I have written

910
01:03:24,210 --> 01:03:27,900
is that this 2 by
2 matrix, if it

911
01:03:27,900 --> 01:03:33,390
acts on the displacement delta
x1 delta x2 on the right,

912
01:03:33,390 --> 01:03:38,640
and then gets contracted on the
left by the same vector delta

913
01:03:38,640 --> 01:03:42,990
x1, delta x2, for any
choice of the vector delta

914
01:03:42,990 --> 01:03:48,030
x1 delta x2, this
better be positive.

915
01:03:48,030 --> 01:03:52,000
This is for two, but in
general, this expression

916
01:03:52,000 --> 01:03:54,960
is going to be valid
for any number.

917
01:03:54,960 --> 01:03:58,640

918
01:03:58,640 --> 01:04:06,190
Now, what are the constraints
that a matrix must satisfy

919
01:04:06,190 --> 01:04:10,490
so that irrespective of the
choice of this displacement,

920
01:04:10,490 --> 01:04:13,070
you will get a positive result?

921
01:04:13,070 --> 01:04:16,295
Mathematically, that's caused
a positive definite matrix.

922
01:04:16,295 --> 01:04:22,800

923
01:04:22,800 --> 01:04:27,260
So one of the conditions that
is certainly immediately obvious

924
01:04:27,260 --> 01:04:32,470
is that if I choose the case
where delta x2 is 0, then

925
01:04:32,470 --> 01:04:34,710
the diagonal term
that corresponds

926
01:04:34,710 --> 01:04:38,020
to that entity
better be positive.

927
01:04:38,020 --> 01:04:41,780
So a matrix is positive
definite if for every one

928
01:04:41,780 --> 01:04:45,120
of the diagonal elements,
I have this condition.

929
01:04:45,120 --> 01:04:49,490

930
01:04:49,490 --> 01:04:53,140
So this is actually something
that we already saw.

931
01:04:53,140 --> 01:04:55,110
In the case of the
gas, this would

932
01:04:55,110 --> 01:04:58,970
be something like minus dP by
dV at constant temperature--

933
01:04:58,970 --> 01:04:59,980
be positive.

934
01:04:59,980 --> 01:05:03,450
This is the nature of
these isotherms always

935
01:05:03,450 --> 01:05:07,395
going one way that
I was drawing.

936
01:05:07,395 --> 01:05:08,390
AUDIENCE: Question.

937
01:05:08,390 --> 01:05:09,360
PROFESSOR: Yes.

938
01:05:09,360 --> 01:05:12,720
AUDIENCE: You're not summing
over alpha now, are you?

939
01:05:12,720 --> 01:05:13,680
Or are you--

940
01:05:13,680 --> 01:05:14,970
PROFESSOR: Yes, I'm not.

941
01:05:14,970 --> 01:05:17,000
And I don't know how
to write. [LAUGHS]

942
01:05:17,000 --> 01:05:20,660
Let's call it delta J1, x1.

943
01:05:20,660 --> 01:05:24,130

944
01:05:24,130 --> 01:05:29,720
So all diagonal elements
must be positive,

945
01:05:29,720 --> 01:05:31,920
but that's not enough.

946
01:05:31,920 --> 01:05:36,640
So I can write a matrix that
has only positive elements

947
01:05:36,640 --> 01:05:40,920
along the diagonal, but
off-diagonal elements such

948
01:05:40,920 --> 01:05:43,610
that the combination
will give you

949
01:05:43,610 --> 01:05:49,210
negative values of
this culmination

950
01:05:49,210 --> 01:05:53,070
for appropriate choices
of delta x1 and delta x2.

951
01:05:53,070 --> 01:05:55,310
The thing that
you really need is

952
01:05:55,310 --> 01:06:03,226
that all eigenvalues
must be positive.

953
01:06:03,226 --> 01:06:08,090

954
01:06:08,090 --> 01:06:12,720
It is not that easy to
write down mathematically

955
01:06:12,720 --> 01:06:18,000
in general what that means
in terms of multiple response

956
01:06:18,000 --> 01:06:19,760
functions.

957
01:06:19,760 --> 01:06:23,620
That's why this
form of writing is

958
01:06:23,620 --> 01:06:25,510
much more compact and effective.

959
01:06:25,510 --> 01:06:29,490
It basically says, no matter
what combination of elements

960
01:06:29,490 --> 01:06:32,780
you choose, you are at the
bottom of the potential.

961
01:06:32,780 --> 01:06:35,310
You should not be able
to go away from it.

962
01:06:35,310 --> 01:06:38,125

963
01:06:38,125 --> 01:06:39,616
AUDIENCE: So the
second statement

964
01:06:39,616 --> 01:06:42,610
is necessary and
sufficient condition?

965
01:06:42,610 --> 01:06:47,580
PROFESSOR: This statement is
necessary but not sufficient.

966
01:06:47,580 --> 01:06:50,810
This statement is
equivalent to the meaning

967
01:06:50,810 --> 01:06:53,230
of what is a positive
definite matrix.

968
01:06:53,230 --> 01:06:54,320
It is sufficient.

969
01:06:54,320 --> 01:06:55,510
Yes?

970
01:06:55,510 --> 01:06:58,080
AUDIENCE: Is there any
physical interpretation

971
01:06:58,080 --> 01:07:00,370
to diagonalizing that matrix?

972
01:07:00,370 --> 01:07:04,750

973
01:07:04,750 --> 01:07:08,010
PROFESSOR: It depends on
the kind of the formations

974
01:07:08,010 --> 01:07:09,210
that you make.

975
01:07:09,210 --> 01:07:12,880
Like if I was really
thinking about a gas

976
01:07:12,880 --> 01:07:16,970
where I allow some exchange
with the surface or something

977
01:07:16,970 --> 01:07:21,310
else so that delta
N is not equal to 0,

978
01:07:21,310 --> 01:07:24,590
the general form of this
matrix in the case where

979
01:07:24,590 --> 01:07:27,280
I have pressure volume would
be something like this.

980
01:07:27,280 --> 01:07:29,860
I would have minus dP by dV.

981
01:07:29,860 --> 01:07:32,740
I would have d mu by dN.

982
01:07:32,740 --> 01:07:36,700
I would have dP by dN.

983
01:07:36,700 --> 01:07:40,360
I would have d mu by dV.

984
01:07:40,360 --> 01:07:45,500
And I would have to make sure
that this matrix is positive

985
01:07:45,500 --> 01:07:49,670
definite, so I know certainly
that the compressibility

986
01:07:49,670 --> 01:07:51,440
by itself has to be positive.

987
01:07:51,440 --> 01:07:54,720
d mu by dN also
has to be positive.

988
01:07:54,720 --> 01:07:57,820
But if I allow
changes both in volume

989
01:07:57,820 --> 01:08:00,590
and the number of
particles, then because

990
01:08:00,590 --> 01:08:04,720
of these off-diagonal terms,
I have additional constraints.

991
01:08:04,720 --> 01:08:07,080
So for appropriate
changes, it is

992
01:08:07,080 --> 01:08:10,180
sufficient to look at
the diagonal terms.

993
01:08:10,180 --> 01:08:14,786
But for the entire set of
possible transformations,

994
01:08:14,786 --> 01:08:17,800
it's not.

995
01:08:17,800 --> 01:08:19,572
Yes?

996
01:08:19,572 --> 01:08:22,893
AUDIENCE: If you look at
different ways to do the work

997
01:08:22,893 --> 01:08:25,760
on the system-- so in
this case, change volume,

998
01:08:25,760 --> 01:08:31,870
change number of particles,
if you're saying forces have

999
01:08:31,870 --> 01:08:36,609
different dimensions, and
if diagonalization of these

1000
01:08:36,609 --> 01:08:41,010
metrics corresponds to inventing
some new ways of acting

1001
01:08:41,010 --> 01:08:45,890
on the system which are a linear
combination of old ways like

1002
01:08:45,890 --> 01:08:48,330
change volume a little
and change particles--

1003
01:08:48,330 --> 01:08:51,810
PROFESSOR: Change the
number of particles, yes.

1004
01:08:51,810 --> 01:08:56,020
AUDIENCE: Those are pretty
weird manipulations,

1005
01:08:56,020 --> 01:08:58,760
in terms of how to
physically explain

1006
01:08:58,760 --> 01:09:00,747
what is diagonalization matrix.

1007
01:09:00,747 --> 01:09:01,330
PROFESSOR: OK.

1008
01:09:01,330 --> 01:09:04,330

1009
01:09:04,330 --> 01:09:06,220
There is a question
of dimensions,

1010
01:09:06,220 --> 01:09:07,930
and there's a
question of amount.

1011
01:09:07,930 --> 01:09:09,630
So maybe what you
are worried about

1012
01:09:09,630 --> 01:09:16,140
is that if I make one them
volume and the other pressure,

1013
01:09:16,140 --> 01:09:18,399
one is extensive,
one is intensive,

1014
01:09:18,399 --> 01:09:20,090
and the response
functions would have

1015
01:09:20,090 --> 01:09:23,689
to carry something that
is proportional to size.

1016
01:09:23,689 --> 01:09:25,439
But then there are
other conditions,

1017
01:09:25,439 --> 01:09:28,710
let's say, where all of the
variables are intensive.

1018
01:09:28,710 --> 01:09:31,939
And then it's moving
around in the space that

1019
01:09:31,939 --> 01:09:36,060
is characterized, let's say, by
pressure and chemical potential

1020
01:09:36,060 --> 01:09:38,560
from one point to another point.

1021
01:09:38,560 --> 01:09:43,620
And there is some generalized
response that you would have,

1022
01:09:43,620 --> 01:09:46,100
depending on which
direction you go to.

1023
01:09:46,100 --> 01:09:48,810
So essentially,
I could certainly

1024
01:09:48,810 --> 01:09:52,640
characterize various
thermodynamic functions

1025
01:09:52,640 --> 01:09:55,050
in this space P mu.

1026
01:09:55,050 --> 01:10:00,720
And what I know is that I cannot
write any function in this

1027
01:10:00,720 --> 01:10:04,140
two-parameter space that
would be consistent with

1028
01:10:04,140 --> 01:10:07,460
the thermodynamic
principles of stability.

1029
01:10:07,460 --> 01:10:12,110
I have to have some
kind of convexity.

1030
01:10:12,110 --> 01:10:16,070
So certainly, what
typically-- let's say again,

1031
01:10:16,070 --> 01:10:21,070
going back to my favorite
example here-- what people do

1032
01:10:21,070 --> 01:10:25,020
and what we will
do later on is we

1033
01:10:25,020 --> 01:10:29,650
specify that you have
some particular pressure

1034
01:10:29,650 --> 01:10:33,370
and volume of an
interacting gas.

1035
01:10:33,370 --> 01:10:35,810
And there is no way
that we can calculate

1036
01:10:35,810 --> 01:10:41,080
the thermodynamic properties
of that system exactly.

1037
01:10:41,080 --> 01:10:43,640
We try to construct
a free energy

1038
01:10:43,640 --> 01:10:47,250
as a function of
these variables.

1039
01:10:47,250 --> 01:10:50,900
And that free energy
will be approximate based

1040
01:10:50,900 --> 01:10:52,630
on some ideas.

1041
01:10:52,630 --> 01:10:58,650
And then we will see that that
free energy that I construct

1042
01:10:58,650 --> 01:11:04,390
in some places will violate
the conditions of stability.

1043
01:11:04,390 --> 01:11:08,280
And then I know that that free
energy that I constructed,

1044
01:11:08,280 --> 01:11:12,930
let's say on the assumption
that the box that I have

1045
01:11:12,930 --> 01:11:16,520
is uniformly covered
by some material,

1046
01:11:16,520 --> 01:11:18,500
that assumption is violated.

1047
01:11:18,500 --> 01:11:21,105
And maybe what has happened
is that part of the volume

1048
01:11:21,105 --> 01:11:24,310
is a liquid and part
of the volume is gas.

1049
01:11:24,310 --> 01:11:28,770
So we will do precisely
that exercise later on.

1050
01:11:28,770 --> 01:11:31,080
We will choose a
pair of variables.

1051
01:11:31,080 --> 01:11:36,750
We will try to construct a
free energy function out.

1052
01:11:36,750 --> 01:11:40,590
We make the assumption that
the box is uniformly occupied

1053
01:11:40,590 --> 01:11:42,490
with the gas at some density.

1054
01:11:42,490 --> 01:11:46,230
Based on that, we estimate its
energy, entropy, et cetera.

1055
01:11:46,230 --> 01:11:49,280
We construct a free
energy function out.

1056
01:11:49,280 --> 01:11:54,810
And then we will see that
that energy function violates

1057
01:11:54,810 --> 01:11:58,760
the compressibility
condition at some point.

1058
01:11:58,760 --> 01:12:01,870
So we know that, oh,
we made a mistake.

1059
01:12:01,870 --> 01:12:04,914
The assumption that it is
uniformly occupying this box

1060
01:12:04,914 --> 01:12:05,414
is wrong.

1061
01:12:05,414 --> 01:12:12,530

1062
01:12:12,530 --> 01:12:15,730
All right, any other questions?

1063
01:12:15,730 --> 01:12:16,811
Yes?

1064
01:12:16,811 --> 01:12:18,644
AUDIENCE: One of the
things that confuses me

1065
01:12:18,644 --> 01:12:21,328
about thermodynamics
in general is

1066
01:12:21,328 --> 01:12:23,768
the limits of its applicability.

1067
01:12:23,768 --> 01:12:25,720
We think of it as
being very general,

1068
01:12:25,720 --> 01:12:27,184
applying to many
different systems.

1069
01:12:27,184 --> 01:12:30,577

1070
01:12:30,577 --> 01:12:33,116
For example, if we
had two bar magnets

1071
01:12:33,116 --> 01:12:35,170
and we had the north pole
close to the south pole

1072
01:12:35,170 --> 01:12:37,559
of the other one, and I
thought of the magnetic field

1073
01:12:37,559 --> 01:12:39,890
in between, is it
conceivable to think

1074
01:12:39,890 --> 01:12:41,850
of that as a
thermodynamic system?

1075
01:12:41,850 --> 01:12:44,560

1076
01:12:44,560 --> 01:12:47,300
PROFESSOR: You can
think of the magnet

1077
01:12:47,300 --> 01:12:51,440
as a thermodynamic system.

1078
01:12:51,440 --> 01:12:55,070
And you can certainly
look at two of them,

1079
01:12:55,070 --> 01:12:59,320
including the electromagnetic
field that surrounds them.

1080
01:12:59,320 --> 01:13:01,640
And there would be
some conditions,

1081
01:13:01,640 --> 01:13:04,800
like if you were to jiggle
these things with respect

1082
01:13:04,800 --> 01:13:08,370
to each other-- and you would
maintain the entire thing

1083
01:13:08,370 --> 01:13:12,630
in a box that prevented escape
of electromagnetic waves

1084
01:13:12,630 --> 01:13:15,540
or heat to the outside--
that eventually

1085
01:13:15,540 --> 01:13:19,430
this jiggling of the two magnets
will lead to their coming

1086
01:13:19,430 --> 01:13:23,610
to equilibrium but at
a higher temperature.

1087
01:13:23,610 --> 01:13:25,090
So this is what I'm saying.

1088
01:13:25,090 --> 01:13:30,260
So imagine that you have-- let's
for simplicity think of, again,

1089
01:13:30,260 --> 01:13:32,820
one big magnet that
you don't move,

1090
01:13:32,820 --> 01:13:40,220
and one small magnet here that
you externally jiggle around.

1091
01:13:40,220 --> 01:13:44,000
And then you remove
the source of work.

1092
01:13:44,000 --> 01:13:45,730
You wait for a while.

1093
01:13:45,730 --> 01:13:47,410
And then you find
that after a while,

1094
01:13:47,410 --> 01:13:50,470
this thing comes
to an equilibrium.

1095
01:13:50,470 --> 01:13:55,490
You ask, where did that
kinetic energy go to?

1096
01:13:55,490 --> 01:13:58,560
You had some kinetic energy
that you had set into this,

1097
01:13:58,560 --> 01:14:00,790
and the kinetic
energy disappeared.

1098
01:14:00,790 --> 01:14:03,470
And I assume that
there are conditions

1099
01:14:03,470 --> 01:14:09,240
that the kinetic energy could
not have escaped from this box.

1100
01:14:09,240 --> 01:14:11,120
So the only thing that
could have happened

1101
01:14:11,120 --> 01:14:14,305
is that that kinetic
energy heated up

1102
01:14:14,305 --> 01:14:18,440
the air of the box, the
magnet, or whatever you have.

1103
01:14:18,440 --> 01:14:23,370
So thermodynamics is applicable
in that sense to your system.

1104
01:14:23,370 --> 01:14:27,220
Now, if you ask me, is
it applicable to having

1105
01:14:27,220 --> 01:14:33,580
a single electron spin
that is polarized because

1106
01:14:33,580 --> 01:14:35,935
of the magnetic field
and some condition

1107
01:14:35,935 --> 01:14:40,950
that I set it to oscillate,
then it's not applicable,

1108
01:14:40,950 --> 01:14:44,470
because then I know some
quantum mechanical rules that

1109
01:14:44,470 --> 01:14:45,750
govern this.

1110
01:14:45,750 --> 01:14:47,570
There may be in
some circumstances

1111
01:14:47,570 --> 01:14:51,780
some conservation law that says
this with oscillate forever.

1112
01:14:51,780 --> 01:14:53,720
If it doesn't oscillate
forever, there's

1113
01:14:53,720 --> 01:14:55,690
presumably a
transition matrix that

1114
01:14:55,690 --> 01:14:59,620
says it will admit a particular
photon of a particular energy.

1115
01:14:59,620 --> 01:15:02,830
And I have to ask what
happened to that photon.

1116
01:15:02,830 --> 01:15:07,210
So what we will see later
on is that thermodynamics

1117
01:15:07,210 --> 01:15:12,730
is very powerful, but it's
not a fundamental theory

1118
01:15:12,730 --> 01:15:14,750
in this sense that
the quantum mechanics

1119
01:15:14,750 --> 01:15:16,990
that governs the spins.

1120
01:15:16,990 --> 01:15:20,300
And the reason that this works
for the case of the magnet

1121
01:15:20,300 --> 01:15:24,060
is because the magnet has
billions of spins in it--

1122
01:15:24,060 --> 01:15:27,100
billions and billions
of spins, or whatever.

1123
01:15:27,100 --> 01:15:32,870
So it is the of large
numbers that ultimately

1124
01:15:32,870 --> 01:15:38,550
allows you to translate
the rules of motion

1125
01:15:38,550 --> 01:15:41,590
into a well-defined
theory that is

1126
01:15:41,590 --> 01:15:42,980
consistent with thermodynamics.

1127
01:15:42,980 --> 01:15:45,556

1128
01:15:45,556 --> 01:15:47,681
AUDIENCE: I know that my
question seemed off-topic.

1129
01:15:47,681 --> 01:15:50,141
But the reason why I asked
about this particular system

1130
01:15:50,141 --> 01:15:53,040
is because I would
think that it doesn't

1131
01:15:53,040 --> 01:15:55,330
have a positive compressibility.

1132
01:15:55,330 --> 01:16:00,960
Like if you pull the two
magnets apart very slowly,

1133
01:16:00,960 --> 01:16:04,320
the force pulling them
together decreases

1134
01:16:04,320 --> 01:16:06,610
instead of increasing
like with the rubber band.

1135
01:16:06,610 --> 01:16:11,998

1136
01:16:11,998 --> 01:16:12,660
PROFESSOR: OK.

1137
01:16:12,660 --> 01:16:18,460
It may be that you have to
add additional agents in order

1138
01:16:18,460 --> 01:16:20,150
to make the whole system work.

1139
01:16:20,150 --> 01:16:24,390
So it may be that the
picture that I had in mind

1140
01:16:24,390 --> 01:16:26,330
was that you have
the magnet that

1141
01:16:26,330 --> 01:16:28,230
is being pushed up and down.

1142
01:16:28,230 --> 01:16:31,570
And I really do put
a number of springs

1143
01:16:31,570 --> 01:16:35,570
that make sure there exists
an equilibrium position.

1144
01:16:35,570 --> 01:16:38,800
If there is no mechanical
equilibrium position,

1145
01:16:38,800 --> 01:16:43,900
then the story is not whether
or not to thermodynamic applies.

1146
01:16:43,900 --> 01:16:47,590
It's a question of whether or
not there is an equilibrium.

1147
01:16:47,590 --> 01:16:52,400
So in order for me to
talk about thermodynamics

1148
01:16:52,400 --> 01:16:56,590
I need to know that there
is an eventual equilibrium.

1149
01:16:56,590 --> 01:17:00,520
and that's why I actually recast
the answer to your question,

1150
01:17:00,520 --> 01:17:03,560
eventually getting rid of
the kinetic energy of this

1151
01:17:03,560 --> 01:17:06,190
so that I have a system
that is in equilibrium.

1152
01:17:06,190 --> 01:17:09,790
If there are things that
are not in equilibrium,

1153
01:17:09,790 --> 01:17:10,930
all bets are off.

1154
01:17:10,930 --> 01:17:14,420
Thermodynamics is the science of
things that are in equilibrium.

1155
01:17:14,420 --> 01:17:22,689

1156
01:17:22,689 --> 01:17:23,355
Other questions?

1157
01:17:23,355 --> 01:17:26,140

1158
01:17:26,140 --> 01:17:31,960
All right, so there is only one
minor part of our description--

1159
01:17:31,960 --> 01:17:35,240
rapid survey of
thermodynamics-- that is left.

1160
01:17:35,240 --> 01:17:38,190
And that's the third
law of thermodynamics

1161
01:17:38,190 --> 01:17:45,270
and that has to do something
with the behavior of things

1162
01:17:45,270 --> 01:17:48,220
as you go towards
the limit of the zero

1163
01:17:48,220 --> 01:17:50,670
of thermodynamic temperature.

1164
01:17:50,670 --> 01:17:57,890
And I think we will spend maybe
10, 15 minutes next lecture

1165
01:17:57,890 --> 01:18:01,210
talking about the third
law of thermodynamics.

1166
01:18:01,210 --> 01:18:03,750
Probably from my
perspective, that

1167
01:18:03,750 --> 01:18:07,000
is enough time with
respect towards the time

1168
01:18:07,000 --> 01:18:09,720
we spend on the other
laws of thermodynamics,

1169
01:18:09,720 --> 01:18:12,690
because as I will
emphasize to you also,

1170
01:18:12,690 --> 01:18:15,760
the third law of
thermodynamics is in some sense

1171
01:18:15,760 --> 01:18:17,870
less valid than the others.

1172
01:18:17,870 --> 01:18:20,230
It's certainly correct
and it is valid.

1173
01:18:20,230 --> 01:18:24,730
It is just that its validity
rests on other things than what

1174
01:18:24,730 --> 01:18:26,660
I was emphasizing
right now, which

1175
01:18:26,660 --> 01:18:29,850
is the large number
of degrees of freedom.

1176
01:18:29,850 --> 01:18:31,940
You could have a large
number of degrees

1177
01:18:31,940 --> 01:18:34,200
of freedom in a world
that is governed

1178
01:18:34,200 --> 01:18:38,090
by classic laws of physics, and
the third law of thermodynamics

1179
01:18:38,090 --> 01:18:39,520
would be violated.

1180
01:18:39,520 --> 01:18:41,730
So it is a condition
that we live

1181
01:18:41,730 --> 01:18:45,610
in a world that is governed by
quantum mechanics that tells us

1182
01:18:45,610 --> 01:18:49,670
about the third law, as
we will discuss next time.

1183
01:18:49,670 --> 01:18:52,132