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RUMEN DANGOVSKI: Hi, everybody.

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I'm Rumen, and today I'll
talk about three things:

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optical trapping, the Boltzmann
constant, and Brownian motion.

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The goal of the
lab is to extract

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Boltzmann's constant
out of Brownian motion

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and there are two key
components to think about.

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First one is the
Boltzmann's constant,

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which is prevalent in
different types of science.

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For example, we can
have it in biophysics

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where people use the Boltzmann's
constant to try to understand

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forces in a cellular level.

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We have it in thermodynamics
with the famous equipartition

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theorem.

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How does this relate
to the Brownian motion?

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The key thing is to think
about length scales.

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Brownian motion is relevant
in terms of microns.

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And at the same time, if you
look at a cellular level,

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for example, if we
look at our hair,

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we have micron-sized hair.

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All right.

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So there are different ways
to measure the Boltzmann's

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constant.

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Some people measure the
speed of sound in argon gas,

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others do optical
trapping in air.

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We will do slightly
different optical trapping,

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as you will see,
but the consensus

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among the scientific community
is that it is challenging.

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Our plan is to prepare
Brownian particles

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and control their
Brownian motion.

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We take spherical
glass beads of diameter

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of 3.2 microns and
the main tool is

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to concentrate a highly-focused
lasers on top of these beads.

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There's interesting
physics going on:

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light carries momentum
thus it generates force.

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We have that the net
gradient force opposes

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the motion of the beam while the
net scattering force goes along

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the motion of the beam.

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And when these two forces
balance each other,

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we have a bead that
is at the center.

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When you push this bead
a little bit to the left

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or to the right, then we have
that the gradient forces are

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pulling me back in the
center and essentially we

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observe a simple
harmonic motion.

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In a more concrete example,
what we did is we took samples

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and we confined everything
into a two-dimensional plane.

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This is very important.

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We have two directions:
the x direction

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and the y direction
for the beads.

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We put them into water and
a source of Brownian motion

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comes from the collisions
between our bead

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with the molecules
into the water.

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These are thermal collisions
that generate Brownian motion.

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As you can see, this lonely
bead has it's Brownian motion.

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What's interesting is when we
shine light on top of a bead,

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we trap this bead and we
can find the Brownian motion

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so it's feasible to
measure the motion.

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And the theory behind
this is very beautiful.

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It's about the
equipartition theorem,

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which relates the
kinetic energy coming

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from the simple harmonic
motion on the left hand side

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with the thermal energy due
to the degrees of freedom.

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Now let us recall that we have
a two-dimensional confinement

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so we have a direction
in x and direction in y.

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So it means that in
each of the directions,

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we have one degree
of freedom, which

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is correlated with this
equipartition theorem.

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Now an interesting thing
about the statistical motion

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of the molecules is that we
have the simple harmonic motion.

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However, the things
are moving into water

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so there is a drag
force that dominates.

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So we simplified
the left hand side.

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What is very interesting
is the f factor, which

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comes from the collisions
between the bead

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with the molecules, this
generates forcing and driving

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of the simple harmonic motion.

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I would like to
point out one thing

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about the scales of the forces.

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We have piconewtons, which is
relevant for optical trapping

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and for Brownian motion.

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This is the first
observation that we did,

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and actually Einstein did this
observation a long time ago.

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He observed the white noise--

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essentially the
collisions-- they

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generate uncorrelated forcing.

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And we can think
of it as something

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that is not biased
with any distribution,

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it's just uniform.

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As you can see on
these slides, we

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have the position
plotted in terms of time

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and it exhibits a uniform
distribution of the spectrum.

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Another thing that I
would like to point out

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is to look at this plot of
fluctuations in x and y,

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and the power which is linear
with the current of the lasers.

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As you can see, as
the power becomes big,

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this means that we
are trapping more

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so we have less fluctuations,
which is something as expected.

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All right, so let's
figure out what

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we want to do with this lab.

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We have the
equipartition theorem

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and essentially we
have three components

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that we would like to measure
in order to extract Kb.

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We need to find the
fluctuations, which I just

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presented to you.

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We need to find a stiffness
coefficient, alpha,

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which is related to the
spring constant of motion.

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And then we need to
measure the temperature, t.

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The apparatus that we use
has two main components.

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The two components are concerned
with two types of light

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that we use in our experiment.

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We use a laser
that shines on top

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of the confined
two-dimensional samples

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and tries to trap a bead.

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The scattered light from
the laser goes into a QPD--

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a quadrant photo detector--

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which is an
ultra-fast camera that

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manages to quickly
digitalize the content

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and give us the position
of the scattered light.

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So when we trap
the bead, we know

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where it is by observing
the feedback from the QPD.

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The other interesting
part of the apparatus

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is the LED light, which
illuminates the sample

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and then it brings
it to the CCD camera,

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and this is very important.

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So the CCD camera is very slow.

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It cannot measure
Brownian motion.

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What it can do is it can
tell us where are the beads.

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It can allow us to
look at the water

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and find the beads
we want to trap.

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So these two components
in combination

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are crucial for the
success of our research.

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There is interesting
electronics coming behind this.

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Essentially, the
signals from the QPD

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and from the stage position
where we put our sample

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are given in terms of volts.

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So a natural
question that arises

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is how are we going to
remember what is important?

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There are two important
things that we

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like to keep track
of, two positions.

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The first one is the
position of the stage

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that gives us a relative
point of consideration.

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And the second one
is the position

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of the QPD, which,
as you can recall,

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it measures the place of the
bead that we have trapped.

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These inputs are given
in terms of volts,

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so the natural thing
to do is to find

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the calibration that
converts these volts

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into actual distances.

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Here you can see how we do this.

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We plot the stage x position--
we can plot the stage y,

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it's completely analogous to
the QPD the positions here.

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And the conversion factor
hides along the slopes here.

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As you can see, we have
two different slopes here

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with different absolute values,
so this type of measurement

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is prone to errors.

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How we approach this
problem is that we

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fault by choosing
more data points

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and trying to increase the
statistic thus hopefully

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reducing the systematics
of this measurement.

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The second step is to start
getting the components

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that we need in
order to extract kb

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is to measure the stiffness
coefficient alpha.

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Now the main idea is to
take the equation of motion

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and to make a
[? free entrance ?]

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from in order to get
the positions in terms

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of frequencies.

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There is mathematics behind
this and essentially, the power

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spectral distribution,
and that's

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the distribution
of this motion here

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that we expect obtains
this form here.

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From where we can extract
the characteristic frequency,

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F naught, and F
naught gives us alpha,

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which is what we really need.

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This is a log log plot,
as you can see here.

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Another thing I
would like to mention

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is look at the
proportionality here.

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As we increase
the power, we also

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increase the stiffness
coefficient alpha

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because we're
trapped more closely,

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so it means that F
naught has to increase.

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And indeed, when we
increase the power,

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we are shifting F
naught to the right.

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Having all of these
components, we

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can put this into
the big picture.

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And the big picture is the
extraction of kb over here.

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I showed you x squared,
the fluctuations.

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I showed you alpha.

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We can measure the temperature,
t, and then we can get kb.

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The essence of
this measurement is

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to look at the inverse
proportionality

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between fluctuations and
the trapped stiffness.

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Then we can make a fit with
a reasonable chi-squared

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probability, and from
here we can extract kb.

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The result is
presented on the slide.

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We also show you the
measurement that we

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extract from literature.

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Our result is within 2 sigma
of the accepted value of kb.

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Actually we're very close to
one sigma from the accepted

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value of a kb.

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What is driving this
unfortunate outcome?

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The thing that drives
this unfortunate outcome

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is the systematic
errors on our picture.

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This is our starting error.

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This is the uncertainty in kb.

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And there are different factors
that contribute to this.

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As you saw, the calibration
is very difficult.

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We have error from the
laser hitting the water

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and changing the temperature.

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We have systematic error of the
electronics, which we safely

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ignore because the first
two factors dominate this.

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Our electronics
were very precise.

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And then we have the
statistical uncertainties

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that we use which correspond to
some error propagating tricks.

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OK, now let's do
our investigation.

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The first one is
about the calibration.

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As you can see, all of these
slopes should be valid,

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but they're actually not.

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So what we do is we take them
into account, we average them,

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then we take more data
points, we average again,

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and then we propagate
errors in order

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to reduce the systematics.

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The second one is due to
the heating of the laser.

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Essentially what happens is
when the laser hits the water,

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it starts heating the vicinity.

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And this is unfortunate
because yes, we

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can measure the room temperature
by using the thermometer,

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but actual uncertainty
on the temperature

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is much bigger because we have
extra heating due to the laser.

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We tried diligently
to avoid this

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by moving the laser
constantly so that it doesn't

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stay in one place
and heat up a lot,

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but it's very hard to quantify
how exactly it heats up

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the water.

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And the third one is
concerned with our fit

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with the fluctuations in terms
of the trapped stiffness.

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Initially, when we did the
fit with the PSD method,

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we get a probability of
chi squared equals zero.

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And then we quickly
realized that essentially

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what we need to do is take into
account the horizontal errors.

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And here what we do is we
transform the horizontal errors

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into the vertical errors.

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And this thing we can do by
using addition in quadrature

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and propagation of
the errors, which

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gave us a reasonable
realistic Chi squared of 0.33.

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In conclusion, I'll
start with limitations.

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As you saw, it's very
hard to distinguish

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between systematic errors and
statistical uncertainties.

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The reason for this
is, as you saw,

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is that we are fighting
with the systematics

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by introducing more statistics
and everything mixes up

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in the propagation of errors.

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The second thing that was
very difficult in this lab

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is that we need to move
the laser constantly.

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So one of the people
working on this lab

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has to keep track of
where the laser is

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and whether you're
trapping things

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that you may not want to trap.

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The third thing is
the need to find

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better ways of calibration.

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And there are people who are
actively working on this topic

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here.

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As you saw, the
main source of error

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came from the
calibration, actually.

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But the plus is that we can
give a reasonable estimate

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to Boltzmann constant
and at the same time,

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we can have a lot of
fun while doing so.

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I'd like to thank
to my partner Emma.

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And I think collaboration
is very important for JLab

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and more specifically for
this particular experiment.

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This experiment is impossible
to be done without a partner

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because it's very difficult.
It's very difficult

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to keep track of where
you want to put the laser

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and also what is going
on around the laser.

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While one of the people
is moving the laser,

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the other one should be
looking at the CCD camera

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and telling where are
we going and what are we

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actually trapping?

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What do we want to avoid?

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Like we don't want these
guys in our picture.

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I would also like
to thank the staff

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of this class for
their useful feedback

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and their valuable help.

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And finally, for your attention.

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[APPLAUSE]

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AUDIENCE: So in
this experiment, we

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found it useful to consider
the trap being elliptical

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00:14:07,076 --> 00:14:08,438
rather than circular.

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00:14:08,438 --> 00:14:11,370
And so it strengthens
alpha, being

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00:14:11,370 --> 00:14:14,758
different in the x
and y directions,

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00:14:14,758 --> 00:14:18,150
did you do that in
your calculation?

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RUMEN DANGOVSKI: Yeah,
so we have a lot of this

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00:14:21,820 --> 00:14:27,670
should be trapped stiffness
versus this is the current.

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And here, actually I
haven't shown that.

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00:14:30,762 --> 00:14:32,470
The trapped stiffness
differs whether you

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are looking in the x
direction or the y direction.

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00:14:35,140 --> 00:14:37,270
The actual measurements
are analogous because

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the mathematics is the same.

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00:14:38,860 --> 00:14:41,410
But as you can see here, we
have different data points

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00:14:41,410 --> 00:14:43,810
for the two cases.

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00:14:43,810 --> 00:14:46,370
We account this into
our considerations, yes.

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00:14:49,840 --> 00:14:50,375
Yes?

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AUDIENCE: So you mentioned
earlier that [INAUDIBLE]

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00:14:53,515 --> 00:14:55,768
with air or in air [INAUDIBLE].

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00:14:55,768 --> 00:14:57,934
Is there any benefit-- or
what's the main difference

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00:14:57,934 --> 00:14:59,410
between [INAUDIBLE]?

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00:15:04,007 --> 00:15:05,590
RUMEN DANGOVSKI:
Everything boils down

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00:15:05,590 --> 00:15:07,975
to this consideration here.

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00:15:12,196 --> 00:15:15,780
Let me just find my explanation.

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00:15:18,300 --> 00:15:23,374
It boils down to the
viscosity that dominates.

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00:15:23,374 --> 00:15:25,290
So in our case, we have
the viscosity of water

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00:15:25,290 --> 00:15:26,310
that dominates.

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00:15:26,310 --> 00:15:28,305
When you look in
different mediums,

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00:15:28,305 --> 00:15:30,180
you might have different
types of viscosities

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00:15:30,180 --> 00:15:32,490
which would change the motions.

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00:15:32,490 --> 00:15:36,090
So I have the paper,
I didn't actually

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00:15:36,090 --> 00:15:39,490
read through the whole
details of how exactly

316
00:15:39,490 --> 00:15:42,150
the mathematics
changes, probably does.

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00:15:42,150 --> 00:15:45,930
Also probably another issues
that may not arise or may arise

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00:15:45,930 --> 00:15:47,640
is the laser, like in this case.

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00:15:47,640 --> 00:15:52,410
The laser heats up
the water, but I

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00:15:52,410 --> 00:15:56,010
don't know how exactly the
laser would react with the air.

321
00:15:56,010 --> 00:15:57,720
Maybe it's going to
heat up a little bit,

322
00:15:57,720 --> 00:16:02,020
but how is this heat going
to be distributed in space?

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00:16:02,020 --> 00:16:04,470
I'm not very knowledgeable
of this as of now.

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00:16:08,853 --> 00:16:12,057
PROFESSOR: Any other questions?

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00:16:12,057 --> 00:16:13,723
AUDIENCE: Right there,
what does it say?

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00:16:13,723 --> 00:16:17,654
Only one degree of
freedom [INAUDIBLE]??

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00:16:17,654 --> 00:16:19,570
RUMEN DANGOVSKI: So the
equipartition theorem,

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00:16:19,570 --> 00:16:21,470
it breaks into components.

329
00:16:21,470 --> 00:16:23,399
It breaks into the
component of x,

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00:16:23,399 --> 00:16:24,940
where you have one
degree of freedom,

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00:16:24,940 --> 00:16:26,523
and it breaks into
the component of y,

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00:16:26,523 --> 00:16:28,800
where you have another
degree of freedom.

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00:16:28,800 --> 00:16:30,930
Each of these considerations
is concerned only

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00:16:30,930 --> 00:16:34,764
within one movement.

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00:16:34,764 --> 00:16:38,677
AUDIENCE: So you have
two degrees of freedom?

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00:16:38,677 --> 00:16:41,260
RUMEN DANGOVSKI: OK, well, you
can think about it in this way.

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00:16:41,260 --> 00:16:43,660
We have two degrees
of freedom in total,

338
00:16:43,660 --> 00:16:45,184
but in the actual
directions that

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00:16:45,184 --> 00:16:46,600
are useful for our
considerations,

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00:16:46,600 --> 00:16:48,058
we have only one
degree of freedom.

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00:16:55,770 --> 00:16:57,770
AUDIENCE: You said you
were using the 3.2 micron

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00:16:57,770 --> 00:16:58,270
[INAUDIBLE].

343
00:16:58,270 --> 00:16:59,738
There were other
sizes available.

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00:16:59,738 --> 00:17:01,706
It looks like some
of your photographs

345
00:17:01,706 --> 00:17:02,690
from the [INAUDIBLE].

346
00:17:02,690 --> 00:17:08,038
Did you try data from both the
small beads and the big beads?

347
00:17:08,038 --> 00:17:11,460
Curious how they
compared in quality.

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00:17:11,460 --> 00:17:15,139
RUMEN DANGOVSKI:
This is 3.2 microns.

349
00:17:20,069 --> 00:17:24,030
Later on, you saw
the one microns.

350
00:17:24,030 --> 00:17:29,220
I think our data is concerned
with the 3.2 microns.

351
00:17:29,220 --> 00:17:31,410
My results are not related
with the one micron.

352
00:17:31,410 --> 00:17:33,630
We just played with
it and tried this.

353
00:17:33,630 --> 00:17:38,230
We also tried trapping
cells from onions.

354
00:17:38,230 --> 00:17:40,650
It was very fun to play
with, but unfortunately, we

355
00:17:40,650 --> 00:17:44,770
couldn't get any valuable
quantitative results there.

356
00:17:44,770 --> 00:17:47,970
So even though it's quite a
lot of fun, we have some clips,

357
00:17:47,970 --> 00:17:49,662
it's not worth for
this presentation,

358
00:17:49,662 --> 00:17:50,870
which concentrates on the kb.

359
00:18:00,770 --> 00:18:03,220
[APPLAUSE]