Show that a small object immersed in a fluid at temperature...

Show that a small object immersed in a fluid at temperature $T$ will undergo a random motion, due to collisions with the molecules of the fluid, such that the mean-square displacement in any direction satisfies $$ \left\langle(\Delta x)^2\right\rangle=T t / \lambda, $$ where $t$ is the elapsed time, and $\lambda$ is a constant proportional to the viscosity of the fluid.

Show that a small object immersed in a fluid at temperature $T$ will undergo a random motion, due to collisions with the molecules of the fluid, such that the mean-square displacement in any direction satisfies
$$
\left\langle(\Delta x)^2\right\rangle=T t / \lambda,
$$
where $t$ is the elapsed time, and $\lambda$ is a constant proportional to the viscosity of the fluid.

Key Concepts

Brownian Motion

Brownian motion refers to the irregular, random movement of microscopic particles suspended in a fluid, resulting from countless collisions with the fluid’s molecules. This phenomenon is a fundamental example of a stochastic process in statistical mechanics, illustrating how microscopic thermal fluctuations lead to observable macroscopic behavior.

Mean-Square Displacement

Mean-square displacement is a statistical measure that quantifies the average squared distance a particle travels as a function of time due to its random motion. In diffusion processes, the linear relationship between mean-square displacement and time is a hallmark of normal diffusion, providing a means to connect microscopic random motion with macroscopic transport behavior.

Diffusion Coefficient

The diffusion coefficient is a parameter that characterizes how fast particles spread out over time in a medium. It relates microscopic properties, like the temperature-induced random collisions and frictional forces (or viscosity), to macroscopic observations of particle transport, often through the Einstein relation which connects it to temperature and friction.

Langevin Dynamics

Langevin dynamics is a theoretical framework used to describe the motion of particles under the influence of both systematic forces, such as viscous drag, and random forces arising from thermal fluctuations. By incorporating both deterministic and stochastic influences, the Langevin equation provides a way to derive statistical properties like the mean-square displacement in systems experiencing Brownian motion.

Fluctuation-Dissipation Theorem

The fluctuation-dissipation theorem establishes a fundamental link between the random thermal fluctuations in a system and the dissipative forces associated with friction or viscosity. This theorem ensures that the energy input from thermal fluctuations is balanced by dissipation, which is essential in deriving relationships like the one between mean-square displacement, temperature, and viscosity.

Transcript

00:02 Okay, this problem is to discuss the browning motion.

00:07 So what's the browning motion? which is if this is the surface, a water surface, there's a lot of the water molecule.

00:18 So here is a particle, a particle.

00:21 And we want to discuss the particle behavior, moving behavior around the water molecules.

00:31 Okay, what is the particle? the part a is to want to find out the rubin square of this particle.

00:43 If these particles behave is like the ideal gas, so the looming square, the speed of loomings will be equal to 1 .73 m0 kb t.

01:13 Then what's the m0? so kb t.

01:25 M0 is if this is a sphere, particle is a four -thirds high radia half radia cubic low low is the density so this could be 1 .73 now one point both main constant time and the temperature now is 20 celsius so this is 293 and 34 pi now 2d3 because we assume the diameter of this particle is d.

02:20 So now the density of this particle is 100 kilograms cubic meter.

02:28 So the result is the top side is 4 .81 times 12.

02:42 And the button is diameter d, square root d.

02:47 So, meter.

02:52 So, this is the luminous square of the particle with the diameter d.

03:00 Now, what's the time interval of the particle move a distant d? so, the time interval for this particle move a distant d and with this rooming square speed.

03:26 So the d is d, the l is d.

03:28 The ruby square speed is 4 .81 minus 12 and this should be d, and then this could be 2 .0a times 10, 11, d squared, square root so this is the unit second.

04:00 So this is the second.

04:01 So this is the time interval for this particle, move a distant d.

04:10 D is his own diameter.

04:13 Now, what's the c? the c is fine out if the particle's diameter is equal to 3 .mikometer, 3 .0 micrometer.

04:29 And what's the ruby square? at this condition, what is rubin square? now plug in the values to part a.

04:41 So, right here is 4 .81 times 10, 12, 11, and the d is 3 mickel, square root, 3 mic, so the value will be 9 .27, 10 minus 4th.

05:16 Okay, this is the rubin square speed for the diameter.

05:23 Is 3 microtor.

05:25 Phoidal diameter is 3 micro.

05:27 And now, at this condition, what's the interval d, or interval delta t, the t will be plug into the number, the part b.

05:42 So now, now we can plug in the number, the d, square, and square root.

06:22 So, and finally, the number will be, the result will be 3 .2.

06:28 24 times 10 minus 3 second.

06:36 This is this tells us that if the vitamin is d, so the ruby square is this one, and the interval is this one.

06:44 It's very short time.

06:48 And now, let's discuss the part of d.

06:53 D is that if the m0, if the particle's mass, it's 70 kilograms, what's the room square speed and time interval.

07:12 Now m0 is 70 kilogram.

07:15 What's loomis square? okay.

07:21 Now if lumi square is 1 .73, the m0 .0 is 70...

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