Reading: Gardiner Sec. 1.2

Homework 2 due Tuesday, March 17 at 2 PM.

The friction constant depends on the dynamic viscosity of the fluid and the size and shape of the particle.

How shall we model the force representing the thermal fluctuations?

The thermal fluctuation force represents the effective randomness of the solvent molecules. For large solute particles, there would be large numbers of collisions with the solvent molecules, and by LLN/CLT the random force would become small relative to their average effect (which is encoded in the friction term). But for particles m, the number of effective collisions with the solvent molecules fails to be large enough for the LLN to be in force, and so the random component should be explicitly incorporated.

The time scale on which the solute particle changes momentum is, depending on its size, s to ; we'll show how to compute this later.

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The typical time scale over which a water molecule moves between collisions (with other water

molecules) is

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The time scale over which a water molecule interacts with the solute particle (when they do) is

•

Let's think about what properties the thermally fluctuating force should have. To this end, it's relevant to take into account some more physical time scales

In summary, the time scale on which the momentum of the solute particle fluctuates is much longer (by at least a factor of ) than the time scale on which the force it feels from the collision with solvent molecules is correlated.

Langevin Equation Model for Brownian MotionFriday, March 13, 20152:04 PM

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(We are also assuming here that the water molecules act roughly independently of each other, which is a reasonably good approximation with caveats from low Reynolds number hydrodynamics.)

The time scale separation also tells us that we can model as Gaussian because CLT resulting from many solvent particle collisions per resolved time scale.

Therefore, if we agree to only ask questions about the behavior of the Brownian particle over time

scales , then the thermal fluctuation force can be modeled as continuous white noise, i.e., with no memory between different moments of time. The way this can be encoded in the physicist-style language is to say the force is delta-correlated in time:

Now, what is the nature of the matrix

We expect the thermal forces to be statistically stationary (time-homogenous), so the correlation function should only depend on the time difference , not the times separately.

Statistical isotropy of the thermal forces:

Actually we need to be a little careful about statistical isotropy, because the current

velocity of the Brownian particle actually breaks isotropy. In fact, the

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velocity of the Brownian particle actually breaks isotropy. In fact, the deterministic friction force is just the mean of the force of collision with the solvent particles. Implicitly, we imagine that this mean effect has been subtracted out, and that the remaining randomness in the solvent particle motion is decoupled from the motion of the large Brownian solute particle.

Therefore, we arrive at the physicist's version of the Langevin equation:

mean •

correlation function •

We will first treat this model in physicist-mode (see Risken Sections 1.1, 3.1), calculate the solution for the velocity and some basic properties, and then later show how these calculations are translated into the language of SDEs.

where is a Gaussian random function with:

Almost always, the thermal force is modeled as independent of .

Let's solve for the random velocity . Just same way as for deterministic force; use integrating factor:

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What are the properties of the resulting random function ?

Most basic property is the mean:

Note that expectation commutes with integrals over deterministic domain by linearity (and Fubini's theorem).

This is exactly expressing the loss of momentum due to friction, and is the same result as for a purely deterministic setting.

The effects of the randomness will be seen through the next order description of the random velocity function, namely the correlation function.

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Note that different dummy variables are used in each argument of the covariance; this will facilitate the next step, where we deploy the bilinearity of covariance:

Now, we replace:

We use the property that if is a continuous function, then

What if we integrate delta functions over finite intervals?

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What if we integrate delta functions over finite intervals?

The upshot is that the integrand is continuous everywhere except at so you can show the delta-function integration rule still works provided that . But if the delta function is located at or , the answer is not well-defined. Calculation continued next time...

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