phys212ln7 - Physics 212: Statistical mechanics II, Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture VII Our first picture of Brownian motion will be as a stochastic (i.e., random) process. A heavy particle in a fluid undergoes random forces as a result of collisions with the light particles of the fluid. You may be familiar with the idealization of the resulting motion as a random walk, but since this is a physics course, we will start from Newton’s equation of motion. The assumption of random forcing leads to a Langevin equation of motion. Our strategy is as follows: we obtain a constraint on the random forcing from the requirement that it reproduce thermal equilibrium, then use this to describe the behavior of the system under nonrandom forcing (e.g., an applied electric field to a charged particle). In writing equations in this lecture, the motion will be assumed to be one-dimensional so that no vector notation is necessary. Including the drag force from Stokes’ law gives: m du dt = - 6 πμau + F ( t ) ≡ - mγu + F ( t ) . (1) Here we defined γ = 6 πμa m . (2) This equation will simplify depending on our assumptions about F . For now we leave it in this general form. The first goal of this lecture is, given statistical information about F ( t ), to carry it through the Langevin equation (29) to get a statistical picture of x ( t ). This sort of random problem appears frequently in statistical physics: an example of great importance in condensed matter physics is, given a random potential V ( x ), to solve Schrodinger’s equation in that potential and get statistical information about its eigenstates. There are even many applications beyond physics: stochastic calculus, the formalization of some ideas in this lecture, is of considerable importance in signal processing and quantitative finance, to name just two fields. Since the Langevin equation (29) is linear, a natural approach is to take the Fourier transform of both sides. It will be useful to imagine that we are observing the system for times 0 t T , for some very large time T . The reason for doing this rather than just going to T = is that we can average quantities by 1 /T and get finite answers in the limit T → ∞ for quantities like the power spectrum, even though the underlying functions like u ( t ) do not fall off sufficiently rapidly at for the Fourier transform to be well defined. These definitions will be familiar to any of you with experience in signal processing. Then the expansion of u ( t ) is u ( t ) = X n u n e n t , ω n = 2 πn T . (3) and so on for the other quantities in the Langevin equation. The inverse formula is u n = 1 T Z T 0 u ( t ) e - n T . (4) The reality of u ( t ) then restricts u n * = u - n .
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phys212ln7 - Physics 212: Statistical mechanics II, Fall...

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