Homework 3 due Friday, April 26. A couple of examples where the averaging principle from last time can be done analytically.

Transcription

1 Stochastic Averaging Examples Tuesday, April 23, :01 PM Homework 3 due Friday, April 26. A couple of examples where the averaging principle from last time can be done analytically. Based on the alternative nondimensionalization of the Langevin equation (Klein- Kramers equation) from last time, we are set up for the averaging theorem through the following identification of variables: Now applying the averaging theorem, we first want to solve for the quasi-equilibrium distribution of the fast variable, given the slow variable: AppSDE13 Page 1

2 We note this is the same PDE that we've encountered before when we did the reduction to Smoluchowski dynamics; the solution is: The only difference is we can specify completely the function c(x') because we now have a conditional probability density for V', not a joint probability density. This quasi-equilibrium distribution should always be completely specified (no free constants). But what we see here, that is not always the case (see the next example), is that the quasi-equilibrium distribution for the fast variable V' is in fact independent of the slow variable X'. AppSDE13 Page 2

3 Now we average the drift coefficient in the slow equation against this quasiequilibrium probability density for the fast variable. So the averaging theory tells me that: This says that on this nondimensionalized time scale of order 1, the position isn't changing, to leading order. This is a classic symptom of a bad nondimensionalization choice. If one gets results that blow up in the perturbation theory, then probably the reference time scale was chosen too large, so that "the action was jumped over." If one gets boring or trivial results, then the reference time scale was likely chosen to be too short, so that the action didn't start yet. For our Langevin equation example: AppSDE13 Page 3

4 The ratio of the reference time scale in this alternative nondimensionalization to the reference time scale in the good nondimensionalization was The good nondimensionalization is therefore equivalent to rescaling time in the alternative nondimensionalization by That would put the equation in a form that isn't suitable for the averaging theorem. But there are other slow-fast theorems that you can find in Arnold, "Hasselmann's Program Revisited: The Analysis of Stochasticity in Deterministic Climate Models" E, Liu, and Vanden-Eijnden, "Analysis of Multiscale Methods for Stochastic Differential Equations" Liu, "Strong Convergence of Principle of Averaging for Multiscale Stochsatic Dynamical Systems" or compute a perturbation theory from scratch like we did. Let's consider a prototype model for overdamped stochastic dynamics (Smoluchowski dynamics) in a potential: These variables should be thought of as nondimensionalized. intelligently nondimensionalized potential energy The noise term on the slow variables could be thermal, but we've made it more general; noise coefficient AppSDE13 Page 4

5 The noise term on the slow variables could be thermal, but we've made it more general; noise coefficient The idea of this system is that one has taken a given complex system, identified "slow modes" and "fast modes," whose time scales are well separated (by a factor of Write differential equations for these slow and fast variables, nondimensionalize. For example, think in molecular dynamics of: Translation, rotation, large collective modes as slow Bond vibrations as fast Often one deals with systems where order parameters are invoked; these are typically what one thinks of as slow variables. Or one can think more simply of motion of a particle in a potential in multiple dimensions where the potential is steeper in some directions than others (before nondimensionalization.) First we solve the stationary Fokker-Planck equation for the fast variables to get the quasi-equilibrium distribution. Solve by integrating: We conclude: AppSDE13 Page 5

6 We conclude: Now one argues that in fact if the right hand size is not zero, one would not get a normalizable probability density. Easy to show for a constant right hand side. Wave hands about the absence of curl (appeal to some uniqueness theorem about elliptic operators on unbounded domains; I haven't found a precise reference yet) AppSDE13 Page 6

7 Normalization condition determines: In dimensional variables, this would appear as: This looks like a partition function in terms of the fast y variables, for a given value of the slow x variable. And we see that the conditional probability density for the y variable has the form of the canonical distribution with respect to the fast y variable, for a given value of the slow x variable. Now what is the averaged dynamics for the slow variables X? AppSDE13 Page 7

8 So our slow variable dynamics can (in nondimensional variables again) be approximated as follows: AppSDE13 Page 8

9 AppSDE13 Page 9 At least for overdamped, thermally forced physical systems, the effective (averaged) dynamics for slow variables, with the behavior of the coupled fast variables averaged out, is governed by a drift that is proportional to minus the gradient of the free energy (of the fast variables relative to the slow variables.) This kind of picture is common in theoretical physics, and we have simply showed how one can arrive at such descriptions (flow down the free energy) in terms of systematic averaging.