To work with stochastic differential equations, we need to establish a stochastic calculus.

For the purpose of evaluating the properties of the solutions to these kinds of problems, we can use the following formal rules (which can be rigorously justified by the more formal stochastic integration theory we'll discuss later).

behaves as a Gaussian random variable

Cov

We'll start with a simplified version of stochastic integration that we'll generalize later. This

theory will work just fine whenever we are only working with integrals of the form

where is deterministic, which typically arise when the noise on a stochastic differential equation is just additive:

To see how this works, we are formally thinking about as a

limit of the finite increment . The finite increment has the following properties which we showed a few lectures before:

is a Gaussian random variable

Cov ( = I

Now let's apply these simple stochastic integration rules to solve our Langevin

Stochastic CalculusFriday, March 20, 20152:05 PM

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Now let's apply these simple stochastic integration rules to solve our Langevin equation, but now in the language of stochastic differential equations.

which is independent of

One can use similar manipulations for solving stochastic differential equations as one does for ordinary differential equations, with the caveat that the chain rule changes for SDE's and in particular the product rule changes when two quantities depending on the noise are multiplied together.

Let's just try an integrating factor as before.

Solve for :

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For elementary operations, stochastic integrals behave much like regular integrals; same linearity properties, and same ability to factor in or out quantities that are not being integrated.

This is what's known as a strong solution of the stochastic differential

equation, meaning that each realization of the noise process driving the equation is mapped to a particular realization of the state

variable . By contrast, the notion of weak solution is very useful in practice, and just means that you have a complete statistical

description of the stochastic process , without necessarily having a pathwise mapping from the noise to the solution. Particularly useful concept in numerical approximation; weak numerical methods are almost always good enough for practical purposes.

To understand the solution of a stochastic differential equation, one often wants to compute its statistics. We'll show how to compute the mean and correlation function of the velocity, and see it agrees with the formal delta-correlated force calculation from before, but this stochastic calculus method will generalize better.

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Which is now a deterministic expression which is the same as what we obtained for the delta-correlated force calculation.

Finally, if we look at the strong solution, and recall that behaves like a Gaussian random function

So if is either deterministic or Gaussian, then is Gaussian because it is a deterministic linear operation on Gaussian random variables and functions.

So to summarize (carrying over the calculus with delta functions that works identically with the delta-correlated force calculation):

Cov

is Gaussian random function if is Gaussian random variable or deterministic.

We note that ; any average initial momentum is eventually forgotten due to dissipation.

There is a similar simplification to the autocorrelation function of the velocity at large times, but there are two time arguments, so we need to organize the long-time appropriately. The standard way to think about correlation functions is to have one time variable express the lag between the observations:

Cov

=

Note that the initial conditions are also forgotten in the autocorrelation function at long times. And in particular, the autocorrelation function only depends on the lag

variable , not on the time chosen to do the observation . Therefore, becomes weakly statistically stationary at long times (weak just means that the autocorrelation function has time-homogeneity). All of this are standard results for dissipative autonomous SDE's. And as we'll discuss later, this is an essential assumption in many Monte Carlo simulations, particularly for molecular dynamics. The idea is that the equations are usually considered to be reliable (at least within the field), but the initial conditions are often treated as artificial, subjective, biased and that they contaminate the statistics. One hopes that some chaos or better yet dissipation causes the system to eventually forget the initial conditions, and then it will settle into a ``natural'' state which is often called the statistically stationary state or quasi-equilibrium or non-equilibrium steady state (NESS) of the system, which only depends on the properties of the equation (and the driving noise) but not the initial conditions. If one believes this,

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the equation (and the driving noise) but not the initial conditions. If one believes this, then the strategy in computation is to allow a burn-in period until one thinks the initial conditions have been forgotten, and then only study the statistics of the system after this burn-in period. How does one know how long a burn-in period to take? Have to have some deep understanding of the field, but one necessary condition is that if the equations are autonomous then the statistics should be statistically stationary after the initial conditions have been forgotten. For our simple system here, we see the burn in period is a few multiples of the characteristic time over which the friction acts.

In fact, if we look at the strong solution, we see that in a strong sense the initial condition is forgotten after times large compared to . And so we can say in a strong sense that:

The point of this is that if we look at the approximation

to the solution, it can be shown (by

doing the stochastic calculus calculations) that:

mean •

autoccorelation function

•

Gaussianity together with weak statistically stationarity implies normal strong statistically stationarity (time-homogeneity).

is a Gaussian random function with:

for large time, so we think of as describing the statistically stationary state (or quasi-equilibrium or NESS) of the Langevin equation.

We can now make contact with equilibrium statistical mechanics.

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