Lecture: Classical Modeling Classical Modeling: Stochastic Methods

Stochastic methods are based on the Markov Chain. A Markov process is characterized by a lack of memory (sound familiar?). The statistical properties of the immediate future are uniquely determined from the present, independent of the past. There are several flavors that exist to implement this approach:

Brownian Dynamics
The Brownian dynamics method computes phase space trajectories of a collection of particles that individually obey Langevin equations in a field of force.
  1. Assign an initial position and velocity.
  2. Get a random number from a Gaussian distribution
  3. Integrate velocity at n+1.
  4. Add random component to the velocity to mimic the interaction of the system with the environment.
Generalized Langevin Equations
The generalized Langevin approach is based on non-equilibrium statistical thermodynamics. A stochastic frictional force is included which is related to the second fluctuation dissipation theorem.
The general equations are

eqnarray13

where

Monte Carlo Methods
Representing the solution of a problem as a parameter of a hypothetical population, and using a random sequence of numbers to construct a sample of the population, from which statistical estimates of the parameter can be obtained.
In particular, the Metropolis Algorithm (Canonical Monte Carlo Method) is commonly used.
  1. Specify an initial configuration
  2. Generate a new configuration
  3. Compute energy and difference
    If difference < 0 accept the new configuration
    If difference > 0, randomly accept new configuration



Author: Ken Flurchick