Page 364 - Numerical Methods for Chemical Engineering

Markov chains and processes; Monte Carlo methods 353

But, since the random motion in each direction is independent,
2 2 2 2
( r) = ( x) + ( y) + ( z) (7.221)
and from the stochastic properties of the Wiener process, we have
2 2 2
( x) = 2Dt = ( y) = ( z) (7.222)
Therefore, in three dimensions,
2
( r) = 2Dt + 2Dt + 2Dt = 6Dt (7.223)
2
Similarly, for diffusion in two dimensions, ( r) = 4Dt.
In general, we relate the Langevin and Fokker–Planck equations as follows. Let us have
an ensemble whose members propagate according to an SDE

dx = a(t, x)dt + B(t, x) · dW t (7.224)
with a state-dependent drift a(t, x) and a state-dependent tensor B(t, x). Then, defining
from B(t, x) the positive-semidefinite diffusion tensor

T
D(t, x) = B(t, x) · B (t, x) (7.225)
the probability distribution of the ensemble follows the Fokker–Planck equation
∂p 1
=−∇ · [a(t, x)p(t, x)] + ∇∇:[D(t, x)p(t, x)] (7.226)
∂t 2
Given D(t, x), a corresponding B(t, x) may be obtained by Cholesky factorization; however,
this choice of B is not unique as BQ, for any orthogonal Q, also satisfies (7.225).

Markov chains and processes; Monte Carlo methods

In our discussion of polymerization and random walks, we have been using the concept
of Markov chains. Many simulation and computational methods are based on the random
generation of states (events) according to a defined probability distribution. Because these
techniques involve random number generation, they are known generally as Monte Carlo
methods, after the famous casino in Monaco.

Markov chains
Let us consider a stochastic system whose state is characterized by a vector q. Using the
laws of conditional probability, we write the probability of observing a particular sequence
[0]
[1]
of states q , q , ... , q [N] as

[0] [1] [N] [0] [1] [0] [2] [1] [0]
P q , q ,..., q = P q × P q q × P q q , q

[N] [N−1] [1] [0]
×··· × P q q ,..., q , q
(7.227)
In a Markov process of order m, each conditional probability of adding a new value q [k]