BBT = 2ζkB T , (8.74)

· 0 (t) = 0, (8.75)

· 0 (t0 )(· 0 )T (t0 + t) = 1 δ(t). (8.76)

In simulations the velocity can be eliminated and the positions can be up-

dated by a simple Euler step:

√

q(t + ”t) = q(t) + ζ ’1 F(t)”t + ζ ’1 Br ”t, (8.77)

where r is a vector of random numbers, each drawn independently from a

probability distribution (conveniently, but not necessarily, Gaussian) with

r2 = 1.

r = 0, (8.78)

Note that “ as the dynamics is non-inertial “ the mass does not enter in the

dynamics of the system anymore. Apart from coupling through the forces,

the coupling between degrees of freedom enters only through mutual friction

coe¬cients.

For the simple Brownian dynamics with diagonal friction matrix, and

using the di¬usion constant Di = kB T /ζii , this equation reduces to

D

qi (t + ”t) = qi (t) + Fi (t)”t + ξ, (8.79)

kB T

where ξ is a random number, drawn from a probability distribution with

ξ 2 = 2D”t.

ξ = 0, (8.80)

One can devise more sophisticated forms that use the forces at half steps in

order to integrate the drift part of the displacement to a higher order, but

the noise term tends to destroy any higher-order accuracy.

¿From (8.77) it is seen that friction scales the time: decreasing the friction

(or increasing the di¬usion constant) has the same e¬ect as increasing the

time step. It is also seen that the displacement due to the force is propor-

tional to the time step, but the displacement due to noise is proportional to

the square root of the time step. This means that slow processes that allow

longer time steps are subjected to smaller noise intensities. For macroscopic

averages the noise will eventually become negligible.

8.8 Probability distributions and Fokker“Planck equations

In Section 8.6 we used a Fokker“Planck equation to derive the probability

distribution for the velocity in the case of the simple pure Langevin equation

(see page 267). This led to the satisfactory conclusion that the simple pure

270 Stochastic dynamics: reducing degrees of freedom

Langevin equation leads to a Maxwellian distribution. In this section we for-

mulate Fokker“Planck equations for the more general Markovian Langevin

equation and for the Brownian dynamics equation. What we wish to gain

from the corresponding Fokker“Planck equations is insight into the steady-

state and equilibrium behavior in order to judge their compatibility with

statistical mechanics, and possibly also to obtain di¬erential equations that

can be solved analytically.

Stochastic equations generate random processes whose distribution funct-

ions behave in time according to certain second-order partial di¬erential

equations, called Fokker“Planck equations. They follow from the master

equation that describes the transition probabilities of the stochastic pro-

cess. The Fokker“Planck equation is similar to the Liouville equation in

statistical mechanics that describes the evolution of density in phase space

resulting from a set of equations of motion; the essential di¬erence is the

stochastic nature of the underlying process in the case of Fokker“Planck

equations.

8.8.1 General Fokker“Planck equations

We ¬rst give the general equations, and apply these to our special cases.

Consider a vector of variables x generated by a stochastic equation:13

x(t) = a(x(t)) + B· 0 (t),

™ (8.81)

with · 0 (t) independent normalized white noise processes, as speci¬ed by

(8.58). The variables may be any observable, as coordinates or velocities

or both. The ¬rst term is a drift term and the second a di¬usion term.

The corresponding Fokker“Planck equation in the Itˆ interpretation for the

o

distribution function ρ(x, t) (van Kampen, 1981; Risken, 1989) is in matrix

notation

‚ρ 1

= ’∇T (aρ) + tr (∇x ∇T BBT ρ), (8.82)

x x

‚t 2

or for clarity written in components:

‚2

‚ρ ‚ 1

=’ (ai ρ) + Bik Bjk ρ. (8.83)

‚t ‚xi 2 ‚xi ‚xj

i ij k

13 We made a remark on this mathematically incorrect form of a stochastic di¬erential equation

on page 253 in relation to (8.4). The proper equation is dx = a dt + B dw, where w is a vector

of Wiener processes.

8.8 Probability distributions and Fokker“Planck equations 271

8.8.2 Application to generalized Langevin dynamics

Let us now apply this general equation to the general Markovian Langevin

equation (8.51):

™

q = v, (8.84)

v = M’1 F(q) ’ M’1 ζv + M’1 B· 0 .

™ (8.85)

The single-column matrix x consists of a concatenation of q and v. The

single-column matrix a then consists of a concatenation of v and M’1 F(q)’

M’1 ζv. Carefully applying (8.82) to this x (and assuming B to be constant)

yields

‚ρ

= ’vT ∇q ρ ’ FT M’1 ∇v ρ + tr (M’1 ζ)ρ

‚t

1

+vT ζM’1 ∇v ρ + tr (M’1 BBT M’1 ∇q ∇q ρ). (8.86)

2

Note that the noise coe¬cient is related to the friction tensor by (8.60) on

page 266:

BBT = 2ζkB T. (8.87)

This rather awesome multidimensional equation can of course be solved

numerically and will give the same results as a simulation of the original

stochastic equation. More insight is obtained when we reduce this equation

to one dimension and obtain the rather famous Kramers equation (Kramers,

1940):14

ζkB T ‚ 2 ρ

‚ρ ‚ρ F ‚ρ ρv ‚ρ ζ

= ’v ’ + + ρ+ . (8.88)

m2 ‚v 2

‚t ‚q m ‚v m ‚v m

Even this much simpler equation cannot be solved analytically, but it can

be well approximated to obtain classical rates for barrier-crossing processes.

Kramer™s theory has been used extensively to ¬nd damping corrections to

the reaction rates derived from Eyring™s transition state theory. It is easy to

¬nd the equilibrium distribution ρeq (q, v) by setting ‚ρ/‚t = 0 (see Exercise

8.5):

mv 2 V (q)

ρeq (q, v) ∝ exp ’ exp ’ , (8.89)

2kB T kB T

where V is the potential de¬ned by F = ’dV /dq. Again, this is a satisfac-

tory result compatible with the canonical distribution.

14 A generalization to colored noise and friction and with external noise has been given by Banik

et al. (2000).

272 Stochastic dynamics: reducing degrees of freedom

8.8.3 Application to Brownian dynamics

For Brownian dynamics the stochastic equations (8.73) and (8.74) are a

function of q only. The corresponding Fokker“Planck equation is

‚2ρ

‚ρ ‚

[(ζ ’1 F)i ρ] + kB T ’1

=’ (ζ )ij . (8.90)

‚t ‚qi ‚qi ‚qj

i ij