equations (only coupled through the forces), the stochastic equation and the

corresponding Fokker“Planck equation read

D

(F + B· 0 ),

q=

™ (8.91)

kB T

‚2ρ

‚ρ D‚

=’ (ρF ) + D 2 . (8.92)

‚t kB T ‚q ‚q

Setting ‚ρ/‚t = 0 and writing F = ’dV /dq, we ¬nd the equilibrium solution

V (q)

ρ(q) ∝ exp ’ , (8.93)

kB T

which again is the canonical distribution. In order to obtain the canonical

distribution by simulation using the stochastic Brownian equation, it is nec-

essary to take the time step small enough for F ”t to be a good approxima-

tion for the step made in the potential V . If that is not the case, integration

errors will produce deviating distributions. However, by applying an accep-

tance/rejection criterion to a Brownian step, a canonical distribution can be

enforced. This is the subject of the following section.

8.9 Smart Monte Carlo methods

The original Metropolis Monte Carlo procedure (Metropolis et al., 1953)

consists of a random step in con¬guration space, followed by an acceptance

criterion ensuring that the accepted con¬gurations sample a prescribed dis-

tribution function. For example, assume we wish to generate an ensemble

with canonical probabilities:

w(r) ∝ e’βV (r) . (8.94)

Consider a random con¬gurational step from r to r = r + ”r and let the

potential energies be given by

E = V (r), (8.95)

E = V (r ). (8.96)

8.9 Smart Monte Carlo methods 273

The random step may concern just one coordinate or one particle at a time,

or involve all particles at once. The sampling must be homogeneous over

space. The transition probabilities W’ from r to r and W← from r to r

should ful¬ll the detailed balance condition:

w(r)W’ = w(r )W← , (8.97)

leading to the ratio

W’ w(r )

= e’β(E ’E) .

= (8.98)

W← w(r)

This is accomplished by accepting the step with a probability pacc :

’

E ’ E ¤ 0 : W’ = pacc = 1,

for (8.99)

’

E ’ E > 0 : W’ = pacc = e’β(E ’E) ,

for (8.100)

’

as is easily seen by considering the backward transition probability:

E ’ E < 0 : W← = eβ(E ’E) ,

for (8.101)

E ’ E ≥ 0 : W← = 1,

for (8.102)

which ful¬lls (8.98). The acceptance with a given probability pacc < 1 is

’

realized by drawing a uniform random number 0 ¤ · < 1 and accepting the

step when · < pacc . When a step is not accepted, the previous step should

’

be counted again.

In the “smart Monte Carlo” procedure, proposed by Rossky et al. (1978),

a Brownian dynamic step is attempted according to (8.79) and (8.80), sam-

pling ξ from a Gaussian distribution. We denote the con¬guration, force

and potential energy before the attempted step by r, F and E and after the

attempted step by r , F and E :

r = r + βD”tF + ξ. (8.103)

The transition probability is not uniform in this case, because of the bias

introduced by the force:

(r ’ r ’ βD”tF )2

∝ exp ’ pacc ,

W’ (8.104)

’

4D”t

because this is the probability that the random variable ξ is chosen such that

this particular step results. Now imposing the detailed balance condition

(8.98):

W’

= e’β(E ’E) , (8.105)

W←

274 Stochastic dynamics: reducing degrees of freedom

we ¬nd for the forward/backward acceptance ratio:

(r ’ r ’ βD”tF )2 ’ (r ’ r ’ βD”tF )2

pacc

’

= exp ’β(E ’ E) +

pacc 4D”t

←

= e’β” , (8.106)

with

1 1

” = E ’ E + (r ’ r) · (F + F ) ’ βD”t(F 2 ’ F 2 ). (8.107)

2 4

Note that ” for the forward and backward step are equal in magnitude and

opposite in sign. The acceptance is realized, similar to (8.98) and (8.100),

by choosing:

” ¤ 0 : pacc = 1,

for (8.108)

’

” > 0 : pacc = e’β” .

for (8.109)

’

The latter acceptance is implemented by accepting the step when a homo-

geneous random number 0 ¤ · < 1 is smaller than exp(’β”). When a step

is not accepted, the previous step should be counted again.

The rejection of a step does destroy the dynamical continuity of the Brow-

nian simulation, but ensures that the proper canonical distribution will be

obtained. In practice, the time step “ or rather the product D”t “ can be

chosen such that almost all steps are accepted and the dynamics remains

valid, at least within the approximations that have led to the Brownian

stochastic equation.

8.10 How to obtain the friction tensor

How can the friction tensor “ or, equivalently, the noise correlation matrix

“ be obtained for use in Langevin or Brownian simulations?

There are essentially three di¬erent routes to obtain the friction tensor:

(i) from theoretical considerations,

(ii) from empirical data,

(iii) from detailed MD simulations.

The route to be chosen depends on the system and on the choice of ”irrele-

vant” degrees of freedom over which averaging should take place. In general

the accuracy required for the friction tensor is not very high: it only in-

¬‚uences the dynamical behavior of the system but not the thermodynamic

equilibria. This is seen from the Fokker“Planck equation that appears to

8.10 How to obtain the friction tensor 275

yield a canonical distribution in con¬guration space, even for the rather in-

accurate Brownian dynamics, which is independent of the applied friction

coe¬cients as long as the ¬‚uctuation“dissipation balance is maintained. It