as a sequence of delta functions with random amplitudes. Every delta function

causes a jump in y and the resulting function y(t) is not di¬erentiable, so that the

notation dy/dt is mathematically incorrect. Instead of a di¬erential equation (8.4),

we should write a di¬erence equation for small increments dt, dy, dw:

dy = f (y) dt + c dw(t), (8.5)

where w(t) is the Wiener“L´vy process, which is in fact the integral of a white

e

noise. The Wiener-L´vy process (often simply called the Wiener process is non-

e

stationary, but its increments dw are stationary normal processes. See, for example,

Papoulis (1965) for de¬nitions and properties of random processes. While modern

mathematical texts avoid the di¬erential equation notation,2 this notation has been

happily used in the literature, and we shall use it as well without being disturbed

by the mathematical incorrectness. What we mean by a stochastic di¬erential

equation as (8.4) is that the increment of y over a time interval can be obtained

by integrating the right-hand side over that interval. The integral r of the random

process ·(t)

t+”t

r= ·(t ) dt (8.6)

t

is a random number with zero mean and variance given by a double integral over

the correlation function of ·:

t+”t t+”t

2

r = dt dt ·(t )·(t ) ; (8.7)

t t

in the case of a white noise ·(t )·(t ) = δ(t ’ t ) and therefore r2 = ”t.

The other remark concerns a subtlety of stochastic di¬erential equations with a

y-dependent coe¬cient c(y) in front of the stochastic white-noise term: the widely

2 See, e.g., Gardiner (1990). An early discussion of the inappropriateness of stochastic di¬erential

equations has been given by Doob (1942).

254 Stochastic dynamics: reducing degrees of freedom

debated Itˆ“Stratonovich “dilemma.”3 Solving the equation in time steps, the

o

variable y will make a jump every step and it is not clear whether the coe¬cient

c(y) should be evaluated before or after the time step. The equation therefore has

no meaning unless a recipe is given how to handle this dilemma. Itˆ™s recipe is to

o

evaluate c(y) before the step; Stratonovich™s recipe is to take the average of the

evaluations before and after the step. The stochastic equation is meant to de¬ne a

process that will satisfy a desired equation for the distribution function P (y, t) of

y. If that equation reads

1 ‚2

‚P ‚

= ’ f (y)P + c(y)2 P, (8.8)

2

‚t ‚y 2 ‚y

Itˆ™s interpretation appears to be correct. With Stratonovich™s interpretation the

o

last term is replaced by

1‚ ‚

c(y) c(y)P. (8.9)

2 ‚y ‚y

Hence the derivation of the equation will also provide the correct interpretation.

Without that interpretation the equation is meaningless. As van Kampen (1981,

p. 245) remarks: “no amount of physical acumen su¬ces to justify a meaningless

string of symbols.” However, the whole “dilemma” arises only when the noise

term is white (i.e., when its time correlation function is a delta function), which

is a mathematical construct that never arises in a real physical situation. When

the noise has a non-zero correlation time, there is no di¬erence between Itˆ™s and

o

Stratonovich™s interpretation for time steps small with respect to the correlation

time. So, physically, the “dilemma” is a non-issue after all.

In the following sections we shall ¬rst investigate what is required for the

potential of mean force in simulations that are meant to preserve long-time

accuracy. Then we describe how friction and noise relate to each other and

to the stochastic properties of the velocities, both in the full Langevin equa-

tion and in the simpler pure Langevin equation which does not contain the

systematic force. This is followed by the introduction of various approxima-

tions. These approximations involve both temporal and spatial correlations

in the friction and noise: time correlations can be reduced to instantaneous

response involving white noise and spatial correlations can be reduced to lo-

cal terms, yielding the simple Langevin equation. In Section 8.7 we average

the Langevin dynamics over times long enough to make the inertial term

negligible, yielding what we shall call Brownian dynamics.4

3 See van Kampen (1981) and the original references quoted therein.

4 The term Brownian dynamics in this book is restricted to approximations of particle dynamics

that are inertia-free but still contain stochastic forces. There is no universal agreement on this

nomenclature; the term Brownian dynamics is sometimes used for any dynamical method that

contains stochastic terms.

8.3 The potential of mean force 255

8.3 The potential of mean force

In a system with reduced dimensionality it is impossible to faithfully re-

tain both thermodynamic and dynamic properties on all time and length

scales. Since the interest is in retaining properties on long time scales and

with coarse space resolution, we shall in almost all cases be primarily in-

terested in a faithful representation of the thermodynamic properties at

equilibrium, and secondarily in the faithful representation of coarse-grained

non-equilibrium behavior. If we can maintain these objectives we should be

prepared to give up on accurate local and short-time dynamical details.

The criterium of retaining thermodynamic accuracy prescribes that the

partition function generated by the reduced dynamics should at least be

proportional to the partition function that would have been generated if all

degrees of freedom had been considered. Assuming canonical ensembles, this

implies that the probability distribution w(r ) in the primed con¬gurational

space should be proportional to the integral of the Boltzmann factor over

the double-primed space:

e’βV (r ,r ) dr .

w(r ) ∝ (8.10)

Now we de¬ne the potential of mean force as

e’βV (r ,r ) dr ,

V mf (r ) = ’kT ln (8.11)

which implies that

w(r ) dr ∝ e’βV

mf (r )

dr . (8.12)

It follows by di¬erentiation that the forces derived as a gradient of V mf equal

the exact force averaged over the ensemble of primed coordinates:

(‚V (r , r )/‚r ) exp[’βV (r , r )] dr

∇r V mf = . (8.13)

exp[’βV (r , r ) dr ]

Note that V mf also contains the direct interactions between r ™s, which are

separable from the integral in (8.11). It is a free energy with respect to

the double-primed variables (beware that therefore V mf is temperature-

dependent!), but it still is a function of the primed coordinates. It de-

termines in a straightforward manner the probability distribution in the

primed space. Note that V mf is not a mean potential over an equilibrium

double-primed ensemble:

V mf = V (r , r ) . (8.14)

Whereas several methods are available to compute potentials of mean force

256 Stochastic dynamics: reducing degrees of freedom

from simulations, as is treated in detail in Chapter 7, Section 7.5 on page 227,

empirical validation and generally adjustments are always necessary; the

best results are often obtained with completely empirical parametrization

because the model can then be ¬ne-tuned to deliver the thermodynamic

accuracy required for the application. In the next section we consider the

special case of superatom models.

8.4 Superatom approach

A special form of coarse graining is the representation of local groups of

atoms by one particle, called a superatom. Superatoms are especially useful

in chain molecules as polymers and lipids, where they typically represent

three to ¬ve monomer units. This is a compromise between accuracy and

simulation e¬ciency. Several superatom de¬nitions have been published and