teristic time scale of the friction, which is around m/ζ, will also be handled

with reasonable accuracy. In many applications one is more interested in

obtaining a fast sampling of con¬guration phase than in accurately repro-

ducing the real dynamics; in such cases one may choose a rather low friction

in order to obtain a faster dynamical behavior. Ultimately one may choose

not to add any friction or noise at all and obtain a fast dynamic sampling

by just simulating Hamiltonian molecular dynamics of a reduced system

with a proper potential of mean force. This is a quite common procedure in

simulations based on ”superatoms.”

In the following we consider a few examples of friction tensors.

8.10.1 Solute molecules in a solvent

The most straightforward application of stochastic dynamics is the simu-

lation of solute molecules in a solvent. In a dilute solution the friction is

determined solely by the di¬erence between the velocity v of the solute

particle and the bulk velocity u of the solvent:

F fr = ’ζ(v ’ u), (8.110)

and the friction tensor can at most be a 3 — 3 matrix for a non-spherical

particle. For a spherical particle the friction tensor must be isotropic and

equal to ζ1. We have not introduced terms like ”bulk velocities” of the

bath particles before, implying that such velocities are assumed to be zero.

Langevin and Brownian dynamics do not conserve momentum (and conserve

energy only as an average) and should not be applied in the formulation given

here when the application requires momentum and/or energy conservation.

The friction coe¬cient ζ follows from the di¬usion coe¬cient D of the

particle and the temperature by the Einstein relation

kB T

ζ= . (8.111)

D

D can be obtained from experiment or from a simulation that includes the

full solvent. The friction coe¬cient can also be obtained from hydrody-

namics if the solvent can be approximated by a continuum with viscosity ·,

yielding Stokes™ law for a spherical particle with radius a:

ζ = 6π·a. (8.112)

276 Stochastic dynamics: reducing degrees of freedom

When the solution is not dilute, the most important addition is an interac-

tion term in the systematic force; this can be obtained by thermodynamic

integration from detailed simulations with pairs of particles at a series of

constrained distances. But the friction force on solute particle i will also

be in¬‚uenced by the velocity of nearby solute particles j. This in¬‚uence

is exerted through the intervening ¬‚uid and is called the hydrodynamic in-

teraction. It can be evaluated from the Navier“Stokes equations for ¬‚uid

dynamics. The hydrodynamic interaction is a long-range e¬ect that decays

with the inverse distance between the particles. The 1/r term in the inter-

action, averaged over orientations, is expressed as a mobility matrix, which

forms the interaction part of the inverse of the friction matrix; this is known

as the Oseen tensor. The equations are

ζ ’1 = H, (8.113)

1

Hii = , (8.114)

6π·a

rrT

1

Hij = 1+ 2 , (8.115)

8π·r r

where r = ri ’ rj and r = |r|. Each element of H, de¬ned above, is a

3 — 3 cartesian matrix; i, j number the solute particles. Hydrodynamic in-

teractions are often included in stochastic modelling of polymers in solution,

where the polymer is modelled as a string of beads and the solution is not

modelled explicitly. Meiners and Quake (1999) have compared di¬usion

measurements on colloidal particles with Brownian simulations using the

Oseen tensor and found excellent agreement for the positional correlation

functions.

8.10.2 Friction from simulation

In cases where theoretical models and empirical data are unavailable the

friction parameter can be obtained from analysis of the “observed” forces in

constrained simulations with atomic detail. If detailed simulations are done

with the “relevant” degrees of freedom q constrained, the forces acting on

the constrained degrees of freedom are the forces from the double-primed

subsystem and “ if carried to equilibrium “ will approximate the sum of

the systematic force and the random force that appear in the Langevin

equation. The friction force itself will not appear as there are no velocities

in the primed coordinates. The average of the constraint force F c will be

the systematic force, which on integration will produce the potential of mean

force. The ¬‚uctuation ”F c (t) will be a realization of the random force. If

Exercises 277

the second ¬‚uctuation“dissipation theorem (8.26) holds, then

”F c (t0 )”F c (t0 + t) = kB T ζ(t). (8.116)

However, we have simpli¬ed the noise correlation function to a δ-function

and the friction to a constant, which implies that

∞ ∞

1

”F c (t0 )”F c (t0 + t) dt.

ζ= ζ(t) dt = (8.117)

kB T

0 0

One may also de¬ne the friction in terms of the di¬usion constant D =

kB T /ζ, so that

(kB T )2

D= ∞ . (8.118)

”F c (t0 )”F c (t0 + t) dt

0

In the multidimensional case, the cross correlation matrix of the constraint

forces will similarly lead to the friction tensor.

Exercises

Solve mv = ’ζv + ·(t) for the velocity v, given the velocity at t = 0,

8.1 ™

to yield (8.65).

Compute v 2 (t) when friction and noise are switched on at t = 0 by

8.2

taking the square of (8.65).

8.3 Show that (8.69) follows from (8.69). Do this by showing that the

time derivative of (”v)2 equals B.

8.4 Write (8.86) out in components.

8.5 Find the equilibrium solution for the Kramers equation (8.88) by

separating variables, considering ρ as a product of f (q) and g(v).

This splits the equation; ¬rst solve for the g(v) part and insert the

result into the f (q) part.

9

Coarse graining from particles to ¬‚uid dynamics

9.1 Introduction

In this chapter we shall set out to average a system of particles over space and

obtain equations for the variables averaged over space. We consider a Hamil-

tonian system (although we shall allow for the presence of an external force,

such as a gravitational force, that has its source outside the system), and

“ for simplicity “ consider a single-component ¬‚uid with isotropic behavior.

The latter condition is not essential, but allows us to simplify notations by

saving on extra indexes and higher-order tensors that would cause unneces-

sary distraction from the main topic. The restriction to a single component

is for simplicity also, and we shall later look at multicomponent systems.

By averaging over space we expect to arrive at the equations of ¬‚uid

dynamics. These equations describe the motion of ¬‚uid elements and are

based on the conservation of mass, momentum and energy. They do not

describe any atomic details and assume that the ¬‚uid is in local equilibrium,

so that an equation of state can be applied to relate local thermodynamic

quantities as density, pressure and temperature. This presupposes that such

thermodynamic quantities can be locally de¬ned to begin with.

For systems that are locally homogeneous and have only very small gra-

dients of thermodynamic parameters, averaging can be done over very large

numbers of particles. For the limit of averaging over an in¬nite number

of particles, thermodynamic quantities can be meaningfully de¬ned and we

expect the macroscopic equation to become exact. However, if the spatial

averaging procedure concerns a limited number of particles, thermodynamic

quantities need to be de¬ned also in terms of spatial averages and we ex-

pect the macroscopic equations to be only approximately valid and contain

unpredictable noise terms.

The situation is quite comparable to the averaging over “unimportant