8.6 Langevin dynamics 263

approximation that even preserves the slow dynamics of the system; if they

are not fast, a Markovian Langevin simulation will perturb the dynamics,

but still preserve the thermodynamics.

8.6 Langevin dynamics

In this section we start with the generalized Langevin equation (8.1), which

we ¬rst formulate in general coordinates. Then, in view of the discussion in

the previous section, we immediately reduce the equation to the memory-

free Markovian limit, while keeping the multidimensional formulation, and

check the ¬‚uctuation“dissipation balance. Subsequently we reduce also the

spatial complexity to obtain the simple Langevin equation.

8.6.1 Langevin dynamics in generalized coordinates

Consider a full Hamiltonian dynamical system with n degrees of freedom,

expressed in 2n generalized coordinates and momenta z = {q, p}. The mo-

menta are connected to the coordinates by the n — n mass tensor M (see

(15.16) on page 401):

™

p = Mq, (8.38)

with inverse

q = M’1 p.

™ (8.39)

The coordinates are distinguished in relevant coordinates q and irrelevant

coordinates q . We partition the inverse mass tensor (as is done in the

discussion on constraints, Section 17.9.3 on page 501) as

XY

M’1 = , (8.40)

YT Z

so that

XY p

™ ™

(q q )= . (8.41)

YT Z p

The next step is to ¬nd a Langevin equation of motion for q by averaging

over a canonical distribution for the double-primed subsystem:

™

q = X p + Yp , (8.42)

‚V

p =’

™ + friction + noise. (8.43)

‚q

264 Stochastic dynamics: reducing degrees of freedom

The canonical averaging is de¬ned as

A(z) exp[’βH(z)] dz

def

A = . (8.44)

exp[’βH(z)] dz

We recognize the r.h.s. of (8.43) as the Langevin force, similar to the carte-

sian Langevin force of (8.1), but the l.h.s. is not equal to the simple mi vi™

™

of the cartesian case. Instead we have, in matrix notation, and denoting q

by v,

t

‚V mf

d ’1

v(t) = ’ ’ ζ(„ )v(t ’ „ ) d„ + ·(t). (8.45)

X

dt ‚q 0

Here we have omitted the second term in (8.42) because it is an odd function

of p that vanishes on averaging. In principle, X can be a function of

primed coordinates, in which case the equations become di¬cult to solve.

But practice is often permissive, as we shall see.

Let us have a closer look at the matrix X. Its elements are (Fixman,

1979)

1 ‚qk ‚ql

·

Xkl = . (8.46)

mi ‚r i ‚r i

i

Typical “relevant” degrees of freedom are cartesian coordinates of “super-

atoms” that represents a cluster of real atoms: the radius vector of the

center of mass of a cluster of atoms, some linear combination of cartesian

coordinates that represent collective motions (principal modes or principal

components of a ¬‚uctuation matrix), etc. Other cases (e.g., reaction coordi-

nates) may involve distances between two particles or between two groups

of particles. The inverse mass matrix X is particularly simple in these

cases. For example, if the relevant coordinates are components of vectors

Rk = i ±ki r i , the inverse mass tensor is diagonal with constant terms

12

Xkl = δkl ±. (8.47)

mi ki

i

In the case that the relevant degree of freedom is a distance r12 between two

particles with mass m1 and m2 , the inverse mass tensor has one element

equal to (1/m1 ) + (1/m2 ), which is the inverse of the reduced mass of the

two particles. The evaluation is equivalent to the evaluation in the case

of constraints, treated in Section 17.9.3; see (17.201) on page 502. In all

these cases the inverse mass tensor is constant and does not depend on time-

dependent coordinates. We shall from hereon restrict ourselves to such cases

8.6 Langevin dynamics 265

and write M for the inverse of X, yielding the general Langevin equation:11

t

Mv = F (q ) ’ ζ(„ )v(t ’ „ ) d„ + ·(t),

s

™ (8.48)

0

™

v = q, (8.49)

M = X’1 . (8.50)

For X see (8.46). Of course, before applying this equation the user should

check that the inverse mass tensor is indeed time-independent. For degrees

of freedom involving angles this may not always be the case.

We note that the formulation given in (8.48) includes the simple case

that the relevant degrees of freedom are the cartesian coordinates of selected

particles; the matrix M is then simply the diagonal matrix of particle masses.

8.6.2 Markovian Langevin dynamics

We shall now consider the dissipation“¬‚uctuation balance for the case of

generalized coordinates including a mass tensor. But since we cannot guar-

antee the validity of the second dissipation“¬‚uctuation theorem for the time-

generalized equation (8.48), we shall restrict ourselves to the Markovian

multidimensional Langevin equation

Mv = Fs (q ) ’ ζv(t) + ·(t).

™ (8.51)

Here ζ is the friction tensor.

Dissipation“¬‚uctuation balance

Consider the generalized equipartition theorem, treated in Section 17.10 and

especially the velocity correlation expressed in (8.16) on page 503:

vvT = M’1 kB T. (8.52)

This equation is valid for the primed subsystem, where M is the inverse of

X as de¬ned above. In order to establish the relation between friction and

noise we follow the arguments of the previous section, leading to (8.35) on

page 262, but now for the multidimensional case.

The change due to friction is given by

d

vvT = vvT + vvT = 2 vvT

™ ™ ™

dt

= ’2M’1 ζ vvT . (8.53)

The notation M should not cause confusion with the same notation used for the mass tensor of

11

the full system, which would be obviously meaningless in this equation. Note that M used here