be computed as the inverse of X.

266 Stochastic dynamics: reducing degrees of freedom

In the ¬rst line we have used the symmetry of vvT . The change in a step

”t due to noise is given by

t+”t t+”t

T ’1

· T (t ) dt M’1

” vv = ·(t ) dt

M

t t

= M’1 AM’1 ”t. (8.54)

The matrix A is the noise correlation matrix:

·i (t0 )·j (t0 + t) = Aij δ(t). (8.55)

We used the symmetry of M’1 . Balancing the changes due to friction and

noise, it is seen that friction and noise are related by

A = 2kB T ζ. (8.56)

This is the multidimensional analog of (8.35). It appears that the noise

terms for the di¬erent degrees of freedom are not independent of each other

when the friction tensor is not diagonal, i.e., when the velocity of one degree

of freedom in¬‚uences the friction that another degree of freedom under-

goes. We see from (8.56) that the friction tensor must be symmetric, as the

Markovian noise correlation is symmetric by construction.

It is also possible, and for practical simulations more convenient, to ex-

press the noise forces as linear combinations of independent normalized white

0

noise functions ·k (t) with the properties

0

·k (t) = 0, (8.57)

·k (t0 )·l0 (t0 + t) = δkl δ(t),

0

(8.58)

Bik ·k (t) or · = B· 0 .

0

·i (t) = (8.59)

k

It now follows that

BBT = A = 2ζkB T. (8.60)

In order to construct the noise realizations in a simulation, the matrix B

must be solved from this equation, knowing the friction matrix. The solu-

tion of this square-root operation is not unique; a lower trangular matrix is

obtained by Choleski decomposition (see Engeln-M¨llges and Uhlig, 1996,

u

for algorithms).

Simple Langevin dynamics

The generalized or the Markovian Langevin equation can be further approx-

imated if the assumption can be made that the friction acts locally on each

8.6 Langevin dynamics 267

degree of freedom without mutual in¬‚uence. In that case the friction tensor

is diagonal and the simple Langevin equation is obtained:

(Mv)i = Fis ’ ζi vi (t) + ·(t),

™ (8.61)

with

·i (t0 )·j (t0 + t) = 2kB T ζi δij δ(t). (8.62)

Although there is no frictional coupling, these equations are still coupled if

the mass tensor M is not diagonal. In the common diagonal case the l.h.s.

is replaced by

™

(Mv)i = mi vi . (8.63)

In the ultimate simpli¬cation with negligible systematic force, as applies

to a solute particle in a dilute solution, the simple pure Langevin equation

is obtained:

mv = ’ζv + ·(t).

™ (8.64)

As this equation can be exactly integrated, the properties of v can be calcu-

lated; they serve as illustration how friction and noise in¬‚uence the velocity,

but are not strictly valid when there are systematic forces as well. The

solution is

1 t ’ζ„ /m

’ζt/m

·(t ’ „ ) d„,

v(t) = v(0)e + e (8.65)

m0

which, after a su¬ciently long time, when the in¬‚uence of the initial velocity

has died out, reduces to

∞

1

e’ζ„ /m ·(t ’ „ ) d„.

v(t) = (8.66)

m 0

We see from (8.65) that in the absence of noise the velocity decays expo-

nentially with time constant m/ζ, and we expect from the ¬rst dissipation“

¬‚uctuation theorem (page 259) that the velocity autocorrelation function

will have the same exponential decay. This can be shown directly from

(8.66) in the case that ·(t) is a white noise with intensity 2ζkB T . It follows

that, if ·(t) is stationary, v(t) is also stationary; when the noise intensity is

2ζkB T , the variance of the velocity is kB T /m. Note that it is not necessary

to specify the distribution function of the random variable.

We can also compute the probability distribution ρ(v) for the velocity

when equilibrium has been reached. To do this we need an equation for

ρ(v, t) as it is generated by the stochastic process de¬ned by (8.64). Such

268 Stochastic dynamics: reducing degrees of freedom

equations are Fokker“Planck equations,12 of which we shall see more exam-

ples in the following section. In this one-dimensional case the Fokker“Planck

equation is

ζkB T ‚ 2 ρ

‚ρ ζ‚

= (ρv) + . (8.67)

m2 ‚t2

‚t m ‚v

The equation is an expression of the conservation of total probability, leading

to a continuum equation ‚ρ/‚t = ’∇v (J), where J is the probability ¬‚ux

consisting of a drift term due to friction and a di¬usional term due to noise.

The di¬usional term follows from the fact that

‚ρ 1 ‚ρ

=B (8.68)

‚t 2 ‚t

implies that (see exercise 8.3)

(”v)2 = B”t (8.69)

with the variance of the velocity ¬‚uctuation given by (2ζkB T /m2 )”t (see

(8.31), (8.32) and (8.35) on page 262).

The equilibrium case (‚ρ/‚t = 0) has the solution

mv 2

m

exp ’

ρ(v) = , (8.70)

2πkB T 2kB T

which is the Maxwell distribution.

8.7 Brownian dynamics

If systematic forces are slow, i.e., when they do not change much on the

time scale „c = m/ζ of the velocity correlation function, we can average the

Langevin equation over a time ”t > „c . The average over the inertial term

Mv becomes small and can be neglected; as a result the acceleration no

™

longer ¬gures in the equation. We obtain non-inertial dynamical equations:

0 ≈ Fi [q(t)] ’ ζij vj (t) + ·i (t), (8.71)

j

or, in matrix notation:

ζv = F + ·(t), (8.72)

yielding the Brownian equation for the velocities:

v = q = ζ ’1 F + ζ ’1 B· 0 (t),

™ (8.73)

12 See van Kampen (1981) for an extensive treatment of the relation between stochastic equations

and the corresponding Fokker“Planck equations.

8.8 Probability distributions and Fokker“Planck equations 269