d

m C v (t) = ’ ζ(„ )C v (t ’ „ ) d„. (8.24)

dt 0

7 Including v0 in the response means that ¦ is normalized: ¦(0) = 1.

260 Stochastic dynamics: reducing degrees of freedom

Given the friction kernel ζ(„ ) and the initial value C v (0) = v 2 , this equa-

tion determines the equilibrium correlation function C v (t). But this equa-

tion is equivalent to the corresponding equation (8.22) for ¦(t), from which

it follows that

C v (t) = v 2 ¦(t). (8.25)

The ¬rst ¬‚uctuation“dissipation theorem has a solid basis; it applies in gen-

eral to small deviations from equilibrium, also for systems that include sys-

tematic forces (see Section 18.3 on page 511). It is the basis for the derivation

of transport properties from equilibrium ¬‚uctuations. However, it does not

provide the link between friction and noise needed for the implementation

of Langevin dynamics.

The second ¬‚uctuation“dissipation theorem states that

·(t0 )·(t0 + t) = m v 2 ζ(t) = kB T ζ(t). (8.26)

This theorem provides the proper connection between friction and noise,

but it stands on much weaker grounds than the ¬rst theorem. It can be

rigorously proven for a pure Langevin equation without systematic force.

The proof uses Laplace transforms or one-sided Fourier transforms and rests

on the derivation of the stationary velocity autocorrelation function, given

the noise correlation function, which must equal the solution of (8.24). We

refer the reader for this proof to the literature, where it can be found in

several places; a readable discussion is given in the ¬rst chapter of Kubo et

al. (1985). When systematic non-linear forces are present (as is the case

in all simulations of real systems), the theorem can no longer be proven to

be generally valid. Various special cases involving harmonic forces and heat

baths consisting of collections of harmonic oscillators have been considered,8

and modi¬cations for the general case have been proposed.9 While the

matter appears not to be satisfactorily settled, our recommendation is that

time-dependent friction kernels should not be used in cases when intrinsic

relaxation times, determined by the systematic forces, are of the same order

as the characteristic times of the friction kernels.

8 The harmonic-oscillator heat bath was pioneered by Zwanzig (1973) and extended by Cohen

(2002); Hernandez (1999) considered the projection of non-equilibrium Hamiltonian systems.

Adelman and Doll (1974) simpli¬ed Zwanzig™s approach for application to atomic collisions

with a solid surface.

9 Ciccotti and Ryckaert (1981) separate the systematic force and obtain a modi¬ed friction and

noise; Bossis et al. (1982) show that the e¬ect of the systematic force is a modi¬cation of the

second ¬‚uctuation“dissipation theorem by the addition of an extra term equal to the correlation

of velocity and systematic force. McDowell (2000) considers a chain of heat baths and concludes

that an extra bias term should be added to the random force.

8.5 The ¬‚uctuation“dissipation theorem 261

However, the memory-free combination of time-independent friction and

white noise does yield consistent dynamics, also in the presence of systematic

forces, with proper equilibrium ¬‚uctuations. This memoryless approxima-

tion is called a Markovian process,10 and we shall call the corresponding

equation (which may still be multidimensional) the Markovian Langevin

equation.

Consider, ¬rst for the simple one-dimensional case, the change in kinetic

energy due to a Markovian friction force and a white-noise stochastic force.

The equation of motion is

mv = F (t) ’ ζv + ·(t),

™ (8.27)

with

·(t0 )·(t0 + t) = A· δ(t), (8.28)

where A· is the intensity of the noise force. Consider the kinetic energy

K(t) = 1 mv 2 . The friction force causes a decrease of K:

2

dK 2ζ

= mv v = ’ζv 2 = ’

™ K. (8.29)

dt m

friction

The stochastic term causes an increase in K. Consider a small time step

”t, which causes a change in velocity:

t+”t

m”v = ·(t ) dt . (8.30)

t

The change in K is

1 1

(”K)noise = m[(v + ”v)2 ’ v 2 ] = mv”v + m(”v)2 . (8.31)

2 2

We are interested in the average over the realizations of the stochastic pro-

cess. The ¬rst term on the r.h.s. vanishes as ”v is not correlated with v.

The second term yields a double integral:

t+”t t+”t

1 1

”K = dt dt ·(t )·(t ) = A· ”t; (8.32)

noise

2m 2m

t t

therefore the noise causes on average an increase of K:

A·

dK

= . (8.33)

dt 2m

noise

Both of these changes are independent of the systematic force. They balance

10 A Markov process is a discrete stochastic process with transition probabilities between succes-

sive states that depend only on the properties of the last state, and not of those of previous

states.

262 Stochastic dynamics: reducing degrees of freedom

on average when K = A· /(4ζ). Using the equilibrium value at a reference

temperature T0 for one degree of freedom:

1

K = kB T0 , (8.34)

2

it follows that a stationary equilibrium kinetic energy is obtained for

A· = 2ζkB T0 . (8.35)

If the initial system temperature T deviates from T0 = A/(2ζkB ), it will

decay exponentially to the reference temperature T0 set by the noise, with

a time constant m/2ζ:

dT 2ζ

= ’ (T ’ T0 ). (8.36)

dt m

Thus the added friction and noise stabilize the variance of the velocity ¬‚uc-

tuation and contribute to the robustness of the simulation. The ¬‚ow of

kinetic energy into or out of the system due to noise and friction can be

considered as heat exchange with a bath at the reference temperature. This

exchange is independent of the systematic force and does not depend on

the time dependence of the velocity autocorrelation function. It is easy to

see that the latter is not the case for a time-dependent (non-Markovian)

friction: (8.29) then reads

∞

dK

= m v(t)v(t) = ’ ζ(„ ) v(t)v(t ’ „ ) d„,

™ (8.37)

dt 0

friction

which clearly depends on the velocity correlation function, which in turn

depends on the behavior of the systematic force.

The whole purpose of simplifying detailed dynamics by Langevin dynam-

ics is reducing fast degrees of freedom to a combination of friction and noise.

When these “irrelevant” degrees of freedom indeed relax fast with respect

to the motion of the relevant degrees of freedom, they stay near equilibrium

under constrained values of the “relevant” degrees of freedom and in fact

realize a good approximation of the constrained canonical ensemble that is

assumed in the derivation of the systematic force (8.13) and that allows the

simpli¬cation without loosing thermodynamic accuracy. “Fast” means that

the correlation time of the force due to the “irrelevant” degrees of freedom

(the frictional and random force) is short with respect to the relaxation time

within the “relevant” system, due to the action of the systematic force. The

latter is characterized by the velocity correlation function in the absence of

friction and noise. If the forces from the “irrelevant” degrees of freedom