zi ‚zj exp[’βH] dz

= kB T δij .

exp[’βH] dz

Consider the following equality

‚e’βH

‚H ’βH ‚

zi e’βH ’ δij e’βH .

’βzi e = zi =

‚zj ‚zj ‚zj

Integrating over phase space dz, the ¬rst term on the r.h.s. drops out after

partial integration, as the integrand vanishes at the boundaries. Hence

‚H

’β zi = ’δij ,

‚zj

which is what we wish to prove.

™

Applying this theorem to zi = pi = (Mq)i ; zj = pj we ¬nd

™™

(Mq)i qj = δij , (17.204)

or, in matrix notation:

qqT = M’1 kB T.

™™ (17.205)

This is the classical equipartition: for cartesian coordinates (diagonal mass

tensor) the average kinetic energy per degree of freedom equals 1 kB T and

2

velocities of di¬erent particles are uncorrelated. For generalized coordinates

the kinetic energy per degree of freedom 1 pi qi still averages to 1 kB T , and

™

2 2

the velocity of one degree of freedom is uncorrelated with the momentum of

any other degree of freedom.

Another interesting special case is obtained when we apply the theorem

to zi = qi ; zj = qj :

qi pj = ’ qi Fj = kB T δij .

™ (17.206)

504 Review of statistical mechanics

For j = i this recovers the virial theorem (see (17.132) on page 485):

n

1 1

Ξ=’ qi Fi = nkB T = Ekin . (17.207)

2 2

i=1

Including the cases j = i means that the virial tensor is diagonal in equilib-

rium, and the average pressure is isotropic.

Exercises

17.1 Show for the canonical ensemble, where Q is a function of V and β,

that U = ’‚ ln Q/‚β. Show that this equation is equivalent to the

Gibbs“Helmholtz equation, (16.32).

17.2 If the pressure p is de¬ned as the ensemble average of ‚Ei /‚V , then

show that for the canonical ensemble p = β ’1 ‚ ln Q/‚V .

17.3 Derive (17.63) by considering how many points there are in k-space

in a spherical shell between k and k+dk. Transform this to a function

of µ.

17.4 Derive the quantum expressions for energy, entropy and heat capac-

ity of the harmonic oscillator. Plot the heat capacity for the quantum

and classical case.

17.5 Show that the quantum correction to the free energy of a harmonic

oscillator (17.109) equals the ¬rst term in the expansion of the exact

Aqu ’ Acl in powers of ξ.

17.6 See (17.133). Show that S r dS = 3V , with the integral taken over

a closed surface enclosing a volume V . Transform from a surface to

a volume integral over the divergence of r.

Carry out the partial di¬erentiation of A(V, T ) = ’kB T ln Q with

17.7

respect to volume to obtain the isotropic form of (17.138). Assume

a cubic L — L — L lattice and use scaled coordinates r/L.

17.8 Prove that the rotational kinetic energy of a harmonic homonuclear

diatomic molecule (2— 1 mvrot ) equals the contribution to the virialof

2

2

the centripetal harmonic force (2 — mvrot /( 1 d)).

2

2

17.9 Compute the matrix Z for a triatomic molecule with constrained

bond lengths.

18

Linear response theory

18.1 Introduction

There are many cases of interest where the relevant question we wish to

answer by simulation is “what is the response of the (complex) system to

an external disturbance?” Such responses can be related to experimental

results and thus be used not only to predict material properties, but also

to validate the simulation model. Responses can either be static, after a

prolonged constant external disturbance that drives the system into a non-

equilibrium steady state, or dynamic, as a reaction to a time-dependent

external disturbance. Examples of the former are transport properties such

as the heat ¬‚ow resulting from an imposed constant temperature gradient,

or the stress (momentum ¬‚ow) resulting from an imposed velocity gradient.

Examples of the latter are the optical response to a speci¬c sequence of laser

pulses, or the time-dependent induced polarization or absorption following

the application of a time-dependent external electric ¬eld.

In general, responses can be expected to relate in a non-linear fashion

to the applied disturbance. For example, the dielectric response (i.e., the

polarization) of a dipolar ¬‚uid to an external electric ¬eld will level o¬ at

high ¬eld strengths when the dipoles tend to orient fully in the electric ¬eld.

The optical response to two laser pulses, 100 fs apart, will not equal the

sum of the responses to each of the pulses separately. In such cases there

will not be much choice other than mimicking the external disturbance in

the simulated system and “observing” the response. For time-dependent

responses such simulations should be repeated with an ensemble of di¬erent

starting con¬gurations, chosen from an equilibrium distribution, in order to

obtain statistically signi¬cant results that can be compared to experiment.

In this chapter we will concentrate on the very important class of linear

responses with the property that the response to the sum of two disturbances

505

506 Linear response theory

equals the sum of responses to each of the disturbances. To this class belong

all responses to small disturbances in the linear regime; these are then pro-

portional to the amplitude of the disturbance. The proportionality constant

determines transport coe¬cients such as viscosity, thermal conductivity and

di¬usion constant, but also dielectric constant, refractive index, conductiv-

ity and optical absorption. Since the decay of a small perturbation, caused

by an external disturbance, is governed by the same equations of motion

that determine the thermal ¬‚uctuations in the equilibrium system, there is

a relation between the decay function of an observable of the system after

perturbation and the time-correlation function of spontaneous ¬‚uctuations

of a related variable. In the next sections we shall elaborate on these rela-

tions.

In Section 18.2 the general relations between an external disturbance and

the resulting linear response will be considered both in the time and fre-

quency domain, without reference to the processes in the system that cause

the response. In Section 18.3 the relation between response functions and

the time correlation function of spontaneously ¬‚uctuating quantities will

be considered for a classical system of particles that interact according to

Hamilton™s equations of motion.

18.2 Linear response relations

In this section we consider our system as a black box, responding to a

disturbance X(t) with a response Y (t) (Fig. 18.1). The disturbance is an

external force or ¬eld acting on the system, such as an electric ¬eld E(t),

and the response is an observable of the system, for example, a current

density j(t) resulting from the disturbance E(t). Both X and Y may be

vectorial quantities, in which case their relations are speci¬ed by tensors,