1549380323-Statistical Mechanics Theory and Molecular Simulation

508 Classical time-dependent statistical mechanics

(see Problem 13.8). Thus, taking the limitt→∞, an expression for the coefficient of
shear viscosity is obtained:

η=

V

kT

∫∞

0

dτ〈Pxy(0)Pxy(τ)〉=

V

kT

∫∞

0

dt〈Pxy(0)Pxy(t)〉, (13.3.16)

where we have renamed the integration variable “t” instead of “τ” in the final expres-
sion to comply with standard notation. Eqn. (13.3.16) is an example ofaGreen–Kubo
relation, which expresses a transport coefficient in terms of the integral of an equilib-
rium time correlation function. In this case, the coefficient of shearviscosity is given as
the time integral of the autocorrelation function of thexycomponent of the pressure
tensor. Interestingly, despite the fact that〈Pxy〉= 0, the equilibrium time correlation
function ofPxydoes not vanish, suggesting that, even in equilibrium, there are short-
lived anisotropic fluctuations that causePxyto have a nonzero correlation time. The
length of the correlation time depends entirely on the details of the potential and the
external thermodynamic conditions of the equilibrium ensemble.

13.3.2 The diffusion constant

The diffusion constant is a measure of the tendency of particles to drift through a
system under the action of a constant external force (see Fig. 13.3). A microscopic

x

Jx

Fig. 13.3 A model diffusion experiment: A fluid is subject to an externalforcef in the
positivexdirection, which gives rise to a particle currentJx.

description of diffusion can be provided by the simple addition of a constant force of
magnitudefto the negative gradient of the potential as

r ̇i=

pi

mi

p ̇i=Fi+fˆex, (13.3.17)

where we have arbitrarily chosen the constant force to act in the positionxdirection.
Eqns. (13.3.17) conserve the total energy:

H′=

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN)−f

∑N

i=1

xi. (13.3.18)