1549380323-Statistical Mechanics Theory and Molecular Simulation

Green–Kubo relations 509

Eqn. (13.3.18) indicates that the external force arises from an external potential field
of the form
φ(x) =−fx (13.3.19)

since the new term in the Hamiltonian is separable and of the form

∑N

i=1φ(xi). The
potential field has a nonzero gradient given by

∇φ=−fˆex. (13.3.20)

This external potential field causes particles to drift “down” the potential gradient,
which is in the positivexdirection in this example. This drift causes a concentration
gradient∇cto develop. In general, the concentrationc(x) of particles in an external
potentialφ(x) follows a Boltzmann distributionc(x) =c(x= 0) exp(−βφ(x)), where
we takeφ(x= 0) = 0. Assuming thatφ(x) is a weak perturbation and that the con-
centration is defined such thatc(x= 0) = 1 (the standard state), then this expression
can be linearized to givec(x)≈ 1 −βφ(x), whose gradient is

∇c=−

1

kT

∇φ=

f

kT

ˆex

∂c

∂x

=−

1

kT

∂φ

∂x

=

f

kT

(13.3.21)

in a direction opposite to that of the potential gradient. Again, because the concen-
tration is linear inx, the gradient is constant.
The drift of particles in a given direction can be quantified in terms of adrift
velocity averaged over all the particles, which can be described by the following phase
space function:

ux(r,p) =

1

N

∑N

i=1

x ̇i=

1

N

∑N

i=1

pi

mi

·ˆex. (13.3.22)

The average ofux(r,p) over the nonequilibrium ensemble, once a steady state has
been achieved, is denotedJx, the average particle current. Thus,Jxis given by

Jx= lim

t→∞

〈ux〉t. (13.3.23)

The particle currentJxcan be related to the concentration gradient∂c/∂xusing Fick’s
law of diffusion. The latter states that

Jx=D

∂c

∂x

. (13.3.24)

The constant of proportionality is thediffusion constant, denotedD, which has units
of (Length)^2 /Time. Substituting eqn. (13.3.21) into eqn. (13.3.24) gives

Jx=−

D

kT

∂φ

∂x

=

D

kT

f. (13.3.25)

Thus, the diffusion constant can be written in terms of nonequilibriumensemble av-
erage as