1549380323-Statistical Mechanics Theory and Molecular Simulation

Green–Kubo relations 507

Pxy(x) =pxy(r,p) =

1

V

∑N

i=1

[

(pi·ˆex)(pi·ˆey)

mi

+ (ri·ˆex)(Fi·ˆey)

]

. (13.3.9)

The pressure-tensor componentPxyappearing in eqn. (13.3.4) is the average of this
estimator over the nonequilibrium ensemble once a steady state hasbeen achieved,
i.e., after transient behavior has died away in the presence of the external field. Thus,
we can rewrite eqn. (13.3.4) as

η=−lim

t→∞

〈Pxy〉t

γ

. (13.3.10)

We will compute the nonequilibrium average〈Pxy〉tusing eqn. (13.2.27). In order to
use linear response theory, we must first compute the dissipative flux. From eqns.
(13.3.6) and (13.3.7), we identifyCi(r,p) andDi(r,p) as

Ci(r,p) =γ(ri·eˆy)ˆex

Di(r,p) =−γ(pi·ˆey)ˆex. (13.3.11)

Also,Fe(t) = 1. In Cartesian coordinates with a Hamiltonian given by eqn. (13.3.5),
eqn. (13.2.17) for the dissipative flux becomes

j(r,p) =

∑N

i=1

[

Ci(r,p)·Fi−Di(r,p)·

pi

mi

]

. (13.3.12)

Substituting eqn. (13.3.11) into eqn. (13.3.12) gives

j(r,p) =γ

∑N

i=1

[

(ri·ˆey)(ˆex·Fi) + (pi·ˆey)

(

ˆex·

pi

mi

)]

=γVPxy. (13.3.13)

Substituting eqn. (13.3.13) into eqn. (13.2.27) yields for〈Pxy〉t

〈Pxy〉t=〈Pxy〉−βγV

∫t

0

ds〈Pxy(0)Pxy(t−s)〉. (13.3.14)

Finally, introducing the change of variablesτ=t−sinto the integral gives

〈Pxy〉t=〈Pxy〉−βγV

∫t

0

dτ〈Pxy(0)Pxy(τ)〉. (13.3.15)

Since the first term involves the average ofPxyover an equilibrium ensemble, which
describes an isotropic system, it is not difficult to see, again on purelyphysical grounds,
that〈Pxy〉= 0; however, this can also be proved analytically from the virial theorem