1549380323-Statistical Mechanics Theory and Molecular Simulation

506 Classical time-dependent statistical mechanics

the form of eqns. (13.2.1) so that we can apply the linear response formula in eqn.
(13.2.29). We consider a system with Cartesian coordinatesr 1 ,...,rN and conjugate
momentap 1 ,...,pN, described by a Hamiltonian of the usual form

H=

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN), (13.3.5)

whereU(r 1 ,...,rN) is the potential. The equation of motion forr ̇ican be understood
on simple physical grounds. Since the shearing force induces a flow fieldu(ri) at the
positionriof a particle, we expect each velocityr ̇ito have a contribution from this
flow field in addition to a contributionpi/mifrom the usual mechanical kinetic energy.
Thus, we can write the equation of motion forrias

r ̇i=

pi

mi

+u(ri) =

pi

mi

+γyiˆex

=

pi

mi

+γ(ri·ˆey)ˆex. (13.3.6)

If we averager ̇iover an equilibrium distribution such as the canonical distribution, the
quantity〈pi/mi〉vanishes, leaving only the overall flow componentγ〈yi〉ˆex, indicating
that, on average, the net flow of the system follows the flow field, as expected from
hydrodynamics. The form of the momentum equation must now be chosen such that
the overall phase space compressibility is zero. The only possible choice consistent with
this requirement is

p ̇i=Fi−γpyieˆx

=Fi−γ(pi·eˆy)ˆex, (13.3.7)

whereFi=−∂U/∂ri.
Eqns. (13.3.6) and (13.3.7) constitute the microscopic equations ofmotion for a
system subject to a shearing force and have the conserved energy

H′=

∑N

i=1

1

2 mi

(pi+miγyiˆex)^2 +U(r 1 ,...,rN). (13.3.8)

Eqns. (13.3.6) and (13.3.7) could, in fact, be used in a molecular dynamics calculation
to simulate the effect of a shearing force. In such a calculation, additional couplings to
a thermostat and possibly a barostat would be included in order generate a canonical
or isothermal-isobaric equilibrium distribution. We will describe such simulations in
detail in Section 13.5. For the purposes of the present analysis, wewill assume an
initial equilibrium distribution and focus on the influence of the external field terms.
In order to apply eqn. (13.3.4), we need to cast it into a form usefulfor microscopic
analysis. Recall from eqn. (5.7.1) that the microscopic estimator for thexycomponent
of the pressure tensor is given by the phase space function