1549380323-Statistical Mechanics Theory and Molecular Simulation

522 Classical time-dependent statistical mechanics

pi←−pi+

∆t

2

Fi

xi←−xi+ ∆t

[

pxi

mi

+γyi

]

+

∆t^2

2 mi

γpyi

yi←−yi+ ∆t

pyi

mi

zi←−zi+ ∆t

pzi

mi

Update Forces

pi←−pi+

∆t

2

Fi. (13.5.9)

The action of the operator exp(iLmNHC∆t/2) is similar to the standard Nos ́e–Hoover
chain operator exp(iLNHC∆t/2) discussed in Section 4.11 (cf. eqn. (4.11.17)). The
presence of theγ-dependent operator requires only a slight modification. The operator

exp

{

−

δα

2

pη 1

Q 1

pi·

∂

∂pi

}

in eqn. (4.11.17) is replaced by

exp

{

−

δα

2

[

pη 1

Q 1

pi·

∂

∂pi

+γpyi

∂

∂pxi

]}

due to theγ-dependent term iniLmNHC. The action of this operator can be derived
by solving the three coupled differential equations

p ̇xi=−

pη 1

Q 1

pxi−γpyi, p ̇yi=−

pη 1

Q 1

pyi, p ̇zi=−

pη 1

Q 1

pzi. (13.5.10)

The solutions forpyiandpziare

pyi(t) =pyi(0)e−pη^1 t/Q^1

pzi(t) =pzi(0)e−pη^1 t/Q^1. (13.5.11)

Substituting the solution forpyi(t) into the equation forpxigives

p ̇xi=−

pη 1

Q 1

pxi−γpyi(0)e−pη^1 t/Q^1. (13.5.12)

This equation can be solved using exp[pη 1 t/Q 1 ] as an integrating factor, which yields
the solution
pxi(t) = [pxi(0)−γtpyi(0)] e−pη^1 t/Q^1. (13.5.13)

Evaluating the solutions forpxi(t),pyi(t), andpzi(t) att=δα/2 then gives the mod-
ification to the Suzuki-Yoshida Nos ́e–Hoover chain evolution necessary for treating
planar Couette flow.