1549380323-Statistical Mechanics Theory and Molecular Simulation

Isokinetic ensemble 203

not canonical is of little consequence. Nevertheless, since the momentum- and position-
dependent parts of the distribution are separable, the isokinetic partition function can
be trivially related to the true canonical partition function by

Q(N,V,T) =

(1/N!)VN(2πmkT/h^2 )^3 N/^2

(1/N!)(K 0 /K)(1/Γ(3N/2))VN(2πmK/h^2 )^3 N/^2

Q(N,V,T,K)

=

Qideal(N,V,T)

Ωideal(N,V,K)

Q(N,V,T,K), (4.12.2)

where ΩidealandQidealare the ideal gas partition functions in the microcanonical and
canonical ensembles, respectively.
Equations of motion for the isokinetic ensemble were first written down by D. J.
Evans and G. P. Morriss (1980) by applying Gauss’s principle of least constraint. The
equations of motion are obtained by imposing a kinetic-energy constraint

∑N

i=1

mir ̇^2 i=

∑N

i=1

p^2 i

mi

= 2K (4.12.3)

on the Hamiltonian dynamics of the system. According to the discussion in Section 1.9,
eqn. (4.12.3) is a nonholonomic constraint, but one that can be expressed in differential
form. Thus, the Lagrangian form of the equations of motion is

d

dt

(

∂L

∂r ̇i

)

−

∂L

∂ri

=αmir ̇i, (4.12.4)

which can also be put into Hamiltonian form

r ̇i=

pi

mi

p ̇i=Fi−αpi. (4.12.5)

Here,α is the single Lagrange multiplier needed to impose the constraint. Using
Gauss’s principle of least constraint gives a closed-form expressionforα. We first
differentiate eqn. (4.12.3) once with respect to time, which yields

∑N

i=1

pi

mi

·p ̇i= 0. (4.12.6)

Thus, substituting the second of eqns. (4.12.5) into eqn. (4.12.6) gives

∑N

i=1

pi

mi

·[Fi−αpi], (4.12.7)

which can be solved forαgiving