1549380323-Statistical Mechanics Theory and Molecular Simulation

194 Canonical ensemble

Here, we have introduced the variableηc=

∑M

j=2ηjas a convenience since this par-
ticular combination of theηvariables comes up frequently. From the compressibility,
we see that the phase space metric is

√

g= exp [dNη 1 +ηc]. (4.10.5)

Using eqns. (4.10.3) and (4.10.5), proving that the Nos ́e–Hoover equations generate
a canonical ensemble is analogous to eqns. (4.9.24) to (4.9.26) for the Nos ́e–Hoover
equations and, therefore, will not be repeated here but left as anexercise at the end
of the chapter (see problem 4.3).

∑An important property of the Nos ́e–Hoover chain equations is thefact that when
N
i=1Fi= 0, the equations of motion still generate a correct canonical distribution in
all variables except the magnitude of the center-of-mass momentumP(see problem
3). When there are no external forces, eqn. (4.9.27) becomes

K=Peη^1. (4.10.6)

In order to illustrate this for the simple cases considered in Figs. 4.9 and 4.10, Fig. 4.11
shows the momentum distribution of the one-dimensional free particle coupled to a
Nos ́e–Hoover chain, together with the correct canonical distribution. The figure shows
that the correct distribution is, indeed, obtained. In addition, Fig.4.12 also shows

-4 -2 0 2 4

p

0

0.1

0.2

0.3

0.4

0.5

f(p

)

Fig. 4.11Momentum distribution obtained by integrating the Nos ́e–Hoover chain equations
for a free particle withm= 1,Q= 1,kT= 1. Here,p(0) = 1,ηk(0) = 0,pηk(0) = 1. The
solid line is the distribution obtained from the simulation(see Problem 4.3), and the circles
are the correct distributionf(p) = exp(−p^2 / 2 mkT)/

√

2 πmkT.