1549380323-Statistical Mechanics Theory and Molecular Simulation

Non-Hamiltonian statistical mechanics 191

quantity. Unfortunately, this is not the typical situation. In the absence of external
forces, Newton’s third law requires that

∑N

i=1Fi= 0, which leads to an additional
conservation law

Peη=K, (4.9.27)

whereP =

∑N

i=1piis the center-of-mass momentum of the system andKis an
arbitrary constant vector inddimensions. When this additional conservation law
is present, the Nos ́e–Hoover equations do not generate the correct distribution (see
Problem 4.3). Fig. 4.9 illustrates the failure of the Nos ́e–Hoover equations for a
single free particle in one dimension. The distributionf(p) should be a Gaussian
f(p) = exp(−p^2 / 2 mkT)/

√

2 πmkT, which it clearly is not. Finally, Fig. 4.10 shows
that the Nos ́e–Hoover equations also fail for a simple harmonic oscillator, for which
eqn. (4.9.27) does not hold. Problem 4.4 suggests that an additionalconservation law
different from eqn. (4.9.27) is a possible culprit in the failure of the Nos ́e–Hoover
equations for the harmonic oscillator.

-4 -2 0 2 4

p

0

0.5

1

1.5

f(

p)

Fig. 4.9 Momentum distribution obtained by integrating the Nos ́e–Hoover equations
p ̇= −(pη/Q)p,η ̇ = pη/Q,p ̇η = p^2 /m−kT for a free particle withm = 1,Q = 1,
kT = 1,p(0) = 1,η(0) = 0,pη(0) = 1. The solid line is the distribution obtained from
the simulation (see Problem 4.3), and the dashed line is the expected analytical distribution
f(p) = exp(−p^2 / 2 mkT)/

√

2 πmkT.