1549380323-Statistical Mechanics Theory and Molecular Simulation

Nos ́e–Hoover chains 195

-4 -2 0 2 4

x

-4

-2

0

2

4

p

-4 -2 0 2 4

x

-4

-2

0

2

4

p

-4 -2 0 2 4

p

0

0.1

0.2

0.3

0.4

0.5

f(

p)

-4 -2 0 2 4

x

0

0.1

0.2

0.3

0.4

0.5

f(

x)

(a) (b)

(c) (d)

Fig. 4.12 Phase space and distribution functions obtained by integrating the Nos ́e–Hoover
chain equations for a harmonic oscillator withm= 1,ω= 1,Q= 1,kT = 1,x(0) = 0,
p(0) = 1,ηk(0) = 0,pη 1 (0) =pη 3 (0) = 1,pη 2 (0) =pη 4 (0) =−1. (a) shows the phase space
pvs.xindependent ofηandpη, (b) shows the phase space forpη 1 ,pη 2 ∈[−ǫ,ǫ], where
ǫ= 0.001, (c) and (d) show distributionsf(p) andf(x) obtained from the simulation (solid
line) together with the correct canonical distributions (circles).

the physical phase space and position and momentum distributions for the harmonic
oscillator coupled to a Nos ́e–Hoover chain. Again, it can be seen that the correct
canonical distribution is generated, thereby solving the failure of the Nos ́e–Hoover
equations. By working through Problem 4.3, it will become clear what mechanism
is at work in the Nos ́e–Hoover chain equations that leads to the correct canonical
distributions and why, therefore, these equations are recommended over the Nos ́e–
Hoover equations.
As one final yet important note, consider rewriting eqns. (4.10.1) such that each
particle has itsownNos ́e–Hoover chain thermostat. This would be expressed in the
equations by adding an additional index to the thermostat variables:

r ̇i=

pi

mi

p ̇i=Fi−

pη 1 ,i

Q 1

pi

η ̇j,i=

pηj,i

Qj

j= 1,...,M