1549380323-Statistical Mechanics Theory and Molecular Simulation

Nos ́e–Hoover chains 193

equations of motion can be expressed as

r ̇i=

pi

mi

p ̇i=Fi−

pη 1

Q 1

pi

η ̇j=

pηj

Qj

j= 1,...,M

p ̇η 1 =

[N

∑

i=1

p^2 i

mi

−dNkT

]

−

pη 2

Q 2

pη 1

p ̇ηj=

[

p^2 ηj− 1

Qj− 1

−kT

]

−

pηj+1

Qj+1

pηj j= 2,...,M− 1

p ̇ηM=

[

p^2 ηM− 1

QM− 1

−kT

]

(4.10.1)

(Martynaet al., 1992). Eqns. (4.10.1) are known as theNos ́e–Hoover chain equa-
tions. These equations ensure that the firstM−1 thermostat momentapη 1 ,...,pηM− 1
have the correct Maxwell-Boltzmann distribution. Note that forM= 1, the equa-
tions reduce to the simpler Nos ́e–Hoover equations. However, unlike the Nos ́e–Hoover
equations, which are essentially Hamiltonian equations in noncanonical variables, the
Nos ́e–Hoover chain equations have no underlying Hamiltonian structure, meaning no
canonical variables exist that transform eqns. (4.10.1) into a Hamiltonian system.
Concerning the parametersQ 1 ,...,QM, Martynaet al.(1992) suggested that an
optimal choice for these is

Q 1 =dNkTτ^2

Qj=kTτ^2 , j= 2,...,M, (4.10.2)

whereτis a characteristic time scale in the system. Since this time scale might not be
known explicitly, in practical molecular dynamics calculations, a reasonable choice is
τ≥20∆t, where ∆tis the time step.
In order to analyze the distribution of the physical phase space generated by eqns.
(4.10.1), we first identify the conservation laws. If

∑N

i=1Fi^6 = 0, then the equations of
motion conserve

H′=H(r,p) +

∑M

j=1

p^2 ηj

2 Qj

+dNkTη 1 +kT

∑M

j=2

ηj, (4.10.3)

which, in general, will be the only conservation law satisfied by the system. Next, the
compressibility of eqns. (4.10.1) is

∇x· ̇x =−dN

pη 1

Q 1

−

∑M

j=2

pηj

Qj

=−dNη ̇ 1 −η ̇c. (4.10.4)