1549380323-Statistical Mechanics Theory and Molecular Simulation

Molecular dynamics 245

η ̇j=

pηj

Qj

, ξ ̇j=

pξj

Q′j

p ̇ηj=Gj−

pηj+1

Qj+1

pηj

p ̇ηM=GM

p ̇ξj=G′j−

pξj+1

Q′j+1

pξj

p ̇ξM=G′M, (5.10.2)

whereP(int)is the internal pressure tensor, whose components are given by eqn. (5.7.1)
or (5.7.29),Iis the 3×3 identity matrix, the thermostat forcesGjare given by eqns.
(4.11.6), and

G′ 1 =

Tr

[

pTgpg

]

Wg

−d^2 kT

G′j=

p^2 ξj− 1

Q′j− 1

−kT. (5.10.3)

The matrixpTg is the transpose ofpg. Eqns. (5.10.2) have the conserved energy

H′=

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN) +

Tr

[

pTgpg

]

2 Wg

+Pdet[h]

+

∑M

j=1

[

p^2 ηj

2 Qj

+

p^2 ξj

2 Q′j

]

+NfkTη 1 +d^2 kTξ 1 +kT(ηc+ξc). (5.10.4)

Furthermore, if

∑

i
F ̃i= 0, i.e., there are no external forces on the system, then when
a global thermostat is used on the particles, there is an additional vector conservation
law of the form
K=hP{det [h]}^1 /Nfeη^1 , (5.10.5)

whereP=

∑

ipiis the center-of-mass momentum.
We will now proceed to show that eqns. (5.10.2) generate the ensemble described
by eqn. (5.6.7). For the purpose of this analysis, we will assume thatthere are no
constraints on the system, so thatNf=dNand that

∑

iFi^6 = 0. The slightly more
complex case that arises when

∑

iFi= 0 will be left for the reader to ponder in
Problem 5.6.
We start by calculating the compressibility of eqns. (5.10.2). Since the matrix mul-
tiplications give rise to a mixing among the components of the position and momentum
vectors, it is useful to write the equations of motion forri,pi,h, andpgexplicitly in
terms of their Cartesian components:

r ̇i,α=

pi,α

mi

+

∑

β

pg,αβ

Wg

ri,β