1549380323-Statistical Mechanics Theory and Molecular Simulation

Jarzynski’s equality 329

equations have an associated phase space metric

√

g = e^3 Nη. Note that the heat
term in eqn. (8.4.4) does not vanish for the Nos ́e–Hoover equations because energy is
exchanged between the physical system and the thermostat. Nevertheless, the work in
eqn. (8.4.6) allows us to construct a conserved energy according to

H ̃(xt,t) =

∑N

i=1

p^2 i

2 mi

+U(r,t) +

p^2 η

2 Q

+ 3NkTη−Wτ(x 0 )

=H′(xt,t)−Wτ(x 0 ). (8.4.13)

Here x is the extended phase space vector x = (r 1 ,...,rN,η,p 1 ,...,pN,pη). According
to the procedure outlined described in Section 4.9, the average of exp[−βWτ(x 0 )] is
computed as

〈

e−βWAB

〉

A=

1

ZT(0)

∫

dx 0 e^3 Nηe−βWτ(x^0 )δ(H′(x 0 ,0)−C), (8.4.14)

where dx 0 = dNp 0 dNr 0 dpη, 0 dη 0 ,Cis a constant andZT(0) is the “microcanonical”
partition function generated by the Nos ́e–Hoover equations forthet= 0 Hamiltonian

ZT(0) =

∫

dx 0 e^3 Nη^0 δ(H′(x 0 ,0)−C). (8.4.15)

In eqns. (8.4.14) and (8.4.15), the ensemble distribution must be thedistribution of the
initialstate, which is the stateA. In Section 4.9, we showed that when eqn. (8.4.15)
is integrated overη, the canonical partition function is obtained in the form

ZT(0) =

eβC

3 NkT

∫

dpη, 0 e−βp

(^2) η, 0 / 2 Q

∫

dNp 0 dNr 0 e−βH(r^0 ,p^0 )

∝QA(N,V,T). (8.4.16)

In order to complete the proof, we need to carry out the integration overηin the
numerator of eqn. (8.4.14). As was done above, we change variables from x 0 to xτ(x 0 ).
Recalling from Section 4.9 that the measure exp(3Nη)dx is conserved, it follows that
exp(3Nη 0 )dx 0 = exp(3Nητ)dxτ. Therefore, the variable transformation leads to

〈

e−βWAB

〉

A=

1

ZT(0)

∫

dxτe^3 Nητe−βWτ(x^0 (xτ))δ(H′(x 0 (xτ),0)−C). (8.4.17)

From eqn. (8.4.13), it follows thatH′(x 0 (xτ),0) =H′(xτ,τ)−Wτ(x 0 (xτ)). Inserting
this into eqn. (8.4.17), we obtain