1549380323-Statistical Mechanics Theory and Molecular Simulation

Jarzynski’s equality 327

H(r,p,t) =

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN,t). (8.4.3)

For time-dependent Hamiltonians, the usual conservation law dH/dt= 0 does not
hold, which can be seen by computing

dH

dt

=∇xtH· ̇xt+

∂H

∂t

, (8.4.4)

where the phase space vector x = (r 1 ,...,rN,p 1 ,...,pN)≡(r,p) has been introduced.
Integrating both sides over time fromt= 0 to an endpointt=τ, we find
∫τ

0

dt

dH

dt

=

∫τ

0

dt∇xtH· ̇xt+

∫τ

0

dt

∂H

∂t

. (8.4.5)

Eqn. (8.4.5) can be regarded as a microscopic version of the first lawof thermodynam-
ics, where the first and second terms represent the heat absorbed by the system and
the work done on the system over the trajectory, respectively.^3 That the work depends
on the initial phase space vector x 0 can be seen by defining the work associated with
the trajectory xt(x 0 ) obtained up to timet=t′as

Wt′(x 0 ) =

∫t′

0

dt

∂

∂t

H(xt(x 0 ),t). (8.4.6)

Note thatWAB(x 0 ) =Wτ(x 0 ).
The derivation of the Jarzynski equality requires the calculation ofthe ensemble
average of exp[−βWAB(x 0 )] = exp[−βWτ(x 0 )] over a canonical distribution in the
initial conditions x 0. Before we examine how this average might be performed along
a molecular dynamics trajectory for a finite system, let us considerthe simpler case
where each initial condition x 0 evolves in isolation according to Hamilton’s equations
as derived from eqn. (8.4.3). If Hamilton’s equations are obeyed, then the first (heat)
term on the right in eqn. (8.4.4) vanishes,∇xtH· ̇xt= 0, and we can express the work
as

Wt′=

∫t′

0

dt

d

dt

H(xt(x 0 ),t) =H(xt′(x 0 ),t′)−H(x 0 ,0). (8.4.7)

Takingt′=τ, and recognizing thatH(x 0 ,0) =HA(x 0 ), we can write the ensemble
average of exp[−βWAB(x 0 )] as

〈

e−βWAB

〉

A=

CN

QA(N,V,T)

∫

dx 0 e−βHA(x^0 )e−β[H(xτ(x^0 ),τ)−HA(x^0 )]

(^3) To see this, consider, for example, a mechanical piston slowly compressing a gas. Such a device
can be viewed as an explicitly time-dependent external agent acting on the system, which can be
incorporated into the potential. For this reason, the second term on the right in eqn. (8.4.5) represents
the work performed on the system. However, even in the absence of an external agent, if the system
interacts with a thermal reservoir (see Section 4.3) then the Hamiltonian is not conserved, and the
first term on the right in eqn. (8.4.5) will be nonzero. Hence,this term represents the heat absorbed.