see Lu et al. (1998). The topic was reviewed by Isralewitz et al. (2001).

7.8 Free energy from non-equilibrium processes 239

tion the non-equilibrium methods are more fully treated and a proof of

Jarzynski™s equation is given.

7.8 Free energy from non-equilibrium processes

In the previous section we have seen how free energies or potentials of mean

force can be computed through perturbation and integration techniques.

The steered dynamics is reminiscent of the slow-growth methods, where an

external agent changes the Hamiltonian of the system during a prolonged

dynamics run, by adding an additional time-dependent potential or force,

or changing the value of a constraint imposed on the system. Thus the

system is literally forced from one state to another, possibly over otherwise

unsurmountable barriers. If such changes are done very slowly, such that

the system remains e¬ectively in equilibrium all the time, the change is a

reversible process and in fact the change in free energy from the initial to

the ¬nal state is measured by the work done to change the Hamiltonian.

In most practical cases the external change cannot be realized in a su¬-

ciently slow fashion, and a partial irreversible process results. The ensemble

“lags behind” the change in the Hamiltonian, and the work done on the

system by the external agent that changes the Hamiltonian, is partially ir-

reversible and converted to heat. The second law of thermodynamics tells

us that the total work W done on the system can only exceed the reversible

part ”A:

W ≥ ”A. (7.78)

This is an inequality that enables us to bracket the free energy change be-

tween two measured values when the change is made both in the forward

and the backward direction, but it does not give any help in quantifying

the irreversible part. It would be desirable to have a quantitative relation

between work and free energy!

Such a relation indeed exists. Jarzynski (1997a, 1997b) has shown that

for an irreversible process the Helmholtz free energy change follows from

the work W done to change Hamiltonian H(») of the system from » = 0 to

» = 1, if averaged over an equilibrium ensemble of initial points for » = 0:

A1 ’ A0 = ’kB T ln e’βW »=0 . (7.79)

This is the remarkable Jarzynski equation, which at ¬rst sight is a counterin-

tuitive expression, relating a thermodynamic quantity to a rather ill-de¬ned

and very much process-dependent amount of work. Cohen and Mauzerall

240 Free energy, entropy and potential of mean force

(2004) have criticized Jarzynski™s derivation on the basis of improper hand-

ling of the heat exchange with a heat bath, which induced Jarzynski (2004)

to write a careful rebuttal. Still, the validity of this equation has been con-

¬rmed by several others for various cases and processes, including stochastic

system evolution (Crooks, 2000; Hummer and Szabo, 2001; Schurr and Fu-

jimoto, 2003; Ath`nes, 2004). Since the variety of proofs in the literature is

e

confusing, we shall give a di¬erent proof below, which follows most closely

the reasoning of Schurr and Fujimoto (2003). This proof will enable us to

specify the requirements for the validity of Jarzynskyi™s equation. In this

proof we shall pay extra attention to the role of the temperature, clarifying

what requirements must be imposed on β.

7.8.1 Proof of Jarzynski™s equation

Consider a system of interacting particles with Hamiltonian H0 that has

been allowed to come to equilibrium with an environment at temperature

T0 or Boltzmann parameter β0 = (kB T )’1 and has attained a canonical

distribution

exp[’β0 H0 (z)]

p0 (z) = . (7.80)

exp[’β0 H0 (z )] dz

Here z stands for the coordinates (spatial coordinates and conjugate mo-

menta) q1 , . . . , p1 , . . . of a point in phase space. At time t0 we pick a sample

from the system with phase space coordinates z0 . When we speak later

about averaging over the initial state, we mean averaging over the canonical

distribution p0 of z0 .

Now the system undergoes the following treatment (see Fig. 7.5): at time

t0 the Hamiltonian is abruptly changed from H0 to H1 by an external agent;

from t0 to t1 the system is allowed to evolve from z0 to z1 under the (con-

stant) Hamiltonian H1 . The evolution is not necessarily a pure Hamiltonian

evolution of the isolated system: the system may be coupled to a thermostat

and/or barostat or extended with other variables, and the process may be

deterministic or stochastic. The only requirement is that the evolution pro-

cess conserves a canonical distribution exp[’β1 H1 (z)], where H1 (z) is the

total energy of the system at phase point z. Note that we do not require

that the temperature during evolution (e.g., given by the thermostat or the

friction and noise in a stochastic evolution) equals the temperature before

the jump. Now at t1 the external agent changes the Hamiltonian abruptly

from H1 to H2 , after which the system is allowed to evolve under H2 from

t1 to t2 , changing from z1 to z2 . Again, the evolution process is such that it

7.8 Free energy from non-equilibrium processes 241

t0 t1 t2 t3 E

Time

Hamiltonian H0 H1 H2 H3 H4

E E E E

Work W01 W12 W23 W34

E E E E

Evolution in

G0 (β0 ) G1 (β1 ) G2 (β2 ) G3 (β3 ) G4 (β4 )

phase space z0 z1 z2 z3

Figure 7.5 Irreversible evolution with changing Hamiltonian. The Hamiltonian

is abruptly changed by an external agent at times t0 , t1 , . . ., who exerts work

W01 , W12 , . . . on the system. In the intervening time intervals the system is al-

lowed to evolve with the propagator Gi (βi ), when the Hamiltonian Hi is valid. The

points in phase space, visited by the system at t0 , t1 , . . ., comprising all coordinates

and momenta, are indicated by z0 , z1 , . . ..

would conserve a canonical distribution exp[’β2 H2 (z)]. These processes of

autonomous evolution followed by a Hamiltonian change may be repeated

as often as required to reach the desired end state.

Two de¬nitions before we proceed: the work done by the external agent

to change Hi to Hi+1 we de¬ne (following Jarzynski) as Wi,i+1 . But with

changing β we also require the change in work relative to the temperature,

which we shall denote by Fi,i+1 :

def

Hi+1 (zi ) ’ Hi (zi ),

Wi,i+1 = (7.81)

def

βi+1 Hi+1 (zi ) ’ βi Hi (zi ).

Fi,i+1 = (7.82)

The sum of these quantities over more than one step is similarly denoted.

For example, W0,i is the total work done in all steps up to and including

the step to Hi , and F0,i is the total relative work done in all steps up to

and including the step to Hi . These quantities have speci¬ed values for each

realization; useful results require averaging over an initial distribution.

In the following we shall prove that after each Hamiltonian jump to Hi ,

the free energy Ai is given by

βi Ai ’ β0 A0 = ’ ln e’F0,i p0 , (7.83)

Averaging is done over the initial canonical distribution p0 of z0 , and also

242 Free energy, entropy and potential of mean force

over all possible stochastic evolutions. The latter averaging is automatically

ful¬lled when the original distribution is su¬ciently sampled.

This is the generalized Jarzynski equation. It reduces to the original

Jarzynski equation when all βi are equal:

Ai ’ A0 = ’kB T ln e’βW0,i p0 . (7.84)

Hi can be taken as the required end state; the process may contain any num-

ber of steps and the intermediate evolution times may have any value from

zero (no evolution) to in¬nite (evolution to complete equilibrium). So the

allowed processes encompass the already well-known single-step (no evolu-