equilibrations, and the limiting slow-growth process, the latter taking a large

number of steps, each consisting of a small Hamiltonian change followed by

a single MD time step.

The proof follows by induction. Consider the free energy change after the

¬rst step, before any evolution has taken place:

dz0 exp[’β1 H1 (z0 )]

β1 A1 ’ β0 A0 = ’ ln

dz0 exp[’β0 H0 (z0 )]

= e’[β1 H1 (z0 )’β0 H0 (z0 )] = e’F0,1 p0 , (7.85)

p0

which is the generalized Jarzynski™s equation applied to a single step without

evolution.

Now, from t0 to t1 the system is left to evolve under the Hamiltonian

H1 . Its evolution can be described by a propagator G1 (z, t; z0 , t0 ; β1 ) that

speci¬es the probability distribution of phase points z at time t, given that

the system is at z0 at time t0 . For pure Hamiltonian dynamics the path is

deterministic and thus G1 is a delta-function in z; for stochastic evolutions

G1 speci¬es a probability distribution. In general G1 describes the evolution

of a probability distribution in phase space:

p(z, t) = G1 (z, t; z0 , t0 ; β1 )p(z0 , t0 ) dz0 . (7.86)

The requirement that G1 preserves a canonical distribution exp[’β1 H1 (z)]

can be written as

G1 (z, t; z0 , t0 ) exp[’β1 H1 (z0 )] dz0 = exp[’β1 H1 (z)] (7.87)

for all t. In fact, G maps the canonical distribution onto itself.

The actual distribution p0 (z0 ) at t0 is not the canonical distribution for

H1 , but rather the canonical distribution for H0 . So the property (7.87)

7.8 Free energy from non-equilibrium processes 243

z0(t0) G1, H1 z1(t1)

Figure 7.6 Paths extending from sampled points z0 at t0 to z1 at t1 . Each of the

paths is weighted (indicated by line thickness) such that the distribution of weights

becomes proportional to the canonical distribution exp[’β1 H1 (z1 )]. The grey areas

indicate the equilibrium distributions for H0 (left) and H1 (right).

cannot be applied to the actual distribution at t1 . But we can apply a trick,

pictured schematically in Fig. 7.6. Let us give every point z0 a weight such

that the distribution of weights, rather than of points, becomes the canonical

distribution for H1 . This is accomplished if we give the point z0 a weight

exp[’β1 H1 (z0 ) + β0 H0 (z0 )]. Note that this weight equals exp[’F01 ]. Since

the distribution of weights, indicated by pw (z0 ), is now proportional to the

canonical distribution for H1 :

exp[’β1 H1 (z0 )]

pw (z0 ) = p0 (z0 )e’β1 H1 (z0 )+β0 H0 (z0 ) = , (7.88)

dz0 exp[’βH0 (z0 )]

the distribution of weights will remain invariant during the evolution with

G1 to z1 , and hence also

pw (z1 ) = pw (z0 ). (7.89)

¿From this we can derive the unweighted distribution of points z1 by dividing

pw (z1 ) with the weight given to z0 :

p(z1 ) = pw (z1 )eβ1 H1 (z0 )’β0 H0 (z0 )] = pw (z1 )eF01

exp[’β1 H1 (z1 ) + F01 ]

= . (7.90)

dz0 exp[’βH0 (z0 )]

Next the external agent changes H1 to H2 , performing the work W1,2 =

H2 (z1 ) ’ H1 (z1 ) on the system. The relative work is

F1,2 = β2 H2 (z1 ) ’ β1 H1 (z1 ). (7.91)

244 Free energy, entropy and potential of mean force

Equation (7.90) can now be rewritten as

exp[’β2 H2 (z1 ) + F0,1 + F1,2 ]

p(z1 ) = . (7.92)

dz0 exp[’βH0 (z0 )]

If we now ask what the expectation of exp[’F0,2 ] will be, we ¬nd

e’F0,2 = e’(F0,1 +F1,2 ) = dz1 p(z1 )e’(F0,1 +F1,2 ) (7.93)

dz1 exp[’β2 H2 (z1 )]

= e’(β2 A2 ’β1 A0 ) .

= (7.94)

dz0 exp[’βH0 (z0 )]

This is the generalized Jarzynski™s equation after the second step has been

made. The extension with subsequent steps is straightforward: for the next

step we start with p(z1 ) and give the points a weight exp[’F0,2 ]. The weight

distribution is now the canonical distribution for H2 , which remains invari-

ant during the evolution G2 . From this we derive p(z2 ), and “ after having

changed the Hamiltonian at t2 to H3 “ we ¬nd that

e’F0,3 = e’(β3 A3 ’β0 A0 ) . (7.95)

This, by induction, completes the proof of (7.83). Note that, in the case

of varying β during the process, it is

the total relative work, i.e., the change in energy divided by the tempera-

ture, F0,i , that must be exponentially averaged rather than the total work

itself.

7.8.2 Evolution in space only

When the external change in Hamiltonian involves the potential energy V (r)

only (which usually is the case), and the evolution processes are mappings in

con¬gurational space that conserve a canonical distribution (e.g., a sequence

of Monte Carlo moves or a Brownian dynamics), the Jarzynski equation is

still valid. The evolution operator Gi (r i , ti ; r i’1 , ti’1 ; βi ) now evolves r i’1

into r i ; it has the property

Gi (r , t ; r, t; βi ) exp[’βi Vi (r)] dr = exp[’β1 Vi (r )]. (7.96)

Here r stands for all cartesian coordinates of all particles, specifying a point

in con¬gurational space. When we re-iterate the proof given above, replacing

z by r and H by V , we ¬nd the same equation (7.84) for the isothermal

case, but a correction due to the kinetic contribution to the free energy if

7.8 Free energy from non-equilibrium processes 245

the initial and ¬nal temperatures di¬er. Equation (7.83) is now replaced by

3N βi

’ ln e’F0,i

βi Ai ’ β0 A0 = ln p0 . (7.97)

2 β0

7.8.3 Requirements for validity of Jarzynski™s equation

Reviewing the proof given above, we can list the requirements for its validity:

(i) The state of the system at time t is completely determined by the

point in phase space z(t). The propagator G determines the fu-

ture probability distribution, given z(t). This is an expression of the

Markovian character of the propagator: the future depends on the

state at t and not on earlier history. This precludes the use of stochas-

tic propagators with memory, such as the generalized Langevin equa-

tion. It is likely (but not further worked out here) that the Markovian

property is not a stringent requirement, as one can always de¬ne the

state at time t to include not only z(t), but also z at previous times.

However, this would couple the Hamiltonian step with the future

propagation, with as yet unforeseen consequences. Hamiltonian (in-

cluding extended systems), simple Langevin, Brownian and Monte

Carlo propagations are all Markovian.13

(ii) The propagator must have the property to conserve a canonical dis-

tribution. Microscopic reversibility and detailed balance are not pri-

mary requirements.

(iii) The sampling must be su¬cient to e¬ectively reconstruct the canon-

ical distribution after each step by the weighting procedure. This

requires su¬cient overlap between the distribution of end points of

each relaxation period and the canonical distribution after the fol-

lowing step in the Hamiltonian. When the steps are large and the

relaxations are short, su¬cient statistics may not be available. This

point is further discussed in the next subsection.

(iv) As is evident from the proof, there is no requirement to keep the