Reversible Work Theorem

The theorem states that the radial distribution function g(r) determines the reversible work w(r) associated with the process by which two particles in a fluid are moved from an infinite separation to a separation r, according to the following equation:

$$g(r)= (-\beta w(r)).$$ (393)

Note that since the process is reversible at constant T, N and V, $w(r) =\Delta A$, where A is the Helmholtz free energy.

Proof:

Consider the mean force between particles 1 and 2, averaged over the equilibrium distribution of all other particles,

$$-<\frac{d V(r^N)}{dr_1}>_{r_1 r_2} = \frac{-\int \frac{dV}{dr_1} e^{-\beta V(r^N)} dr_3 ... dr_N }{\int e^{-\beta V(r^N)} dr_3 ... dr_N},$$ (394)

which gives

$$-<\frac{d V(r^N)}{dr_1}>_{r_1 r_2} =\frac{1}{\beta} \frac{d}{dr_1} \int e^{-\beta V(r^N)} dr_3 ... dr_N,$$ (395)

or

$$-<\frac{d V(r^N)}{dr_1}>_{r_1 r_2} = +\frac{1}{\beta} \frac{d}{dr_1} \frac{N(N+1)}{Z} + \frac{1}{\beta} \frac{d}{dr_1} \int e^{-\beta V(r^N)} dr_3 ... dr_N,$$ (396)

since both the number of particles N in the system and the partition function Z are constants. Eq. (396) can be simplified according to Eqs. (388) and (389) as follows,

$$-<\frac{d V(r^N)}{dr_1}>_{r_1 r_2} =\frac{1}{\beta} \frac{d}{dr_1} g(r_1).$$ (397)

Integration of the expression for the average force, introduced by Eq. (397), gives the reversible work,

$$w(r_{12}) = \int_{r_{12}}^{\infty} dr_1 (-<\frac{d V(r^N)}{dr_1}>) = k T \int_{r_{12}}^{\infty} dr_1 \frac{d}{dr_1} g(r_1),$$ (398)

Therefore, the reversible work $w(r_{12})$ associated with the process by which particles 1 and 2 are moved from infinite separation to a relative separation $r_{12}$ is

$$w(r_{12}) = k T g(\infty) - k T g(r_{12})=- k T g(r_{12}),$$ (399)

since $g(\infty)=1$.

Finally, note that

$$w(r) = k T \frac{Z(r_{12}=\infty)}{Z(r_{12}=r)} = -(A (r_{12}= \infty) - A(r_{12}=r)),$$ (400)

where $Z(r_{12}=r)$ and $A(r_{12}=r)$ are the canonical partition function and the Helmholtz free energy of the system, subject to the constraint of fixed relative distance $r_{12}=r$ between particles 1 and 2.