Equilibrium Free Energies from Nonequilibrium Processes *

A recent result, relating the (irreversible) work performed on a system during a non-quasistatic process, to the Helmholtz free energy difference between two equilibrium states of the system, is discussed. A proof of this result is given for the special case when the evolution of the system in question is modelled by a Langevin equation in configuration space. My purpose in this talk is to present and discuss a result which relates the free energy difference between two equilibrium states of a system — defined with respect to two values of an external parameter — to the work performed on the system by changing that parameter at a finite rate from one value to the other. Let me begin with a fundamental statement from classical thermodynamics [1]: the total work performed on a system during an isothermal, quasistatic process is equal to the free energy difference between the initial and final equilibrium states of the system. 1 This statement may be understood as follows. Imagine a finite system which depends on some external parameter, λ. Macroscopically, an equilibrium state is the unique state attained by the system by allowing it to come to equilibrium with an infinite heat reservoir at temperature T , holding λ fixed. Such a state may be represented by a single point in the (λ, T)-plane, as shown in Fig.1. With each equilibrium state we may associate a free energy F : F (λ, T) = E − ST, (1) where E and S denote, respectively, the internal energy and the entropy of the system, both functions of the state. If we now prepare the system in a state (λ A , T), and then infinitely slowly change the value of λ from λ A to λ B , always keeping the system thermostatted (i.e. in contact with the heat reservoir) at temperature T , then the system will evolve through a continuous sequence of equilibrium states — represented by the dotted line in Fig.1 — 1 Throughout this talk, the term " free energy " will refer specifically to the Helmoltz free energy.