Nonequilibrium Equality for Free Energy Differences

Consider a finite classical system in contact with a heat reservoir. A central concept in thermodynamics is that of the work performed on such a system, when some external parameters of the system are made to change with time. (These parameters may represent, for instance, the strength of an external field, or the volume of space within which the system is confined, or, more abstractly, some particleparticle interactions which are turned on or off during the course of a molecular dynamics simulation.) When the parameters are changed infinitely slowly along some path g from an initial point A to a final point B in parameter space, then the total work W performed on the system is equal to the Helmholtz free energy difference DF between the initial and final configurations [1]: W ­ DF ; FB 2 FA. [Here FA sFBd refers to the equilibrium free energy of the system, with the parameters held fixed at A sBd.] By contrast, when the parameters are switched along g at a finite rate, then W will depend on the microscopic initial conditions of the system and reservoir, and will, on average, exceed DF: