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Jarzynski equality

The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two equilibrium states and non-equilibrium processes. It is named after the physicist Christopher Jarzynski (then at Los Alamos National Laboratory) who discovered it in 1997.

In thermodynamics, the free energy difference ΔF = FBFA between two states A and B is connected to the work W done on the system through the inequality:

$\Delta F \leq W$,

the equality happening only in the case of a quasistatic process, i.e. when one takes the system from A to B infinitely slowly.

In contrast to the thermodynamic statement above, the JE remains valid no matter how fast the process happens. The equality itself can be straightforwardly derived from the Crooks fluctuation theorem and is presented below, followed by a brief explanation of the terms and notation:

$e^ { -\Delta F / k T} = \overline{ e^{ -W/kT } }.$

Here k is the Boltzmann constant and T is the temperature of the system in the equilibrium state A (or, equivalently, the temperature of the heat reservoir with which the system was thermalized before the process took place).

The over-line indicates an average over all possible realizations of a process that takes the system from the equilibrium state A to the equilibrium state B.1 In the case of an infinitely slow process, the work W performed on the system in each realization is numerically the same, so the average becomes irrelevant and the Jarzynski equality reduces to the thermodynamic equality ΔF = W (see above). In general, however, W depends upon the specific initial microstate of the system, though its average can still be related to ΔF through an application of Jensen's inequality in the JE, viz.

$\Delta F \leq \overline{W},$

in accordance with the second law of thermodynamics.

Since its original derivation, the Jarzynski equality has been verified in a variety of contexts, ranging from experiments with biomolecules to numerical simulations. Many other theoretical derivations have also appeared, lending further confidence to its universality.

An issue of current debate is whether the Jarzynski equality was published already in 1977 by the Russian physicists G.N. Bochkov and Yu. E. Kuzovlev (see Bibliography). These authors proposed a generalized version of the Fluctuation-Dissipation relations which holds even in the presence of arbitrary external time-dependent forces. The generalized Fluctuation-Dissipation relations take on a similar form as the more recently proposed fluctuation theorem.

Notes

1. The Jarzynski equality actually encompasses more general scenarios where the final state of the system is out of equilibrium. In this case, since free energies are generally defined only for equilibrium states, one has to specify exactly what is the quantity FB that appears on the l.h.s. of the JE. This specification requires a precise definition of the process that takes the system from A to B, and is beyond the scope of this presentation.

Bibliography

• C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997)
• C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Phys. Rev. E 56, 5018 (1997)
• G. E. Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 90, 1481 (1998)
• G. Hummer, A. Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments, Proc. Nat. Acad. Sci. 98, 3658 (2001)
• J. Liphardt et al., Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality, Science 296, 1832 (2002)
• D. J. Evans, A non-equilibrium free energy theorem for deterministic systems, Mol. Phys. 101, 1551 (2003)
• A. B. Adib, Entropy and density of states from isoenergetic nonequilibrium processes, Phys. Rev. E 71, 056128 (2005)
• F. Douarche, S. Ciliberto, A. Petrosyan, I. Rabbiosi, An experimental test of the Jarzynski equality in a mechanical experiment, Europhys. Lett. 70 (5), 593 (2005, see also cond-mat/0502395)

For earlier results dealing with the statistics of work in nonequilibrium processes, see:

• G. N. Bochkov and Yu. E. Kuzovlev, Zh. Eksp. Teor. Fiz. 72, 238 (1977); op. cit. 76, 1071 (1979)
• G. N. Bochkov and Yu. E. Kuzovlev, Physica 106A, 443 (1981); op. cit. 106A, 480 (1981)