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 derived it in 1997.

In thermodynamics, the free energy difference $\Delta F=F_{B}-F_{A}$ 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. The JE equality states:

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

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. 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 $\Delta F=W$ (see above). In general, however, W depends upon the specific initial microstate of the system, though its average can still be related to $\Delta F$ 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.

History

A question has been raised about who gave the earliest statement of the Jarzynski equality. For example in 1977 the Russian physicists G.N. Bochkov and Yu. E. Kuzovlev (see Bibliography) proposed a generalized version of the Fluctuation-Dissipation relations which holds in the presence of arbitrary external time-dependent forces. The generalized Fluctuation-Dissipation relations take on a similar form to one of the more recently proposed forms of the fluctuation theorem, namely the Nonequilibrium partition identity.

However the earliest statement of what is now known as the Nonequilibrium partition identity (also known as the Kawasaki identity see Fluctuation Theorem), is due to Yamada and Kawasaki a decade earlier. (The Nonequilibrium Partition Identity is the Jarzynski equality applied to two systems whose free energy difference is zero - like straining a fluid.)

However, these early statements are very limited in their application. Both Bochkov and Kuzovlev as well as Yamada and Kawasaki consider a deterministic time reversible Hamiltonian system. As Kawasaki himself noted this precludes any treatment of nonequilibrium steady states. The fact that these nonequilibrium systems heat up forever because of the lack of any thermostatting mechanism leads to divergent integrals etc. No purely Hamiltonian description is capable of treating the experiments carried out to verify the Crooks fluctuation theorem, Jarzynski equality and the Fluctuation Theorem. These experiments involve thermostated systems in contact with heat baths.

The mathematics for how to describe time reversible deterministic thermostatted systems was not developed until 1982, by Hoover, Evans and later Nose. The first derivation of the Nonequilibrium Partition Identity for reversible thermostatted systems is due to Morriss and Evans 1985.

The Fluctuation Theorem implies the Nonequilibrium Partition Identity. However the Partition Identity does not imply the Fluctuation Theorem (see Carberry et al.). This is mirrored in the relationship between the Crooks fluctuation theorem and Jarzynski. The former implies the latter but the reverse is not true.

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 $F_{B}$ 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

• A. B. Adib, Entropy and density of states from isoenergetic nonequilibrium processes, Phys. Rev. E 71, 056128 (2005)
• D. M. Carberry, S. R. Williams, G. M. Wang, E. M. Sevick and D. J. Evans, "The Kawasaki identity and the fluctuation theorem", Journal of Chemical Physics, 121, 8179 – 8182 (2004).
• G. E. Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 90, 1481 (1998)
• 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)
• D. J. Evans, A non-equilibrium free energy theorem for deterministic systems, Mol. Phys. 101, 1551 (2003)
• G. Hummer, A. Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments, Proc. Nat. Acad. Sci. 98, 3658 (2001)
• 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)
• J. Liphardt et al., Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality, Science 296, 1832 (2002)
• G.P. Morriss and D.J. Evans,"Isothermal response theory", Molecular Physics, '54, 629 (1985).

For earlier results dealing with the statistics of work in adiabatic (ie Hamiltonian) 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)
• K. Kawasaki and J.D. Gunton, Phys. Rev. A, 8, 2048 (1973)
• T. Yamada and K. Kawasaki, Prog. Theo. Phys., 38, 1031 (1967)

See also

• Fluctuation theorem - Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems.
• Crooks fluctuation theorem -Provides a Fluctuation Theorem between two equilibrium states. ImpliesJarzynski equality.
• Nonequilibrium partition identity
• Jarzynski Equality on arxiv.org

• "Fluctuation-Dissipation: Response Theory in Statistical Physics" by Umberto Marini Bettolo Marconi, Andrea Puglisi, Lamberto Rondoni, Angelo Vulpiani, [1]