Jarzynski equality - Citizendia

The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two equilibrium states and non-equilibrium processes. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a System, and is helpful in Engineering It is named after the physicist Christopher Jarzynski (then at Los Alamos National Laboratory) who discovered it in 1997. Los Alamos National Laboratory (LANL (previously known at various times as Site Y, Los Alamos Laboratory, and Los Alamos Scientific Laboratory) is a

In thermodynamics, the free energy difference $Δ F = F B - F A$ 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. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Thermodynamics, a quasistatic process is a Thermodynamic process that happens infinitely slowly 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 Crooks equation (CE is an equation in Statistical Mechanics that relatesthe work done on a system during a non-equilibrium transformation to thefree energy difference between the The JE equality states:

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

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. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics In Thermodynamics a heat reservoir is considered as a constant Temperature source

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 $Δ F = W$ (see above). In general, however, W depends upon the specific initial microstate of the system, though its average can still be related to $Δ F$ through an application of Jensen's inequality in the JE, viz. In Statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system that the system visits in the course of its thermal fluctuations In Mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral

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

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. The fluctuation theorem (FT is a theorem from Statistical mechanics dealing with the relative probability that the Entropy of a system which is currently away from The nonequilibrium partition identity (NPI is a remarkably simple and elegant consequence of the Fluctuation Theorem previously known as the Kawasaki 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 fluctuation theorem (FT is a theorem from Statistical mechanics dealing with the relative probability that the Entropy of a system which is currently away from (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. In Classical mechanics, a Hamiltonian system is a Physical system in which Forces are Velocity invariant 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. The Crooks equation (CE is an equation in Statistical Mechanics that relatesthe work done on a system during a non-equilibrium transformation to thefree energy difference between the The fluctuation theorem (FT is a theorem from Statistical mechanics dealing with the relative probability that the Entropy of a system which is currently away from 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. The fluctuation theorem (FT is a theorem from Statistical mechanics dealing with the relative probability that the Entropy of a system which is currently away from 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. In Mathematics, LHS is informal shorthand for the left-hand side of an Equation. 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)

History

Bibliography

See also