# Nonequilibrium statistical mechanics

Non-equilibrium statistical mechanics has the reputation of being conceptually intricate and technically complicated. The simple and powerful principles of equilibrium theory no longer apply and for a long time general results for non-equilibrium situations were restricted to the linear response regime near equilibrium.

This has changed within the last 15 years with the discovery of several simple, general, and exact results for classical systems driven arbitrarily far from thermal equilibrium. Of central importance are the Jarzynski equation

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relating the free energy difference between two equilibrium states to the statistics of the work necessary to perform an arbitrary non-equilibrium transition between them, and the Crooks relation

connecting the ratio of probabilities of a non-equilibrium trajectory and its time mirror with the dissipated work. These exciting results stimulate novel and interesting lines of research and may pave the way for an improved understanding of non-equilibrium processes. Moreover, they are important practically since they suggest new ways to efficiently extract equilibrium information out of non-equilibrium experiments or simulations.

#### Asympotics of work distribution

A. Engel, Phys. Rev. E 80, 021120

The exponential average in the Jarzynski equation has subtle statistical properties. It is dominated by the left tail of the probability distribution comprising the atypical realizations in which the work is much smaller than the free energy difference . In experiments and simulations this region is rarely sampled and correspondingly the estimate for may show large uncertainties.

Although the full distribution is generally not accessible analytically it may be possible to get analytical information about the asymptotic behaviour of for small by using the method of optimal fluctuation. By fitting this asymptotics to the region of that is still sufficiently sampled by experiment or simulation the resulting estimate for may be significantly improved.

#### Targeted estimators

A.M. Hahn, H. Then, Phys. Rev. E 79 (2009) 011113

Traditional methods often converge extremely slowly making it difficult to estimate free-energy differences. An example is the numerical computation of chemical potentials of fluids and gases in the high density regime. The problem is that after inserting a test particle, the underlying phase space distribution does no longer coincide with its previous form.

Targeted estimators that link the two different phase space distributions to each other converge much faster. In addition, a convergence criterion allows to conclude the accuracy and reliability of the results.

#### Optimal protocols

H. Then, A. Engel, Phys. Rev. E 77 (2008) 041105

A very remarkable feature of the Jarzynski equation is that the average of only depends on equilibrium properties of the initial and final states. It is hence independent of the transition itself and the same for any protocol specifying the change of the Hamiltonian of the system from the initial to the final state. On the other hand, in practical applications the average can only be determined approximately and the quality of the approximation may well depend on the protocol implemented. Useful information on the accuracy of the approximation is conveyed by the average value, , of the dissipated work. Protocols minimizing this average show surprising structure including continuous parts as well as jumps.

#### Jarzynski equation in Bayesian inference

H. Ahlers, A. Engel, Eur. Phys. J. B 62 (2008) 357

Methods of Bayesian statistics play an ever growing role in statistical data analysis. Building on rather general and essentially simple principles the efficiency of Bayesian methods in practical applications depends crucially on the implemented numerical algorithms. A major difficulty common to Bayesian data analysis is the calculation of integrals in high-dimensional spaces which are dominated by contributions from small and labyrinthine regions. It is therefore no surprise that some of the tools of statistical mechanics have found their way into the arsenal of Bayesian statisticians.

A rather successful example of this transfer of methods is thermodynamic integration which, however, is built upon the determination of equilibrium averages for several values of the temperature. In systems with long relaxation times which typically occur if the relevant probability distributions are multimodal it may therefore be exceedingly slow. In such cases a variant of the Jarzynski equation allows to determine the necessary averages much more efficiently.

#### Quantum work theorems

A. Engel, R. Nolte, Europhys. Lett. 79 (2007) 10003

Several proofs for non-equilibrium work and fluctuation theorems are by now available if the microscopic dynamics of the system is described by the laws of classical mechanics. The situation is less clear in the case of dissipative quantum systems where different definitions of the work performed in a non-equilibrium process are conceivable. For some of these the validity of the Jarzynski equation may be proven in close analogy to the classical case, for others the Jarzynski equation in its present form does not hold. For a simple, exactly solvable example the similarities and differences between two representative definitions of quantum work may be highlighted.