A note on non-equilibrium work fluctuations and equilibrium free energies

We consider in this paper, a few important issues in non-equilibrium work fluctuations and their relations to equilibrium free energies. First we show that the Jarzynski identity can be viewed as a cumulant expansion of work. For a switching process which is nearly quasistatic the work distribution is sharply peaked and Gaussian. We show analytically that dissipation given by average work minus reversible work WRWR, decreases when the process becomes more and more quasistatic. Eventually, in the quasistatic reversible limit, the dissipation vanishes. However the estimate of pp, the probability of violation of the second law given by the integral of the tail of the work distribution from −∞−∞ to WRWR, increases and takes a value of 0.5 in the quasistatic limit. We show this analytically employing Gaussian integrals given by error functions and the Callen–Welton theorem that relates fluctuations to dissipation in process that is nearly quasistatic. Then we carry out Monte Carlo simulation of non-equilibrium processes in a liquid crystal system in the presence of an electric field and present results on reversible work, dissipation, probability of violation of the second law and distribution of work.

Research highlights► We show that the Jarzynski identity can be viewed as a cumulant expansion of work. ► We show analytically that dissipation decreases and vanishes in the reversible limit. ► We show analytically that P(W≤ΔF)P(W≤ΔF) increases and goes to one-half in the reversible limit, for Gaussian work distribution. ► Results are demonstrated on a lattice model of liquid crystalline system.