A Simple Derivation of Crooks Relation

The link between properties of a system at equilibrium, in particular free energy difference, to the fluctuations in the work performed during non-equilibrium process is called Crooks relation. This relation, which is a measure of the grade of irreversibility of a process, was elegantly derived based on the equations of motion for a set of particles along with the formal solution of the evolution equation using a distribution function, both solved in a classical and a stochastic way. This technical note, reports on a simple derivation of Crooks formula based on the energy balance and entropy generation in a system undergoing a process in which fluctuations are not neglected.

___

  • R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys, vol. 29, pp. 255-284, 1966.
  • G.E. Crooks, “Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems” J. Stat. Phys, vol. 90, pp. 14811487, 1998.
  • G.E. Crooks, “The entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,” Phys. Rev. E., vol. 60 no. 3, pp. 2721-2726, 1999.
  • G.E. Crooks “Path ensemble averages in systems driven far from equilibrium”, Phys. Rev. E., vol. 61, pp. 2361- 2366, 2000.
  • S.R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, New York: Dover Publication, 1984.
  • L. Onsager, reprinted in, The collected works of Lars Onsager, Singapore: World Scientific, 1996.
  • S. Ciliberto, S. Joubaud, and A. Petrosyan, “Fluctuations in out-of-equilibrium systems: from theory to experiment,” Journal of Statistical Mechanics: Theory and Experiment, vol. P12003 pp. 1-27, 2010.
  • R. Van Zon, S. Ciliberto, E. G. D. Cohen, “Power and Heat Fluctuation Theorems for Electric Circuits” Phys. Rev. Lett., vol. 92, pp. 1-7, 2004.
  • D. J. Evans, E.G.D. Cohen, and G.P. Morris, “Probability of second law violations in shearing steady states,” Phys. Rev. Lett, vol. 71, pp. 2401-2404, 19
  • S. Ciliberto and C. Laroche, J. “An experimental test of the Gallavotti-Cohen fluctuation theorem,” in International Conference on Disorder and Chaos in honour of Giovanni Paladin Physique. vol. 8 pp. 215219, 1998.
  • D. J. Evans and D. J. Searles, “Advances in Physics, The fluctuation theorem,” vol. 51, pp. 1529-1585, 200
  • D. J. Evans and D. J. Searles, Phys. Rev. “Equilibrium microstates which generate second law violating steady states”, vol. 50, pp. 1645-1648, 1994. G. Gallavotti and E. G.D. Cohen, “Dynamical ensembles in nonequilibrium statistical mechanics” Phys. Rev. Lett., vol. 74, 2694-2697, 1995.
  • J. Kurchan, “Fluctuation theorem for stochastic dynamics,” Journal of Physics A: Mathematical and General, vol. 31, no. 16, pp. 3719-3729, 1998.
  • Ruelle, D. “Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,” Journal of Statistical Physics, vol. 95 (1-2), pp. 393-468, 1999. Lebowitz, Joel L., and Herbert Spohn. "A Gallavotti– Cohen-type symmetry in the large deviation functional for stochastic dynamics," Journal of Statistical Physics, vol. 95, pp. 333-365, 1999.
  • Maes, Christian. "The fluctuation theorem as a Gibbs property," Journal of Statistical Physics, vol. 95, pp. 367-392, 1999.
  • T. Hatano and S. Sasa” Steady-State Thermodynamics of Langevin Systems,” Phys. Rev. Lett., vol. 86, pp. 3463-3466, 2001.
  • R. Van Zon, E. G. D. Cohen, “Stationary and transient work-fluctuation theorems for a dragged Brownian particle,” Phys. Rev., vol. 67, pp. 1-20, 2003.
  • D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr. and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature, vol. 437, pp. 231-234, 200 C. Bustamante, J. Liphardt and F. Ritort, “The nonequilibrium thermodynamics of small systems,” Phys Today, vol. 58, no. 7, pp. 43 – 48, 2005.
  • C. Jarzynski, “Nonequilibrium Equality for Free Energy Differences,” Phys. Rev. Lett., vol. 78, pp. 2690-2693, 1997.
  • C. Jarzynski, “Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach,” Phys. Rev. E., vol. 56, 5018 – 5035, 1997.
  • D. Chandler, Introduction to Modern Statistical Mechanics, New York: Oxford University Press, 1987.