A Simple Derivation of Crooks Relation

Abstract

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.

Keywords

• Crooks Relation, fluctuations, first and second laws of thermodynamics

References

1. R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys, vol. 29, pp. 255-284, 1966.
2. G.E. Crooks, “Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems” J. Stat. Phys, vol. 90, pp. 14811487, 1998.
3. 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.
4. G.E. Crooks “Path ensemble averages in systems driven far from equilibrium”, Phys. Rev. E., vol. 61, pp. 2361- 2366, 2000.
5. S.R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, New York: Dover Publication, 1984.
6. L. Onsager, reprinted in, The collected works of Lars Onsager, Singapore: World Scientific, 1996.
7. 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.
8. 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.

9. 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
10. 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.
11. D. J. Evans and D. J. Searles, “Advances in Physics, The fluctuation theorem,” vol. 51, pp. 1529-1585, 200
12. 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.
13. J. Kurchan, “Fluctuation theorem for stochastic dynamics,” Journal of Physics A: Mathematical and General, vol. 31, no. 16, pp. 3719-3729, 1998.
14. 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.
15. Maes, Christian. "The fluctuation theorem as a Gibbs property," Journal of Statistical Physics, vol. 95, pp. 367-392, 1999.
16. T. Hatano and S. Sasa” Steady-State Thermodynamics of Langevin Systems,” Phys. Rev. Lett., vol. 86, pp. 3463-3466, 2001.
17. 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.
18. 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.
19. C. Jarzynski, “Nonequilibrium Equality for Free Energy Differences,” Phys. Rev. Lett., vol. 78, pp. 2690-2693, 1997.
20. C. Jarzynski, “Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach,” Phys. Rev. E., vol. 56, 5018 – 5035, 1997.
21. D. Chandler, Introduction to Modern Statistical Mechanics, New York: Oxford University Press, 1987.