Abstract
Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski’s equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo’s work (Proc. Natl. Acad. Sci. USA 98(7):3658–3661, [2001]) and Hatano-Sasa equality of steady state thermodynamics (Phys. Rev. Lett. 86:3463–3466, [2001]) to the general multidimensional case.
Similar content being viewed by others
References
Baiesi, M., Jacobs, T., Maes, C., Skantzos, N.S.: Fluctuation symmetries for work and heat. Phys. Rev. E 74, 021111 (2006)
Crooks, G.E.: Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 1481–1487 (1998)
Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721–2726 (1999)
Crooks, G.E.: Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E 61(3), 2361–2366 (2000)
Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse. Springer, Berlin (1961)
Dynkin, E.B.: Markov Processes, vols. 1, 2. Springer, Berlin (1965)
Friedman, A.: Partial Differential Equations of Parabolic Type. Englewood Cliffs (1964)
Ge, H., Qian, M.: Generalized Jarzynski’s equality in inhomogeneous Markov chains. J. Math. Phys. 48, 053302 (2007)
Ge, H., Jiang, D.Q., Qian, M.: A simple discrete model of Brownian motors: time-periodic Markov chains. J. Stat. Phys. 123(4), 831–859 (2006)
Ge, H., Jiang, D.Q., Qian, M.: Reversibility and entropy production of inhomogeneous Markov chains. J. Appl. Probab. 43(4), 1028–1043 (2006)
Glansdorff, P., Prigogine, I.: Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley–Interscience, London (1971)
Haken, H.: Synergetics: An Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology. Springer, Berlin (1977)
Haken, H.: Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices. Springer, Berlin (1983)
Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff and Noorrdhoff, Alphen aan den Rijn-Germantown (1980)
Hatano, T., Sasa, S.: Steady-states thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463–3466 (2001)
Hill, T.L.: Studies in irreversible thermodynamics IV. Diagrammatic representation of steady state fluxes for unimolecular systems. J. Theor. Biol. 10, 442–459 (1966)
Hill, T.L.: Free Energy Transduction in Biology. Academic, New York (1977)
Hill, T.L.: Free Energy Transduction and Biochemical Cycle Kinetics. Springer, New York (1995)
Hill, T., Chen, Y.: Stochastics of cycle completions (fluxes) in biochemical kinetic diagrams. Proc. Natl. Acad. Sci. USA 72, 1291–1295 (1975)
Hummer, G., Szabo, A.: Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl. Acad. Sci. USA 98(7), 3658–3661 (2001)
Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)
Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Phys. Rev. E 56, 5018–5035 (1997)
Jarzynski, C.: Microscopic analysis of Clausius-Duhem processes. J. Stat. Phys. 96, 415–427 (1999)
Jarzynski, C.: Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. 98, 77–102 (2000)
Jiang, D.Q., Qian, M., Qian, M.P.: Mathematical Theory of Nonequilibrium Steady States—On the Frontier of Probability and Dynamical Systems. Lect. Notes Math., vol. 1833. Springer, Berlin (2004)
Keizer, J.: Statistical Thermodynamics of Nonequilibrium Processes. Springer, New York (1987)
Karatzas, I., Shreve, E.S.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)
Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Springer, New York (1994)
Liphardt, J., Dumont, S., Smith, S.B., Tinoco, I., Bustamante, C.: Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science 296, 1832–1835 (2002)
Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations. Wiley, New York (1977)
Oono, Y., Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130, 29–44 (1998)
Qian, H.: Mathematical formalism for isothermal linear reversibility. Proc. R. Soc. Lond. Ser. A 457, 1645–1655 (2001)
Qian, H.: Cycle kinetics, steady state thermodynamics and motors-a paradigm for living matter physics. J. Phys. Condens. Matter 17, S3783–S3794 (2005)
Qian, M.P., Qian, M.: Circulation for recurrent Markov chains. Z. Wahrsch. Verw. Geb. 59, 203–210 (1982)
Qian, M.P., Qian, M.: The entropy production and reversibility of Markov processes. Sci. Bull. 30(3), 165–167 (1985)
Qian, M.P., Qian, C., Qian, M.: Circulations of Markov chains with continuous time and the probability interpretation of some determinants. Sci. Sin. (Ser. A) 27(5), 470–481 (1984)
Qian, M.P., Qian, M., Gong, G.L.: The reversibility and the entropy production of Markov processes. Contemp. Math. 118, 255–261 (1991)
Sekimoto, K.: Kinetic characterization of heat bath and the energetics of thermal ratchet models. J. Phys. Soc. Jpn. 66, 1234–1237 (1997)
Strook, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)
van Zon, R., Cohen, E.G.D.: Stationary and transient work-fluctuation theorems for a dragged Brownian particle. Phys. Rev. E 67, 046102 (2003)
van Zon, R., Cohen, E.G.D.: Extension of the fluctuation theorem. Phys. Rev. Lett. 91, 110601 (2003)
van Zon, R., Cohen, E.G.D.: Extended heat-fluctuation theorems for a system with deterministic and stochastic forces. Phys. Rev. E 69, 056121 (2004)
Zhang, Z.S.: Mathematical Analysis, vols. 1, 2, 3. Peking University Press (1991) (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ge, H., Jiang, DQ. Generalized Jarzynski’s Equality of Inhomogeneous Multidimensional Diffusion Processes. J Stat Phys 131, 675–689 (2008). https://doi.org/10.1007/s10955-008-9520-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9520-4