Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem.

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.

[1]  Evans,et al.  Probability of second law violations in shearing steady states. , 1993, Physical review letters.

[2]  R. Chetrite.  Fluctuation relations for diffusion that is thermally driven by a nonstationary bath. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  M. Sasai,et al.  Effects of the DNA state fluctuation on single-cell dynamics of self-regulating gene. , 2007, The Journal of chemical physics.

[4]  Integral fluctuation theorem for the housekeeping heat , 2005, cond-mat/0507420.

[5]  ENERGY FLOW, PARTIAL EQUILIBRATION, AND EFFECTIVE TEMPERATURES IN SYSTEMS WITH SLOW DYNAMICS , 1997, cond-mat/9611044.

[6]  Hong Qian,et al.  Physical origins of entropy production, free energy dissipation, and their mathematical representations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Modified fluctuation-dissipation theorem for non-equilibrium steady states and applications to molecular motors , 2010, 1009.4123.

[8]  J. Parrondo,et al.  Generalized fluctuation-dissipation theorem for steady-state systems. , 2009, Physical review letters.

[9]  Fritz Haake,et al.  Fluctuation-dissipation theorems for classical processes , 1975 .

[10]  Peter Hänggi,et al.  Stochastic processes: Time evolution, symmetries and linear response , 1982 .

[11]  Udo Seifert.  Entropy production along a stochastic trajectory and an integral fluctuation theorem. , 2005, Physical review letters.

[12]  Thermodynamics of a colloidal particle in a time-dependent nonharmonic potential. , 2005, Physical review letters.

[13]  Juan Ruben Gomez-Solano,et al.  Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. , 2009, Physical review letters.

[14]  R. J. Crewther,et al.  Introduction to quantum field theory , 1995, hep-th/9505152.

[15]  G. Hummer,et al.  Free energy reconstruction from nonequilibrium single-molecule pulling experiments , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Evans,et al.  Equilibrium microstates which generate second law violating steady states. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  M. Esposito,et al.  Three faces of the second law. I. Master equation formulation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Thomas Speck,et al.  Fluctuation-dissipation theorem in nonequilibrium steady states , 2009, 0907.5478.

[19]  J. Hasty,et al.  Effective temperature in stochastic kinetics and gene networks. , 2006, Biophysical journal.

[20]  Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Jin Wang,et al.  Kinetic paths, time scale, and underlying landscapes: a path integral framework to study global natures of nonequilibrium systems and networks. , 2010, The Journal of chemical physics.

[22]  Gregory Falkovich,et al.  Fluctuation relations in simple examples of non-equilibrium steady states , 2008, 0806.1875.

[23]  Debra J Searles,et al.  Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. , 2002, Physical review letters.

[24]  R. Kubo.  The fluctuation-dissipation theorem , 1966 .

[25]  Artyom Petrosyan,et al.  Fluctuations and response in a non-equilibrium micron-sized system , 2010, 1006.3196.

[26]  Physical Review , 1965, Nature.

[27]  A. Vulpiani,et al.  Fluctuation-dissipation: Response theory in statistical physics , 2008, 0803.0719.

[28]  Christian Maes,et al.  Fluctuations and response of nonequilibrium states. , 2009, Physical review letters.

[29]  D. Andrieux,et al.  A fluctuation theorem for currents and non-linear response coefficients , 2007, 0704.3318.

[30]  Jin Wang,et al.  Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations , 2008, Proceedings of the National Academy of Sciences.

[31]  C. Jarzynski.  Nonequilibrium Equality for Free Energy Differences , 1996, cond-mat/9610209.

[32]  Takahiro Harada,et al.  Equality connecting energy dissipation with a violation of the fluctuation-response relation. , 2005, Physical review letters.

[33]  T. Hatano,et al.  Steady-state thermodynamics of Langevin systems. , 2000, Physical review letters.