Stochastic dynamics, large deviation principle, and nonequilibrium thermodynamics.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

[1]  H. Qian,et al.  Emergence and Breaking of Duality Symmetry in Thermodynamic Behavior: Repeated Measurements and Macroscopic Limit , 2020 .

[2]  H. Qian,et al.  The statistical foundation of entropy in extended irreversible thermodynamics , 2020, Journal of Physics A: Mathematical and Theoretical.

[3]  H. Qian,et al.  Kinematic basis of emergent energetics of complex dynamics , 2017, EPL (Europhysics Letters).

[4]  Liangrong Peng,et al.  Generalized Onsager's reciprocal relations for the master and Fokker-Planck equations. , 2018, Physical review. E.

[5]  Amir Dembo,et al.  Large deviations theory for Markov jump models of chemical reaction networks , 2017, The Annals of Applied Probability.

[6]  C. Maes.  Frenetic Bounds on the Entropy Production. , 2017, Physical review letters.

[7]  D. Khoshnevisan,et al.  From Lévy-Type Processes to Parabolic SPDEs , 2016 .

[8]  Hong Qian,et al.  Mesoscopic kinetic basis of macroscopic chemical thermodynamics: A mathematical theory. , 2016, Physical review. E.

[9]  Carsten Wiuf,et al.  Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks , 2015, Bulletin of mathematical biology.

[10]  A. C. Barato,et al.  A Formal View on Level 2.5 Large Deviations and Fluctuation Relations , 2014, 1408.5033.

[11]  C. Landim,et al.  Macroscopic fluctuation theory , 2014, 1404.6466.

[12]  Wen-An Yong,et al.  Conservation-dissipation formalism of irreversible thermodynamics , 2014, 1407.5498.

[13]  M. Jara,et al.  The Einstein Relation for the KPZ Equation , 2014, 1407.3153.

[14]  H. Qian,et al.  Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation , 2012, 1205.6052.

[15]  U. Seifert.  Stochastic thermodynamics, fluctuation theorems and molecular machines , 2012, Reports on progress in physics. Physical Society.

[16]  L. Petzold,et al.  Reaction-diffusion master equation in the microscopic limit. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  E. Çinlar.  Probability and Stochastics , 2011 .

[18]  Eric Smith.  Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions , 2011, 1102.3938.

[19]  Hydrodynamical behavior of symmetric exclusion with slow bonds , 2010, 1010.4769.

[20]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[21]  H. Touchette.  The large deviation approach to statistical mechanics , 2008, 0804.0327.

[22]  Stefan Grosskinsky Warwick,et al.  Interacting Particle Systems , 2016 .

[23]  Jin Feng.  Large deviation for diffusions and Hamilton–Jacobi equation in Hilbert spaces , 2006, math/0602655.

[24]  L. Bertini,et al.  Large deviations of the empirical current in interacting particle systems , 2007 .

[25]  David Ruelle,et al.  Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics , 2004 .

[26]  G. Gallavotti,et al.  Nonequilibrium thermodynamics , 2003, 1901.08821.

[27]  C. Landim,et al.  Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States , 2001, cond-mat/0108040.

[28]  Spatial Correlations near Turing Instabilities: Criteria for Wavenumber Selection , 1999 .

[29]  B. Derrida.  AN EXACTLY SOLUBLE NON-EQUILIBRIUM SYSTEM : THE ASYMMETRIC SIMPLE EXCLUSION PROCESS , 1998 .

[30]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[31]  B. Derrida,et al.  Exact Large Deviation Function in the Asymmetric Exclusion Process , 1998, cond-mat/9809044.

[32]  Jianfeng Feng.  The hydrodynamic limit for the reaction diffusion equation — An approach in terms of the GPV method , 1996 .

[33]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[34]  Mark Dykman,et al.  Large fluctuations and optimal paths in chemical kinetics , 1994 .

[35]  Horng-Tzer Yau,et al.  Relative entropy and hydrodynamics of Ginzburg-Landau models , 1991 .

[36]  Graham,et al.  Steady-state ensemble for the complex Ginzburg-Landau equation with weak noise. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[37]  J. Kevorkian,et al.  Partial Differential Equations: Analytical Solution Techniques , 1990 .

[38]  Yoshitsugu Oono,et al.  Large Deviation and Statistical Physics , 1989 .

[39]  Stefano Olla,et al.  Hydrodynamics and large deviation for simple exclusion processes , 1989 .

[40]  George Papanicolaou,et al.  Nonlinear diffusion limit for a system with nearest neighbor interactions , 1988 .

[41]  Joel Keizer,et al.  Statistical Thermodynamics of Nonequilibrium Processes , 1987 .

[42]  Hu.  Stationary solution of master equations in the large-system-size limit. , 1987, Physical review. A, General physics.

[43]  J. Serrin.  New Perspectives in Thermodynamics , 1986 .

[44]  Graham,et al.  Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems. , 1985, Physical review. A, General physics.

[45]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[46]  Robert Graham,et al.  On the weak-noise limit of Fokker-Planck models , 1984 .

[47]  James Serrin,et al.  The Concepts of Thermodynamics , 1978 .

[48]  T. L. Hill,et al.  Free Energy Transduction in Biology: The Steady-State Kinetic and Thermodynamic Formalism , 1977 .

[49]  G. Carrier,et al.  Partial Differential Equations: Theory and Technique , 1976 .

[50]  R. Jackson,et al.  General mass action kinetics , 1972 .

[51]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[52]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[53]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[54]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[55]  L. Onsager.  Reciprocal Relations in Irreversible Processes. II. , 1931 .