Refined Second Law of Thermodynamics for Fast Random Processes

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.

[1]  G. Gallavottia.  Fluctuation theorem and chaos , 2008 .

[2]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[3]  G. Gallavotti.  On thermostats: isokinetic or Hamiltonian? Finite or infinite? , 2008, Chaos.

[4]  G. Gallavotti.  Fluctuation Theorem and Chaos , 2008, 0802.3694.

[5]  C. Jarzynski.  Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale , 2011 .

[6]  Landauer's erasure principle in non-equilibrium systems , 2011, 1112.0898.

[7]  E. Aurell,et al.  Boundary layers in stochastic thermodynamics. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Udo Seifert,et al.  Stochastic thermodynamics: principles and perspectives , 2007, 0710.1187.

[9]  E. Lutz,et al.  Memory erasure in small systems. , 2008, Physical review letters.

[10]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[11]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[12]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[13]  Leo Szilard,et al.  Über die Ausdehnung der phänomenologischen Thermodynamik auf die Schwankungserscheinungen , 1925 .

[14]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[15]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[16]  K. Gawȩdzki,et al.  Fluctuation Relations for Diffusion Processes , 2007, 0707.2725.

[17]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[18]  Paolo Muratore-Ginanneschi,et al.  Optimal protocols and optimal transport in stochastic thermodynamics. , 2010, Physical review letters.

[19]  C. Maes,et al.  On the definition of entropy production, via examples , 2000 .

[20]  J. Parrondo,et al.  Dissipation: the phase-space perspective. , 2007, Physical review letters.

[21]  P. Glansdorff,et al.  Thermodynamic theory of structure, stability and fluctuations , 1971 .

[22]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[23]  D. Andrieux,et al.  Dynamical randomness, information, and Landauer's principle , 2008, 0801.2727.

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

[25]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[26]  The Fluctuation Theorem as a Gibbs Property , 1998, math-ph/9812015.

[27]  B. Andresen.  Current trends in finite-time thermodynamics. , 2011, Angewandte Chemie.

[28]  Dimitri P. Bertsekas,et al.  Network optimization : continuous and discrete models , 1998 .

[29]  Angelo Plastino,et al.  Landauer's Principle and Divergenceless Dynamical Systems , 2009, Entropy.

[30]  U. Seifert,et al.  Optimal finite-time processes in stochastic thermodynamics. , 2007, Physical review letters.

[31]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[32]  Mario Milman,et al.  Monge Ampère equation : applications to geometry and optimization : NSF-CBMS Conference on the Monge Ampère Equation : Applications to Geometry and Optimization, July 9-13, 1997, Florida Atlantic University , 1999 .

[33]  C. Maes,et al.  Steady state statistics of driven diffusions , 2007, 0708.0489.

[34]  L. Peliti,et al.  Heat Release by Controlled Continuous-Time Markov Jump Processes , 2012, 1203.4062.

[35]  Massimiliano Esposito,et al.  Finite-time thermodynamics for a single-level quantum dot , 2009, 0909.3618.

[36]  Udo Seifert,et al.  Optimal protocols for minimal work processes in underdamped stochastic thermodynamics. , 2008, The Journal of chemical physics.

[37]  U. Frisch,et al.  Reconstruction of the early Universe as a convex optimization problem , 2003 .

[38]  C. Maes.  On the origin and the use of fluctuation relations for the entropy , 2003 .

[39]  C. Jarzynski.  Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach , 1997, cond-mat/9707325.

[40]  C. Maes,et al.  On and beyond entropy production: the case of Markov jump processes , 2007, 0709.4327.

[41]  Debra J. Searles,et al.  The Fluctuation Theorem , 2002 .

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

[43]  E. Lutz,et al.  Experimental verification of Landauer’s principle linking information and thermodynamics , 2012, Nature.

[44]  R. Bellman.  Dynamic programming. , 1957, Science.

[45]  Massimiliano Esposito,et al.  Efficiency at maximum power of low-dissipation Carnot engines. , 2010, Physical review letters.

[46]  M. Qian,et al.  Reversibility and entropy production of inhomogeneous Markov chains , 2006, Journal of Applied Probability.

[47]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[48]  Francesco Guerra,et al.  Quantization of Dynamical Systems and Stochastic Control Theory , 1983 .

[49]  C. Jarzynski,et al.  Path-integral analysis of fluctuation theorems for general Langevin processes , 2006, cond-mat/0605471.

[50]  J. Schnakenberg.  Network theory of microscopic and macroscopic behavior of master equation systems , 1976 .

[51]  C. Villani.  Topics in Optimal Transportation , 2003 .

[52]  H. Soner,et al.  Stochastic Control for a Class of Random Evolution Models , 2004 .

[53]  J. Burgers.  The Nonlinear Diffusion Equation , 2013 .

[54]  Udo Seifert,et al.  Efficiency at maximum power: An analytically solvable model for stochastic heat engines , 2007, 0710.4097.

[55]  Massimiliano Esposito,et al.  Second law and Landauer principle far from equilibrium , 2011, 1104.5165.