Nonequilibrium fluctuations in small systems: From physics to biology

In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule force experiments. Next I present a general discussion on path thermodynamics and consider distributions of work/heat fluctuations as large deviation functions. Then I address the topic of glassy dynamics from the perspective of nonequilibrium fluctuations due to small cooperatively rearranging regions. Finally, I conclude with a brief digression on future perspectives.

[1]  Ignacio Tinoco,et al.  Force unfolding kinetics of RNA using optical tweezers. I. Effects of experimental variables on measured results. , 2007, Biophysical journal.

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

[3]  Observable dependence of fluctuation-dissipation relations and effective temperatures. , 2001, Physical review letters.

[4]  J. P. Garrahan,et al.  Coarse-grained microscopic model of glass formers , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Jorge Kurchan,et al.  Fluctuation theorem for stochastic dynamics , 1998 .

[6]  Jeffrey W. Roberts,et al.  遺伝子の分子生物学 = Molecular biology of the gene , 1970 .

[7]  Christopher Jarzynski,et al.  Work distribution for the adiabatic compression of a dilute and interacting classical gas. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Gerhard Hummer,et al.  Free energy surfaces from single-molecule force spectroscopy. , 2005, Accounts of chemical research.

[9]  G. Crooks.  Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  R. Astumian.  The unreasonable effectiveness of equilibrium theory for interpreting nonequilibrium experiments , 2005, cond-mat/0512024.

[11]  Steven M Block,et al.  Passive all-optical force clamp for high-resolution laser trapping. , 2005, Physical review letters.

[12]  The Manipulation of Single Biomolecules , 2001 .

[13]  T. L. Hill,et al.  Muscle contraction and free energy transduction in biological systems. , 1985, Science.

[14]  M. Mézard,et al.  Self induced quenched disorder: a model for the glass transition , 1994, cond-mat/9405075.

[15]  Joshua W. Shaevitz,et al.  Backtracking by single RNA polymerase molecules observed at near-base-pair resolution , 2003, Nature.

[16]  W. Gotze,et al.  Relaxation processes in supercooled liquids , 1992 .

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

[18]  D. Swigon,et al.  Relating single-molecule measurements to thermodynamics. , 2003, Biophysical journal.

[19]  Cohen,et al.  Dynamical Ensembles in Nonequilibrium Statistical Mechanics. , 1994, Physical review letters.

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

[21]  Determination of the chemical potential using energy-biased sampling. , 2005, The Journal of chemical physics.

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

[23]  Ioan Kosztin,et al.  Calculating potentials of mean force and diffusion coefficients from nonequilibrium processes without Jarzynski's equality. , 2006, The Journal of chemical physics.

[24]  Luca Cipelletti,et al.  Slow dynamics in glassy soft matter , 2005 .

[25]  Work distribution and path integrals in general mean-field systems , 2005, cond-mat/0501576.

[26]  I. Tinoco,et al.  How RNA folds. , 1999, Journal of molecular biology.

[27]  G. Biroli,et al.  On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses. , 2004, Journal of Chemical Physics.

[28]  K. Schulten,et al.  Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality , 2003 .

[29]  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.

[30]  Erwin Frey,et al.  Brownian motion: a paradigm of soft matter and biological physics , 2005, Annalen der Physik.

[31]  A. Crisanti,et al.  Intermittency of glassy relaxation and the emergence of a non-equilibrium spontaneous measure in the aging regime , 2003 .

[32]  C. Jarzynski,et al.  Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies , 2005, Nature.

[33]  Mode-dependent nonequilibrium temperature in aging systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  B. Derrida,et al.  Large Deviation of the Density Profile in the Symmetric Simple Exclusion Process , 2001 .

[35]  E. R. Speer,et al.  Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process , 2002 .

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

[37]  J Kurchan,et al.  Metastable states in glassy systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  S. Smith,et al.  Single-molecule studies of DNA mechanics. , 2000, Current opinion in structural biology.

[39]  F. Ritort,et al.  Force unfolding kinetics of RNA using optical tweezers. II. Modeling experiments. , 2007, Biophysical journal.

[40]  E. Sevick,et al.  Fluctuations and irreversibility: an experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap. , 2004, Physical review letters.

[41]  Jorge Kurchan,et al.  Direct evaluation of large-deviation functions. , 2005, Physical review letters.

[42]  I. Tinoco,et al.  Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality , 2002, Science.

[43]  D. Wheatley.  The way of the cell , 2001 .

[44]  S Ciliberto,et al.  Power and heat fluctuation theorems for electric circuits. , 2004, Physical review letters.

[45]  F. Ritort,et al.  The nonequilibrium thermodynamics of small systems , 2005 .

[46]  On the static and dynamical transition in the mean-field Potts glass , 1994, cond-mat/9410093.

[47]  E. Evans.  Probing the relation between force--lifetime--and chemistry in single molecular bonds. , 2001, Annual review of biophysics and biomolecular structure.

[48]  Direct visualization of ageing in colloidal glasses , 2002, cond-mat/0209148.

[49]  G. Adam,et al.  On the Temperature Dependence of Cooperative Relaxation Properties in Glass‐Forming Liquids , 1965 .

[50]  M. Karplus,et al.  Self-guided enhanced sampling methods for thermodynamic averages , 2003 .

[51]  Jean-Philippe Bouchaud,et al.  On the Adam-Gibbs-Wolynes scenario for the viscosity increase in glasses , 2022 .

[52]  K. Schulten,et al.  Calculating potentials of mean force from steered molecular dynamics simulations. , 2004, The Journal of chemical physics.

[53]  S. Cocco,et al.  Theoretical models for single-molecule DNA and RNA experiments: from elasticity to unzipping , 2002, cond-mat/0206238.

[54]  A. Grosberg,et al.  Practical applicability of the Jarzynski relation in statistical mechanics: a pedagogical example. , 2005, The journal of physical chemistry. B.

[55]  M D Ediger,et al.  Spatially heterogeneous dynamics in supercooled liquids. , 2003, Annual review of physical chemistry.

[56]  C. Angell,et al.  Formation of Glasses from Liquids and Biopolymers , 1995, Science.

[57]  Aaron Klug,et al.  A Marvellous Machine for Making Messages , 2001, Science.

[58]  Bryant S. Fujimoto,et al.  Equalities for the Nonequilibrium Work Transferred from an External Potential to a Molecular System. Analysis of Single-Molecule Extension Experiments , 2003 .

[59]  Jean-Philippe Bouchaud,et al.  Mode-coupling approximations, glass theory and disordered systems , 1995, cond-mat/9511042.

[60]  F. Ritort,et al.  Methods and Applications , 2006 .

[61]  F. Ritort,et al.  Thermodynamic and kinetic aspects of RNA pulling experiments. , 2004, Biophysical journal.

[62]  J. Liphardt,et al.  Reversible Unfolding of Single RNA Molecules by Mechanical Force , 2001, Science.

[63]  E. Cohen.  Some Recent Advances in Classical Statistical Mechanics , 2002 .

[64]  Gilles Charvin,et al.  Stretching of macromolecules and proteins , 2003 .

[65]  Martin Karplus,et al.  Optimal estimates of free energies from multistate nonequilibrium work data. , 2006, Physical review letters.

[66]  Ignacio Tinoco,et al.  The effect of force on thermodynamics and kinetics of single molecule reactions. , 2002, Biophysical chemistry.

[67]  Michelle D. Wang,et al.  Force and velocity measured for single molecules of RNA polymerase. , 1998, Science.

[68]  Evan Evans,et al.  Dynamic Force Spectroscopy , 2002 .

[69]  Inherent structures, configurational entropy and slow glassy dynamics , 2001, cond-mat/0110259.

[70]  J. H. Gibbs,et al.  Nature of the Glass Transition and the Glassy State , 1958 .

[71]  U. Seifert,et al.  Probing molecular free energy landscapes by periodic loading. , 2004, Physical review letters.

[72]  Simona Cocco,et al.  The micromechanics of DNA , 2003 .

[73]  Steven M Block,et al.  Resource Letter: LBOT-1: Laser-based optical tweezers. , 2003, American journal of physics.

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

[75]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[76]  C. Broeck,et al.  Jarzynski equality for the Jepsen gas , 2005, cond-mat/0506289.

[77]  Work Fluctuations, Transient Violations of the Second Law and Free-Energy Recovery Methods: Perspectives in Theory and Experiments , 2004, cond-mat/0401311.

[78]  E. Schrödinger,et al.  What is life? : the physical aspect of the living cell , 1946 .

[79]  Stimulated and spontaneous relaxation in glassy systems , 2003, cond-mat/0311370.

[80]  A. Crisanti,et al.  Activated processes and Inherent Structure dynamics of finite-size mean-field models for glasses , 2000 .

[81]  P. Pusey,et al.  Phase behaviour of concentrated suspensions of nearly hard colloidal spheres , 1986, Nature.

[82]  K. Schulten,et al.  Steered molecular dynamics and mechanical functions of proteins. , 2001, Current opinion in structural biology.

[83]  F. Ritort,et al.  Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[84]  Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism , 2004, cond-mat/0405707.

[85]  A. Crisanti,et al.  Potential energy landscape of finite-size mean-field models for glasses , 2000 .

[86]  G. Crooks.  Path-ensemble averages in systems driven far from equilibrium , 1999, cond-mat/9908420.

[87]  E. Weeks,et al.  S ep 2 00 2 Direct visualization of aging in colloidal glasses , 2002 .

[88]  R van Zon,et al.  Extended heat-fluctuation theorems for a system with deterministic and stochastic forces. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[89]  C. Jarzynski,et al.  A “fast growth” method of computing free energy differences , 2001 .

[90]  G. Parisi,et al.  Replica field theory for deterministic models: I. Binary sequences with low autocorrelation , 1994, hep-th/9405148.

[91]  K. Schulten,et al.  Energetics of glycerol conduction through aquaglyceroporin GlpF , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[92]  B Ermentrout,et al.  Dynamics of single-motor molecules: the thermal ratchet model. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[93]  J. Kurchan,et al.  Weak ergodicity breaking in mean-field spin-glass models , 1994, cond-mat/9403040.

[94]  Schofield,et al.  Three-dimensional direct imaging of structural relaxation near the colloidal glass transition , 2000, Science.

[95]  G. Orphanides,et al.  A Unified Theory of Gene Expression , 2002, Cell.

[96]  Aging in the linear harmonic oscillator , 1998, cond-mat/9706303.

[97]  Equilibrium and ageing dynamics of simple models for glasses , 1999, cond-mat/9912207.

[98]  Thomas B Woolf,et al.  Theory of a systematic computational error in free energy differences. , 2002, Physical review letters.

[99]  S. Ciliberto,et al.  Experimental test of the Gallavotti–Cohen fluctuation theorem in turbulent flows , 2004 .

[100]  A. Crisanti,et al.  A glass transition scenario based on heterogeneities and entropy barriers , 2001, cond-mat/0106328.

[101]  Marco Paniconi,et al.  Steady State Thermodynamics , 1998 .

[102]  D. Thirumalai,et al.  RNA and protein folding: common themes and variations. , 2005, Biochemistry.

[103]  C. Maes,et al.  Time-symmetric fluctuations in nonequilibrium systems. , 2006, Physical Review Letters.

[104]  Jorge Kurchan,et al.  Non-equilibrium work relations , 2005, cond-mat/0511073.

[105]  Daniel M. Zuckerman,et al.  Overcoming finite-sampling errors in fast-switching free-energy estimates: extrapolative analysis of a molecular system , 2002 .

[106]  Charles H. Bennett,et al.  Efficient estimation of free energy differences from Monte Carlo data , 1976 .

[107]  S. Ciliberto,et al.  An experimental test of the Gallavotti-Cohen fluctuation theorem , 1998 .

[108]  Stationary and transient work-fluctuation theorems for a dragged Brownian particle. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[110]  Nancy R Forde,et al.  Mechanical processes in biochemistry. , 2004, Annual review of biochemistry.

[111]  C. Jarzynski.  Rare events and the convergence of exponentially averaged work values. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[112]  D. Saad.  Europhysics Letters , 1997 .

[113]  E. Gardner.  Spin glasses with p-spin interactions , 1985 .

[114]  E. R. Speer,et al.  Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process , 2002 .

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

[116]  S Ciliberto,et al.  Nonequilibrium fluctuations in a resistor. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[117]  J. Bouchaud,et al.  The Kovacs effect in model glasses , 2003, cond-mat/0306089.

[118]  C. Bustamante,et al.  Single-molecule study of transcriptional pausing and arrest by E. coli RNA polymerase. , 2000, Science.

[119]  C. Jarzynski,et al.  Exactly solvable model illustrating far-from-equilibrium predictions , 1999 .

[120]  T. Speck,et al.  Dissipated work in driven harmonic diffusive systems: General solution and application to stretching Rouse polymers , 2005 .

[121]  F. Douarche,et al.  An experimental test of the Jarzynski equality in a mechanical experiment , 2005 .

[122]  B. Derrida,et al.  Free energy functional for nonequilibrium systems: an exactly solvable case. , 2001, Physical review letters.

[123]  F. Ritort,et al.  Force-dependent fragility in RNA hairpins. , 2006, Physical review letters.

[124]  On the work distribution for the adiabatic compression of a diluteclassical gas , 2006 .

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

[126]  Daniel K Wójcik,et al.  Classical and quantum fluctuation theorems for heat exchange. , 2004, Physical review letters.

[127]  P. Hänggi,et al.  Reaction-rate theory: fifty years after Kramers , 1990 .

[128]  G. Parisi,et al.  Replica field theory for deterministic models: II. A non-random spin glass with glassy behaviour , 1994, cond-mat/9406074.

[129]  K. Kawasaki,et al.  Nonlinear Effects in the Shear Viscosity of Critical Mixtures , 1967 .

[130]  G. Hummer.  Fast-growth thermodynamic integration: Error and efficiency analysis , 2001 .

[131]  B. Derrida.  Random-Energy Model: Limit of a Family of Disordered Models , 1980 .

[132]  Robert H. Wood,et al.  Systematic errors in free energy perturbation calculations due to a finite sample of configuration space: sample-size hysteresis , 1991 .

[133]  Martin Goldstein,et al.  Viscous Liquids and the Glass Transition: A Potential Energy Barrier Picture , 1969 .

[134]  T Hwa,et al.  Force-induced denaturation of RNA. , 2001, Biophysical journal.

[135]  C Van den Broeck,et al.  Fluctuation and dissipation of work in a Joule experiment. , 2006, Physical review letters.

[136]  Y. Kuzovlev,et al.  Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics , 1981 .

[137]  Nancy R. Forde,et al.  Using mechanical force to probe the mechanism of pausing and arrest during continuous elongation by Escherichia coli RNA polymerase , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[138]  J M Rubí,et al.  The mesoscopic dynamics of thermodynamic systems. , 2005, The journal of physical chemistry. B.

[139]  Giovanni Gallavotti,et al.  Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem , 1995, chao-dyn/9506006.

[140]  D. Gross,et al.  Microcanonical Thermodynamics: Phase Transitions in 'Small' Systems , 2001 .

[141]  T. Speck,et al.  Experimental test of the fluctuation theorem for a driven two-level system with time-dependent rates. , 2005, Physical review letters.

[142]  F. Jülicher,et al.  Modeling molecular motors , 1997 .

[143]  C. Jarzynski.  What Is the Microscopic Response of a System Driven Far From Equilibrium , 2002 .

[144]  L. Peliti,et al.  Evaluation of free energy landscapes from manipulation experiments , 2006 .

[145]  C Jarzynski,et al.  How does a system respond when driven away from thermal equilibrium? , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[146]  P G Wolynes,et al.  Fragilities of liquids predicted from the random first order transition theory of glasses. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[147]  E. Cohen,et al.  A note on the Jarzynski equality , 2004 .

[148]  A. Crisanti,et al.  TOPICAL REVIEW: Violation of the fluctuation dissipation theorem in glassy systems: basic notions and the numerical evidence , 2002 .

[149]  Activated aging dynamics and negative fluctuation-dissipation ratios. , 2005, Physical review letters.

[150]  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.

[151]  C Jarzynski,et al.  Experimental test of Hatano and Sasa's nonequilibrium steady-state equality. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[152]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[153]  W. Kob.  Supercooled Liquids and Glasses , 1999, cond-mat/9911023.

[154]  Robert Landick,et al.  RNA Polymerase as a Molecular Motor , 1998, Cell.

[155]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[156]  R. Zwanzig.  High‐Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases , 1954 .

[157]  Denis J. Evans,et al.  A non-equilibrium free energy theorem for deterministic systems , 2003 .

[158]  Ritort.  Glassiness in a Model without Energy Barriers. , 1995, Physical review letters.

[159]  Some non-perturbative calculations on spin glasses , 1994, cond-mat/9410107.

[160]  H. Qian.  Cycle kinetics, steady state thermodynamics and motors—a paradigm for living matter physics , 2005, Journal of physics. Condensed matter : an Institute of Physics journal.

[161]  Franz,et al.  Glassy transition and aging in a model without disorder. , 1994, Physical Review Letters.

[162]  Intermittency in ageing , 2002, cond-mat/0210490.

[163]  G. Biroli,et al.  Diverging length scale and upper critical dimension in the Mode-Coupling Theory of the glass transition , 2004, cond-mat/0401260.

[164]  E. Cohen,et al.  Extension of the fluctuation theorem. , 2003, Physical review letters.

[165]  T. L. Hill,et al.  Thermodynamics of Small Systems , 2002 .

[166]  W. Greenleaf,et al.  Direct observation of base-pair stepping by RNA polymerase , 2005, Nature.

[167]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[168]  Fluctuation symmetries for work and heat. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[169]  E. Sevick,et al.  Reversibility in nonequilibrium trajectories of an optically trapped particle. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[170]  Work-probability distribution in systems driven out of equilibrium. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[171]  Carlos Bustamante,et al.  Optical-trap force transducer that operates by direct measurement of light momentum. , 2003, Methods in enzymology.

[172]  J. Kirkwood.  Statistical Mechanics of Fluid Mixtures , 1935 .

[173]  Cugliandolo,et al.  Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. , 1993, Physical review letters.

[174]  Estimate of the free energy difference in mechanical systems from work fluctuations: experiments and models , 2005, cond-mat/0504465.

[175]  Spontaneous Relaxation in Generalized Oscillator Models with Glassy Dynamics , 2004, cond-mat/0412650.

[176]  L. Berthier,et al.  Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition , 2005, Science.

[177]  V. I. Mel'Nikov,et al.  The Kramers problem : fifty years of development , 1991 .

[178]  A B Kolomeisky,et al.  The force exerted by a molecular motor. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[179]  F. Ritort,et al.  A two-state kinetic model for the unfolding of single molecules by mechanical force , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[180]  Fluidized granular medium as an instance of the fluctuation theorem. , 2003, Physical review letters.

[181]  Tartaglia,et al.  Extension of the Fluctuation-Dissipation Theorem to the Physical Aging of a Model Glass-Forming Liquid. , 2000, Physical review letters.

[182]  Stillinger,et al.  Statistical mechanics of metastable matter: Superheated and stretched liquids. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[183]  Michael R. Shirts,et al.  Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods. , 2003, Physical review letters.

[184]  Ignacio Tinoco,et al.  Force as a useful variable in reactions: unfolding RNA. , 2004, Annual review of biophysics and biomolecular structure.

[185]  Carlos Bustamante,et al.  Direct Observation of the Three-State Folding of a Single Protein Molecule , 2005, Science.

[186]  Christopher Jarzynski,et al.  Nonequilibrium work theorem for a system strongly coupled to a thermal environment , 2004 .

[187]  J. Lebowitz,et al.  A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics , 1998, cond-mat/9811220.

[188]  Carlos Bustamante,et al.  Grabbing the cat by the tail: manipulating molecules one by one , 2000, Nature Reviews Molecular Cell Biology.

[189]  J. Jäckle,et al.  Models of the glass transition , 1986 .

[190]  C. Calladine,et al.  Understanding DNA: The Molecule & How It Works , 1992 .

[191]  Steady State Thermodynamics , 2004, cond-mat/0411052.