Summary
We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. That mostly concerns the time-antisymmetric part of dynamical fluctuation theory.We also briefly comment on recent attempts to combine that with the time-symmetric sector in a Langrangian or extended Onsager-Machlup approach. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles.
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References
L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states, J. Stat. Phys. 107, 635–675 (2002).
I. Bjelaković, J. D. Deuschel, T. Krüger, S. Seiler, R. Siegmund-Schultze, A. Szkola, A quantum version of Sanov's theorem, Commun. Math. Phys. 260, 659–671 (2005).
I. Bjelaković, J. D. Deuschel, T. Krüger, S. Seiler, R. Siegmund-Schultze, A. Szkola, The Shannon-McMillan theorem for ergodic quantum lattice systems, Invent. Math. 155 (1), 203–202 (2004).
A. Bovier, F. den Hollander and F. Nardi, Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary conditions, Prob. Th. Rel. Fields. 135, 265–310 (2006).
O. Bratelli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume 2, Springer, Berlin (1979).
J. Bricmont, Science of chaos or chaos in science, In: The Flight from Science and Reason, Ann. N.Y. Academy of Science, 79, 731 (1996).
G. E. Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 90, 1481 (1998).
N. Datta, Quantum entropy and quantum information, In: Les Houches, Session LXXXIII, 2005, Eds. A. Bovier, F. Dunlop, F. den Hollander, A. van Enter, and J. Dalibard, pp. 395–465 (2006).
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, Jones and Barlett Publishers, Boston (1993).
W. De Roeck, C. Maes, and K. Netočný, An extension of the Kac ring model, J. Phys. A: Math. Gen. 36, 1–13 (2003).
W. De Roeck, C. Maes, and K. Netočný, H-theorems from macroscopic autonomous equations, J. Stat. Phys. 123, 571–584 (2006).
W. De Roeck, C. Maes, and K. Netočný, Quantum macrostates, equivalence of ensembles and an H-theorem, J. Math. Phys. 47, 073303 (2006).
B. Derrida, J. L. Lebowitz, and E. R. Speer, Free Energy Functional for Nonequilibrium Systems: An Exactly Soluble Case, Phys. Rev. Lett. 87, 150601 (2001); J. Stat. Phys. 107, 599 (2002).
M. D. Donsker, S. R. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time I., Comm. Pure Appl. Math. 28:1–47 (1975).
M. Dresden, F. Feiock, Models in nonequilibrium quantum statistical mechanics, J. Stat. Phys. 4, 111–173 (1972).
D. E. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in steady flows, Phys. Rev. Lett. 71, 2401–2404 (1993). cond-mat/9908420
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett. 74, 2694–2697 (1995); Dynamical ensembles in stationary states, J. Stat. Phys. 80, 931–970 (1995).
P. L. Garrido, S. Goldstein and J. L. Lebowitz, The Boltzmann entropy for dense fluids not in local equilibrium, Phys. Rev. Lett. 92, 050602 (2003).
H.-O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, New York (1988).
S. Goldstein and J. L. Lebowitz, On the (Boltzmann) entropy of nonequilibrium systems, Physica D 193, 53–56 (2004).
S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland Publishing Company (1962).
P. Hänggi, P. Talkner, and M. Borkovec, Reaction rate theory: fifty years after Kramers, Rev. of Mod. Phys. 62, 251–341 (1990).
F. den Hollander, Large Deviations, Field Institute Monographs, Providence, Rhode Island (2000).
R. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press (1978).
C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690–2693 (1997).
E. T. Jaynes, The second law as physical fact and as human inference (1990) (unpublished), download from http://www.etjaynescenter.org /bibliography.shtml.
E. T. Jaynes, The evolution of Carnot's principle, In: Maximum-Entropy and Bayesian Methods in Science and Engineering, 1, Eds. G. J. Erickson and C. R. Smith, Kluwer, Dordrecht, pp. 267–281 (1988).
E. T. Jaynes, Gibbs vs Boltzmann entropies, In: Papers on Probability, Statistics and Statistical Physics, Reidel, Dordrecht (1983); Am. J. Phys. 33, 391–398 (1965).
M. Kac. Probability and Related Topics in Physical Sciences, Interscience Publishers Inc., New York (1959).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin (1999).
R. Kubo, K. Matsuo, and K. Kitahara, Fluctuation and Relaxation of Macrovariables, J. Stat. Phys. 9, 51–95 (1973).
O. E. Lanford III, Time evolution of large classical systems, Lecture Notes in Phys. 38, 1–111 (1975); The hard sphere gas in the Boltzmann-Grad limit, Physica 106A, 70–76 (1981).
J. L. Lebowitz, Microscoic origins of irreversible macroscopic behavior, Physica A 263, 516–527 (1999).
J. L. Lebowitz, C. Maes, and E. R. Speer, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59 117–170 (1990).
J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys. 95, 333–365 (1999).
T. M. Liggett, Interacting Particle Systems, Springer (1985).
R. Lima, Equivalence of ensembles in quantum lattice systems: states, Commun. Math. Phys. 24, 180–192 (1972).
C. Maes, On the origin and the use of fluctuation relations for the entropy, Poincaré Seminar 2003, Eds. J. Dalibard, B. Duplantier and V. Rivasseau, Birkhäuser (Basel), pp. 145–191 (2004).
C. Maes. Fluctuation theorem as a Gibbs property, J. Stat. Phys. 95, 367–392 (1999).
C. Maes and K. Netočný, Time-reversal and entropy, J. Stat. Phys. 110, 269–310 (2003).
C. Maes and K. Netočný, Static and dynamical nonequilibrium fluctuations, cond-mat/0612525.
C. Maes, K. Netočný, and M. Verschuere, Heat conduction networks, J. Stat. Phys. 111, 1219–1244 (2003).
C. Maes, F. Redig, and A. Van Moffaert, On the definition of entropy production via examples, J. Math. Phys. 41, 1528–1554 (2000).
C. Maes, F. Redig, and A. Van Moffaert, The restriction of the Ising model to a layer, J. Stat. Phys. 96, 69–107 (1999).
C. Maes and K. Netočný, The canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhysics Letters 82, 30003 (2008).
C. Maes, K. Netočný and B. Wynants, On and beyond entropy production; the case of Markov jump processes, to appear in Markov Processes and Related Fields (2008).
C. Maes and K. Netočný, Minimum entropy production principle from a dynamical fluctuation law, J.Math.Phys. 48, 053306 (2007).
C. Maes, K. Netočný and B. Wynants, Steady state statistics of driven diffusion, Physica A 387, 2675–2689 (2008).
K. Netočný and F. Redig, Large deviations for quantum spin systems, J. Stat. Phys. 117, 521–547 (2004).
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Translated form the German edition by R. T. Beyer, Princeton University Press, Princeton(1955).
D. Petz, G. A. Raggio, and A. Verbeure, Asymptotics of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys. 121, 271-?282 (1989) .
I. Prigogine, Introduction to Non-Equilibrium Thermodynamics, Wiley-Interscience, New York (1962).
M. Lenci, L. Rey-Bellet, Large deviations in quantum lattice systems: one-phase region, J. Stat. Phys., 119, 715–746 (2005).
D. Ruelle, Smooth dynamics and new rheoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95, 393–468 (1999).
B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press Princeton, (1993).
H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag Heidelberg, (1991).
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Maes, C., Netoˇcn´y, ., Shergelashvili, . (2009). A Selection of Nonequilibrium Issues. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_6
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