Foundations and History of Statistical Mechanics

Last update: 22 Dec 2024 19:09
First version: 12 October 2007 (or earlier?)

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Technical issues: things like, what exactly is a C* algebra? Role of large deviations.

Conceptual issues: Why is it legitimate to treat deterministic mechanical systems with many unstable degrees of freedom as stochastic processes? (My impulse is to appeal to ergodic theory.) When and why do we get convergence to equilibria characterized by only a few macroscopic degrees of freedom? (That sounds like a central limit theorem, some kind of result about how the large-scale limit is insensitive to all but a few aspects of the small scales.)

Historical issues: It's interesting to know how people have argued about this stuff.

See also:
• Statistical Mechanics
• Nonequilibrium Statistical Mechanics
• Maximum Entropy
• Tsallis Statistics
• Large Deviations
• Concentration of Measure
• Dynamical Systems (Including Chaos)

Recommended, big picture:
• David Z. Albert, Time and Chance
• Patrizia Castiglione, Massimo Falcioni, Annick Lesne and Angelo Vulpiani, Chaos and Coarse Graining in Statistical Mechanics [Mini-review]
• A. Greven, G. Keller and G. Warnecke (eds.), Entropy
• A. I. Khinchin, Mathematical Foundations of Statistical Mechanics
• Joel L. Lebowitz, "Statistical mechanics: A selective Review of Two Central Issues", Reviews of Modern Physics 71 (1999): S346--S357, math-ph/0010018 [Abstract: "I give a highly selective overview of the way statistical mechanics explains the microscopic origins of the time-asymmetric evolution of macroscopic systems towards equilibrium and of first-order phase transitions in equilibrium. These phenomena are emergent collective properties not discernible in the behavior of individual atoms. They are given precise and elegant mathematical formulations when the ratio between macroscopic and microscopic scales becomes very large."]
• Lawrence Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics
• Eric Smith, "Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions", arxiv:1102.3938
• Hugo Touchette, "The Large Deviations Approach to Statistical Mechanics", arxiv:0804.0327

Recommended, close-ups:
• Jean Bricmont, "Science of Chaos or Chaos in Science?", chao-dyn/9603009
• Stephen G. Brush, "Foundations of Statistical Mechanics 1845--1915", Archive for the History of Exact Sciences 4 (1966): 145--183
• E. G. D. Cohen, "Entropy, Probability and Dynamics", arxiv:0807.1268
• W. De Roeck, Christian Maes and Karel Netocny, "H-Theorems from Autonomous Equations", cond-mat/0508089 = Journal of Statistical Physics 123 (2006): 571--584 ["If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution."]
• Richard S. Ellis, Entropy, Large Deviations and Statistical Mechanics
• Sheldon Goldstein, "Boltzmann's Approach to Statistical Mechanics," cond-mat/0105242 ["most twentieth-century innovations are thoroughly misguided"]
• Michael C. Mackey
  • "The Dynamic Origin of Increasing Entropy", Reviews of Modern Physics 61 (1989): 981--1015 [Basically a precis of Time's Arrow]
  • Time's Arrow: The Origins of Thermodynamic Behavior [This is a very valuable short introduction to the ergodic theory of Markov operators, which is highly relevant to the origins of irreversibility, etc., but I don't think his favored approach here works, because he focuses on the relative entropy (Kullback-Leibler divergence from the invariant distribution), rather than the Boltzmann entropy or even the Gibbs entropy.]
• Benoit Mandelbrot, "The Role of Sufficiency and of Estimation in Thermodynamics", Annals of Mathematical Statistics 33 (1962): 1021--1038 [Extensive thermodynamic variables as sufficient statistics for the conjugate intensive variables; Gibbs canonical form arising from natural requirements on finite-dimensional sufficient statistics, which can only be achieved for exponential families of probability distributions. Very clever.]
• Sandu Popescu, Anthony J. Short, and Andreas Winter, "Entanglement and the Foundations of Statistical Mechanics", quant-ph/0511225 [Roughly speaking: due to environmental entanglement, most states of a sub-system look "thermalized", no matter what the real state of the whole system is]
• Hans Reichenbach, The Direction of Time [Comments]
• Steven Savitt (ed.), Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time
• Geoffrey Sewell
  • Quantum Mechanics and Its Emergent Macrophysics
  • "On the Question of Temperature Transformations under Lorentz and Galilei Boosts", arxiv:0808.0803 [Punch-line: "there is no law of temperature transformation under either Lorentz or Galilei boosts, and so the concept of temperature stemming from the Zeroth Law is restricted to states of bodies in their rest frames."]
• J. Uffink, "Bluff Your Way in the Second Law of Thermodynamics," cond-mat/0005327
• W. H. Zurek, "Algorithmic Randomness, Physical Entropy, Measurements, and the Demon of Choice," quant-ph/9807007

Modesty forbids:
• CRS, "The Backwards Arrow of Time of the Consistently Bayesian Statistical Mechanic", cond-mat/0410063 [Self-exposition]
• CRS and Cristopher Moore, "What Is a Macrostate?", Foundations of Physics 55 (2025): 2, cond-mat/0303625

To read:
• Walid K. Abou Salem and Jürg Fröhlich, "Status of the Fundamental Laws of Thermodynamics", Journal of Statistical Physics 126 (2007): 1045-1068 ["We describe recent progress towards deriving the Fundamental Laws of thermodynamics (the 0th, 1st, and 2nd Law) from nonequilibrium quantum statistical mechanics in simple, yet physically relevant models."]
• A. E. Allahverdyan and Th. M. Nieuwenhuizen, "Explanation of the Gibbs paradox within the framework of quantum thermodynamics", Physical Review E 73 (2006): 066119 = quant-ph/0507145 [The abstract says many things with which I am sympathetic, most notably coming out against "a direct association of physical irreversibility with lack of information", but I don't know if I'll ever find time to read this...]
• Massimiliano Badino
  • "The Foundational Role of Ergodic Theory", phil-sci/2277
  • "Probability and Statistics in Boltzmann's Early Papers on Kinetic Theory", phil-sci/2276
  • "Was there a statistical Turn? The Interaction between Mechanics and Probability in Boltzmann's Theory of Non Equilibrium (1872-1877)", phil-sci/2878
• Robert W. Batterman, "Why Equilibrium Statistical Mechanics Works: Universality and the Renormalization Group", Philosopy of Science 65 (1998): 183--208 [JSTOR]
• Battimelli et al., (eds.), Proceedings of the Int'l Symposium on Ludwig Boltzmann
• Joseph Berkovitz, Roman Frigg and Fred Kronz, "The Ergodic Hierarchy, Randomness and Hamiltonian Chaos", phil-sci/2927
• Ludwig Boltzmann, Lectures on Gas Theory [Get the Dover reprint]
• Michele Campisi, "Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy", arxiv:0704.2567 [i.e., Boltzmann entropy]
• Michele Campisi and Donald H. Kobe, "Derivation of Boltzmann Principle", arxiv:0911.2070
• Miguel Carrion-Alvarez, "Variations on a theme of Gelfand and Naimark", math.FA/0402150 [Algebras of observables, including C* algebras as a special case]
• Hasok Chang, Inventing Temperature: Measurement and Scientific Progress
• Marius Costeniuc, Richard S. Ellis, Hugo Touchette and Bruce Turkington, "The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble", Journal of Statistical Physics 119 (2005): 1283--1329
• Stefano Curtarolo and Gerbrand Ceder, "Dynamic of a non homogeneously coarse grained system," cond-mat/0106263
• N. D. Hari Dass, S. Kalyana Rama and B. Sathiapalan, "On the Emergence of the Microcanonical Description from a Pure State," cond-mat/0112439
• Kevin Davey, "What Is Gibbs's Canonical Distribution?", phil-sci/4282
• Paul and Tatiana Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics
• Richard S. Ellis, Kyle Haven and Bruce Turkington, "The Large Deviation Principle for Coarse-Grained Processes," math-ph/0012023
• Denis J. Evans, Debra J. Searles, Stephen R. Williams, "A simple mathematical proof of Boltzmann's equal a priori probability hypothesis", arxiv:0903.1480
• Roman Frigg, "Probability in Boltzmannian Statistical Mechanics", phil-sci/3489
• Roman Frigg and Charlotte Werndl, Foundations of Statistical Mechanics
• Alexandre Giraud and Julien Serreau, "Decoherence and Thermalization of a Pure Quantum State in Quantum Field Theory", Physical Review Letters 104 (2010): 230405 ["real-time evolution of a self-interacting O(N) scalar field initially prepared in a pure, coherent quantum state. ... nonequilibrium quantum dynamics from a 1/N expansion of the two-particle-irreducible effective action at next-to-leading order, which includes scattering and memory effects. ... restricting one's attention (or ability to measure) to a subset of the infinite hierarchy of correlation functions, one observes an effective loss of purity or coherence and, on longer time scales, thermalization. .... physics of decoherence is well described by classical statistical field theory."]
• Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghi, "Canonical Typicality", Physical Review Letters 96 (2006): 050403
• H. Grad, "The many faces of entropy", Communications on Pure and Applied Mathematics 14 (1961): 323--354 [Apparently makes the point that the correct entropy function is dependent on the level of description. This is important for revising my paper with Cris Moore...]
• D. H. E. Gross
  • "Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit," cond-mat/0201235
  • "The microcanonical entropy is multiply differentiable. No dinosaurs in microcanonical gravitation: No special 'microcanonical phase transitions'," cond-mat/0403582
  • "On the Microscopic Foundation of Thermo-Statistics," cond-mat/0209482
  • "A New Thermodynamics,From Nuclei to Stars," cond-mat/0302267
  • "Second Law of Thermodynamics, Macroscopic Observables within Boltzmann's Principle but without Thermodynamic Limit," cond-mat/0101281
  • "Thermo-Statistics or Topology of the Microcanonical Entropy Surface," cond-mat/0206341
• Meir Hemmo and Orly Shenker
  • "Quantum Decoherence and the Approach to Equilibrium", Philosophy of Science 70 (2003): 330--358
  • The Road to Maxwell's Demon: Conceptual Foundations of Statistical Mechanics [Review by Charlotte Wendl in NDPR, suggesting a bizarre gross error about the meaning of "absolute continuity"]
• Steven Huntsman, "Effective statistical physics of Anosov systems", arxiv:1009.2127
• Fengping Jin, Thomas Neuhaus, Kristel Michielsen, Seiji Miyashita, Mark Novotny, Mikhail I. Katsnelson, Hans De Raedt, "Equilibration and Thermalization of Classical Systems", arxiv:1209.0995
• Dragi Karevski, "Foundations of Statistical Mechanics: in and out of Equilibrium", cond-mat/0509595 ["The first part of the paper is devoted to the foundations, that is the mathematical and physical justification, of equilibrium statistical mechanics. It is a pedagogical attempt, mostly based on Khinchin's presentation, which purpose is to clarify some aspects of the development of statistical mechanics. In the second part, we discuss some recent developments that appeared out of equilibrium, such as fluctuation theorem and Jarzynski equality."]
• Gerhard Keller, Equilibrium States in Ergodic Theory
• Martin Krieger, Constitutions of Matter: Mathematically Modeling the Most Everyday of Physical Phenomena
• Juraj Kumicak, "Irreversibility in a simple reversible model", Physical Review E 71 (2005): 016115 = nlin.CD/0510016
• David A. Lavis
  • "The spin-echo system reconsidered", cond-mat/0311527
  • "Is Equilibrium a Useful Concept in Statistical Mechanics?", cond-mat/0401061
  • "Boltzmann, Gibbs and the Concept of Equilibrium", arxiv:0710.2052 = phil-sci/3595
• Chuang Liu, "Approximations, Idealizations, and Models in Statistical Mechanics," PITT-PHIL-SCI00000365
• A. Majda, I. Timofeyev and E. Vanden-Eijnden, "Stochastic models for selected slow variables in large deterministic systems", Nonlinearity 19 (2006): 769
• Benoit Mandelbrot, "On the Derivation of Statistical Thermodynamics from Purely Phenomenological Principles", Journal of Mathematical Physics 5 (1964): 164--171 [PDF reprint]
• Stéphane Mischler, Clément Mouhot, "Kac's Program in Kinetic Theory", arxiv:1107.3251 ["his paper is devoted to the study of propagation of chaos and mean-field limit for systems of indistinguable particles undergoing collision processes, as formulated by M. Kac (1956)..."]
• Wayne C. Myrvold, "Probabilities in Statistical Mechanics: What are they?", philsci/9236
• Jill North, "An Empirical Approach to Symmetry and Probability", phil-sci/5192
• Oliver Penrose, Foundations of Statistical Mechanics: A Deductive Treatment
• A. Perez-Madrid, "Gibbs Entropy and Irreversibility", cond-mat/0401532
• E. A. J. F. Peters, "Projection operator formalism and entropy", cond-mat/0703672
• Denes Petz, "Entropy, von Neumann and the von Neumann Entropy," math-ph/0102013
• Peter Reimann
  • "Foundation of Statistical Mechanics under Experimentally Realistic Conditions", Physical Review Letters 101 (2008): 190403
  • "Typicality for Generalized Microcanonical Ensembles", Physical Review Letters 99 (2007): 160404, arxiv:0710.4214 ["For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium."]
• David Ruelle
  • Statistical Mechanics: Rigorous Results
  • Thermodynamic Formalism
• Geoffrey L. Sewell, "Statistical Thermodynamics of Moving Bodies", arxiv:0902.3881
• David Shalloway, "Macrostates of classical stochastic systems", Journal of Chemical PHysics 105 (1996): 9986--10007
• Orly R. Shenker and Meir Hemmo
  • "The Von Neumann Entropy: A Reconsideration", phil-sci/2256
  • "Von Neumann's Entropy Does Not Correspond to Thermodynamic Entropy", phil-sci/3716
• Michael Strevens, Tychomancy: Inferring Probability from Causal Structure [Polite, but to my mind painful, review by Colin Howson in Notre Dame Philosophical Reviews]
• Hal Tasaki
  • "From Quantum Dynamics to the Second Law of Thermodynamics," cond-mat/0005128
  • "The second law of Thermodynamics as a theorem in quantum mechanics," cond-mat/0011321
  • "The approach to thermal equilibrium and "thermodynamic normality" --- An observation based on the works by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi in 2009, and by von Neumann in 1929", arxiv:1003.5424
• Jos Uffink
  • "Insuperable difficulties: Einstein's statistical road to molecular physics", Studies in History and Philosophy of Modern Physics 37 (2006): 36--70
  • "Compendium of the foundations of classical statistical physics", phil-sci/2691
• Charlotte Werndl, "Justifying Typicality Measures of Boltzmannian Statistical Mechanics and Dynamical Systems", arxiv:1310.1573