Foundations and History of Statistical Mechanics

08 Aug 2008 04:18

---

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

Recommended:
• David Z. Albert, Time and Chance
• 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
• 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."]
• Michael C. Mackey, 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 approach 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 [JSTOR; free PDF reprint. 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]
• 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 [blurb, ch. 1]
  • "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."]
• Lawrence Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics
• W. H. Zurek, "Algorithmic Randomness, Physical Entropy, Measurements, and the Demon of Choice," quant-ph/9807007

Modesty forbids:
• CRS and Cristopher Moore, "What Is a Macrotate?" 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]
• Miguel Carrion-Alvarez, "Variations on a theme of Gelfand and Naimark", math.FA/0402150 [Algebras of observables, including C* algebras as a special case]
• 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
• 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
• Roman Frigg, "Probability in Boltzmannian Statistical Mechanics", phil-sci/3489
• Sheldon Goldstein, "Boltzmann's Approach to Statistical Mechanics," cond-mat/0105242 ["most twentieth-century innovations are thoroughly misguided"]
• 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...]
• A. Greven, G. Keller and G. Warnecke (eds.), Entropy
• 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
• 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."]
• 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
• Chuang Liu, "Approximations, Idealizations, and Models in Statistical Mechanics," PITT-PHIL-SCI00000365
• Benoit Mandelbrot, "On the Derivation of Statistical Thermodynamics from Purely Phenomenological Principles", Journal of Mathematical Physics 5 (1964): 164--171 [PDF reprint]
• Oliver Penrose, Foundations of Statistical Mechanics: A Deductive Treatment [blurb]
• 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
• David Ruelle
  • Statistical Mechanics: Rigorous Results
  • Thermodynamic Formalism
• 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
• 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
• Jos Uffink
  • "Bluff Your Way in the Second Law of Thermodynamics," cond-mat/0005327
  • "Insuperable difficulties: Einstein's statistical road to molecular physics", Studies in History and Philosophy of Modern Physics 37 (2006): 36--70