## Statistical Mechanics (and Condensed Matter)

*01 Apr 2003 09:01*

The first mathematical, natural science of emergent properties. (I hedge this way, because one could argue that economics and evolutionary theory are both, also, concerned with emergent properties --- efficient allocation, and adaptation and speciation, respectively, and they preceeded statistical mechanics.) The heart of the subject is figuring out what happens when vast numbers of particles bounce around and into each other, all obeying the laws of mechanics (classical or quantum as the case may be).

*Things I Want to Understand
Better:* Phase transitions and critical
phenoma; the renormalization
group; field-theory
methods; what happens far from equilibrium (more
specifically, are there action principles or the like that govern probability
distributions of trajectories, the way thermodynamic potentials govern
equilibrium configurations); "soft" condensed matter; biological applications;
amorphous materials and glasses; connections between spin glasses and biology
(e.g., perceptrons); technical, conceptual and historical issues
in the foundations of statistical mechanics.

See also: Astrophysics and Cosmology; Atomism; Cellular Automata; Chaos and Non-Linear Dynamics; Complexity; Dissipative Structures; Exponential Families of Probability Measures; Field Theory; Large Deviations; Liquid Crystals; Monte Carlo; Pattern Formation; Ilya Prigogine; Probability; Random Fields; Self-Organization; Stochastic Processes; Tsallis Statistics; Turbulence

Recommended:

<rant> If a non-scientist wants to learn about some large and important part of science, say planetary astronomy or genetics, there are usually a handful of reliable, uncontroversial, well-written, non-technical books about it to be found in the stores and libraries, which will convey at least something of the field's history, problems, results and methods. By this point there must be dozens of good popular books written on evolution, particle physics, cosmology, relativity and quantum mechanics, notwithstanding that the last two are about as abstract and abstruse as science gets. There are even excellent popularizations of mathematics, in a continuous tradition from E. T. Bell (if not before). Writing popularizations is an accepted and even encouraged activity for eminent scientists, and has been since Galileo's Starry Messanger. --- Popularizations are also important in the recruitment and education of scientists, but the only one I know of who's written on this is John Maynard Smith, in Did Darwin Get It Right?

A few months ago, when I was trying to explain some parts of my research to
my father, I realized I was assuming he knew what statistical mechanics was,
and something about how it worked, when in fact he did not. My first thought
was to pass on some popular work about statistical mechanics (it's only fair;
he did it to me constantly when I was younger). A great many thoughts later I
realized I could not think of a single one which didn't stake out some very
peculiar philosophical position, or did more than just blab about the second
law, never mind something as good as Einstein for Beginners or
The First Three Minutes or Does God Play Dice?
Granted that relativity and particles and chaos are sexy, and statistical
mechanics is not, it's peculiar that there's *nothing.* Stat. mech. is,
after all, one of the essential theories of current physics, actually used by
chemists and biologists and materials scientists, etc., the part of physics
most directly applicable to daily life (you could illustrate the core of it
with a coffee cup, and the whole with a kitchen), and bound up with deep
puzzles about why time goes the way it does. This cries out for a remedy.

The undergraduate textbooks on statistical mechanics, like those on most part of physics, are by and large vile. Kittel and Kroemer's Thermal Physics is however decent; if you want a quick-and-dirty guide, and can put up with bad typesetting, try M. G. Bowler's Lectures on Statistical Mechanics. There is nothing analogous to Griffiths's books on electromagnetism, quantum mechanics and particle physics, and if he's got time on his hands...

Chandler's Introduction to Modern Statistical Mechanics is good, as is Landau and Lifshitz's Statistical Physics; the latter is far more comprehensive, but the former is much newer, and easier to learn from. Huang's Statistical Mechanics, one of the other standard texts, is a pedagogic horror.

Having finished this venting of spleen, we turn to the usual list. </rant>

- Recommended, less technical:
- Vinay Ambegaokar, Reasoning about Luck: Probability and Its Uses in Physics [This is intended as a substitute for the usual sort of physics-for-people-who-have-to-fill-a-distribution-requirement course, and I think well enough of it that I'd be willing to teach it, while wild horses couldn't get me to do the standard physics for poets, but it's not really what I'm looking for.]
- David Ruelle, Chance and Chaos [Parts of this approach what I was raving for above, but still doesn't quite hack it, since it doesn't cover enough.]
- Hans Christian von Baeyer, Maxwell's Demon: Why Warmth Disperses and Time Passes [Again, almost makes it]

- Recommended, more technical:
- Philip W. Anderson, Basic Notions of Condensed Matter Physics
- Beck and Schlögl, Thermodynamics of Chaotic Systems [See notice under non-linear dynamics]
- Britney Spears's Guide to Semiconductor Physics
- Chaikin and Lubensky, Principles of Condensed Matter
- Richard S. Ellis, Entropy, Large Deviations and Statistical Mechanics
- K. H. Fischer and J. A. Hertz, Spin Glasses
- Dieter Forster, Hydrodynamic Fluctuations, Broken Symmetry,
and Correlation Functions [An excellent book which looks
*horrible.*Bless Donald Knuth for delivering us from type-writen equations!] - D. H. E. Gross, "Microscopic statistical basis of classical Thermodynamics of finite systems", cond-mat/0505242
- Meir Hemmo and Orly Shenker, "Quantum Decoherence and the Approach
to Equilibrium", Philosophy of Science
**70**(2003): 330--358 - Chris Hillman, Entropy on the World Wide Web
- Mark Kac
- Probability in Physical Sciences and Related Topics
- "Foundations of Kinetic Theory", Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3 (Univ. of Calif. Press, 1956), 171--197

- 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."] - L. D. Landau and E. M. Lifshitz, Statistical Physics [What I was raised on. To be completely honest, it's been about a decade since I read it, and more since it was my constant companion, and I am a little afraid to re-read it, the same way one is sometimes afraid to re-read favorite novels from long ago, lest they have become worse in the meanwhile...]
- Andreas Maurer, "Thermodynamics and Concentration", Bernoulli submitted (2011) [Deriving concentration-of-measure results from statistical-mechanical arguments; very nice. PDF preprint via Dr. Maurer]
- David Selmeczi, Simon F. Tolic-Norrelykke, Erik Schaeffer, Peter H. Hagedorn, Stephan Mosler, Kirstine Berg-Sorensen, Niels B. Larsen and Henrik Flyvbjerg, "Brownian Motion after Einstein: Some new applications and new experiments", physics/0603142
- James Sethna
- "Order Parameters, Broken Symmetry, and Topology", pp. 243--265 in Lynn Nadel and Daniel L. Stein (eds.), 1991 Lectures in Complex Systems [Online version]
- Statistical Mechanics: Entropy, Order Parameters and Complexity [Mini-review; free PDF]

- Geoffrey Sewell, Quantum Mechanics and Its Emergent Macrophysics
- Hugo Touchette, "The Large Deviations Approach to Statistical
Mechanics", Physics Reports
**478**(2009): 1--69, arxiv:0804.0327 - Julia Yeomans, The Statistical Mechanics of Phase Transitions
- Richard Zallen, The Physics of Amorphous Solids

- Modesty forbids:
- CRS and Cristopher Moore, "What Is a Macrotate?" cond-mat/0303625

- To read, historical:
- Stephen Brush
- Statistical Physics and the Atomic Theory of Matter
- The Kind of Motion We Call Heat

- M. E. Cates, "Soft Condensed Matter", cond-mat/0411650 ["I described the evolution of soft matter physics as a discipline during the 20th century"]
- Cyril Domb, The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena
- Albert Einstein, Investigations on the Theory of Brownian Motion
- Martin Niss, "History of the Lenz-Ising Model, 1920--1950: From
Ferromagnetic to Cooperative
Phenomena", Archive
for History of Exact Sciences
**59**(2005): 267--318 ["In the early 1920s, Lenz and Ising introduced the model in the field of ferromagnetism. Based on an exact derivation, Ising concluded that it is incapable of displaying ferromagnetic behavior, a result he erroneously extended to three dimensions. In the next phase, Lenz and Ising's contemporaries rejected the model as a representation of ferromagnetic materials because of its conflict with the new quantum mechanics. In the third phase, from the early 1930s to the early 1940s, the model was revived as a model of cooperative phenomena. ... [I] focus on the development of the model in its capacity as a*model*. ... though the Lenz-Ising model is not fully realistic, it is more useful than more realistic models because of its mathematical tractability... this point of view, important for the modern conception of the model, is novel and that its emergence, while perhaps not a consequence of its study, is coincident with the third phase of its development." Those of us who work with grossly unrealistic but tractable models of complex systems should pay heed...] - Johanna Levelt Sengers, How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes [Blurb]
- D. ter Haar, The Scientific Contributions of H. A. Kramers [blurb]

- To read, teaching:
- Greg Anderson, Thermodynamics of Natural Systems [Blurb]
- Roger Balian
- From Microphysics to Macrophysics: Methods and Applications of Statistical Physics
- RB and Jean-Paul Blaizot, "Stars and Statistical Physics: A Teaching Experience," cond-mat/9909291 [I plan to steal from this wholesale if I teach stat. mech.]
- Giovanni Gallavotti, "Equilibrium Statistical Mechanics", cond-mat/0504790 [56 pp. introductory review]
- Martin and Inge F. Goldstein, The Refrigerator and the Universe
- Donald T. Haynie, Biological Thermodynamics [blurb]
- Josef Honerkamp, Statistical Physics: An Advanced Approach with Applications
- Charles Kittel, Elementary Statistical Physics [1958 textbook now republished in Dover paperback; looks good and cheap; I learned a lot from Kittel and Kromer's textbook as an undergraduate]
- Don S. Lemons, Mere Thermodynamics
- R. A. Minlos, Introduction to Mathematical Statistical Physics [Blurb]
- Anastasios A. Tsonis, Introduction to Atmospheric Thermodynamics [blurb]

- To read, learning:
- Ambjorn, Durhuss and Jonsson, Quantum Geometry [field-theory methods for Brownian motion and higher-dimensional random surfaces]
- John C. Baez, Mike Stay, "Algorithmic Thermodynamics", arxiv:1010.2067
- Franco Bagnoli and Raul Rechtman, "Thermodynamic
entropy and chaos in a discrete hydrodynamical
system", Physical Review E
**79**(2009): 041115 ["thermodynamic entropy density is proportional to the largest Lyapunov exponent of a discrete hydrodynamical systems, a deterministic two-dimensional lattice gas automaton"] - Francois Bardou et al., Levy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest
- Jean-Louis Barrat and Jean-Pierre Hansen, Basic Concepts for Simple and Complex Liquids
- Robert W. Batterman, "The tyranny of scales", phil-sci/8678
- Rodney J. Baxter, Exactly Solved Models in Statistical Mechanics [Blurb]
- Golan Bel and Eli Barkai, "A Random Walk to a Non-Ergodic Equilibrium Concept", cond-mat/0506338 [I've only read the abstract, but it puzzles me. I'd be very interested if we could have a good notion of equilibrium which didn't depend on ergodicity, but in the model they're consdering, they can evidently say things like "in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law", and I'm not sure how such limits are meaningful without some kind of ergodic property. But I should just read the paper.]
- Anton Bovier, Statistical Mechanics of Disordered Systems [Blurb; enthusiastic review in J. Stat. Phys.]
- Anton Bovier, Michael Eckhoff, Veronique Gayrard and Markus Klein, "Metastability and Small Eigenvalues in Markov Chains," cond-mat/0007343
- Todd A. Brun and James B. Hartle, "Entropy of Classical
Histories," Physical Review E
**59**(1999): 6370--6380 - Lapo Casetti, Marco Pettini, E. G. D. Cohen, "Geometric Approach to Hamiltonian Dynamics and Statistical Mechanics," cond-mat/9912092
- Tommaso Castellani and Andrea Cavagna, "Spin-Glass Theory for Pedestrians", cond-mat/0505032
- Sourav Chatterjee, "Chaos, concentration, and multiple valleys", arxiv:0810.4221 ["Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other areas, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a drastic shift of the ground state. We present a rigorous theory of chaos in disordered systems that confirms long-standing physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the existence of multiple valleys in the energy landscape."]
- Amir Dembo and Andrea Montanari, "Gibbs Measures and Phase Transitions on Sparse Random Graphs", arxiv:0910.5460
- Emilio De Santis and Carlo Marinelli, "Stochastic games with infinitely many interacting agents", math.PR/0505608 [Sounds very cool: "We introduce and study a class of infinite-horizon non-zero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove ``fixation'', i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zero-temperature Glauber dynamics on random graphs of possibly infinite volume."]
- Deepak Dhar, "Pico-canonical ensembles: A theoretical description of metastable states," cond-mat/0205011
- Enrico Di Cera, Thermodynamic Theory of Site-Specific Binding Processes in Biological Macromolecules
- Viktor Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems
- Sam F. Edwards and Moshe Schwartz
- "Lagrangian Statistical Mechanics applied to Non-linear Stochastic Field Equations," cond-mat/0012044
- "Statistical Mechanics in Collective Coordinates," cond-mat/0204178

- David Ford and Steven Huntsman, "Descriptive Thermodynamics", cond-mat/0510030
- Surya Ganguli and Haim Sompolinsky, "Statistical Mechanics of
Compressed
Sensing", Physical
Review Letters
**104**(2010): 188701 - Cristian Giardina', Jorge Kurchan, Luca Peliti, "Direct evaluation of large-deviation functions", cond-mat/0511248 ["We introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities, such as current or density, that are local in time. The large-deviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values."]
- G. Gregoire and H. Chate, "Onset of collective and cohesive motion", cond-mat/0401208
- J. Woods Halley, Statistical Mechanics: From First Principles to Macroscopic Phenomena [blurb. Sounds nice.]
- Horsthemke, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology
- Stephen Hyde, Sten Andersson, Kare Larsson, Zoltan Blum, Tomas Landh, Sven Lidin and Barry Ninham, The Language of Shape: The Role of Curvature in Condensed Matter --- Physics, Chemistry, and Biology
- Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory (2 vols.)
- Henrik Jeldtoft Jensen, Elsa Arcaute, "Complexity, Collective Effects and Modelling of Ecosystems: formation, function and stability", arxiv:0709.2015 [" We describe examples where combining statistical mechanics and ecology has led to improved ecological modelling and, at the same time, broadened the scope of statistical mechanics."]
- Richard A. L. Jones, Soft Condensed Matter
- Wouter Kager and Bernard Nienhuis, "A Guide to Stochastic Loewner Evolution and Its Applications", math-ph/0312056
- Daniel Korenblum and David Shalloway, "Macrostate Data Clustering",
Physical Review E
**67**(2003): 056704 [This sounds a lot like spectral clustering and diffusion maps] - Werner Krauth, Statistical Mechanics: Algorithms and Computations
- Stephan Lawi, "A characterization of Markov processes enjoying the time-inversion property", math.PR/0506013
- Frederic Legoll and Tony Lelievre, "Some remarks on free energy and coarse-graining", arxiv:1008.3792
- L. Leuzzi and T. M. Nieuwenhuizen, Thermodynamics of the Glassy State [Favorable review in J. Stat. Phys.]
- Elliott H. Lieb, "Quantum Mechanics, the Stability of Matter and Quantum Electrodynamics", math-ph/0401004
- Valerio Lucarini, "Response Theory for Equilibrium and
Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig
relations", Journal of Statistical Physics
**131**(2008): 543--558, arxiv:0710.0958 - D. Lynden-Bell and R. M. Lynden-Bell, "Relaxation to a Perpetually
Pulsating Equilibrium", cond-mat/0401093
= Journal of Statistical Physics
**117**(2004): 199--209 [A profoundly weird-looking result] - Gerald D. Mahan, Condensed Matter in a Nutshell [Blurb, ch. 1]
- Dörthe Malzahn and Manfred Opper, "A statistical physics approach for the analysis of machine learning algorithms on real data", Journal of Statistical Mechanics: Theory and Experiment (2005): P11001
- Daniel C. Mattis and Robert H. Swendsen, Statistical Mechanics Made Simple [On the principle of supporting local authors...]
- Michel Mitov, Sensitive Matter: Foams, Gels, Liquid Crystals, and Other Miracles
- David R. Nelson, Defects and Geometry in Condensed Matter Physics [blurb]
- J. Ortiz de Sarate and J. V. Sengers, Hydrodynamic Fluctuations [Favorable review in J. Stat. Phys.]
- Hans L. Pécseli, Fluctuations in Physical Systems [blurb]
- Alessandro Pelizzola, "Cluster variation method in statistical
physics and probabilistic graphical models", Journal of Physics
A: Mathematical and General
**38**(2005): R309--R339 = cond-mat/0508216 - Leonard Sander, Advanced Condensed Matter Physics [blurb]
- David Shalloway, "Macrostates of classical stochastic
systems", Journal of Chemical PHysics
**105**(1996): 9986--10007 - Michel Talagrand, Mean Field Models for Spin Glasses: A First Course [110 pp. MS.; thanks to Alessandro Rinaldo for sharing his copy with me]
- Uwe C. Tauber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior
- Y. Vallis, T. Qu, M. Micoulaut, F. Chaimbault and P. Boolchand, "Direct evidence of rigidity loss and self-organisation in silicate glasses", cond-mat/0406509
- David Wales, Energy Landscapes: Applications to Clusters, Biomolecules and Glasses