The large deviation approach to statistical mechanics

By Hugo Touchette. 2008

1,745 citations recorded by the source corpus.

Abstract The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein’s theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

Subfields: Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3836776a75021492c79d5d9befd087a5a14c7b76

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Citations in the corpus

Temperature in and out of equilibrium: A review of concepts, tools and attempts (2017, background): “For the sake of selfconsistence let us briefly remind the basic aspects of this approach [21,22].”
Kinetically Constrained Models (2010, positive): “This phase transitions can be studied by recourse to the large-deviation method (Lecomte et al., 2007; Touchette, 2009).”
Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels (2014, methodology): “If s has a linear or affine parts over (ul, uh), then a similar physical interpretation involving a first-order phase transition also applies, but with discrete phases replaced by a continuum of phases [29].”
Lecture notes on Generalised Hydrodynamics (2019, neutral): “[22].”
Stochastic thermodynamics: From principles to the cost of precision (2017, methodology): “Fluctuations in a non-equilibrium steady state (NESS) in the long-time limit From now on, we focus on the fluctuations in a NESS in the long-time limit for which general results can be derived based on techniques from large-deviation theory as reviewed in [24, 25, 26] and by Touchette in this volume”
Dissipation Bounds All Steady-State Current Fluctuations. (2015, neutral): “First, we turn Iðp; jÞ into a rate function for just j by using the contraction principle [25], which states that IðjÞ 1⁄4 infpIðp; jÞ.”
Nonequilibrium Markov Processes Conditioned on Large Deviations (2014, background): “If these sets are singleton sets, then their unique element correspond to the typical value of BT in the sense of the ergodic theorem [50,99].”
Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics (2017, background): “(24) in general acquires a LD form at long times [10, 41, 26],”
Drude Weight for the Lieb-Liniger Bose Gas (2017, background): “The diagonal entries D ii, the scaled second cumulants of charge transfer, are part of the large-deviation theory for non-equilibrium transport [29, 30].”
Stochastic resetting and applications (2019, background): “the generating function G(k;t) = hetkA ti (8.4) and nding the asymptotic behaviour for large t G(k;t) ˘e(k)t: (8.5) Then I(a) can be found by the Legendre transform I(a) = sup k [ka (k)]; (8.6) see [161] as well as [162] for reviews. Using the renewal approach Meylahn, Sabhapandit and Touchette [163”
Beyond Kolmogorov cascades (2019, methodology): “By Kramer’s theorem (Touchette 2009), one sees that as in the limit `→ 0, ln(L/`)→∞ and we have”
Stochastic thermodynamics, fluctuation theorems and molecular machines (2012, methodology): “For a network in a NESS, currents obey an FT as first derived by Gaspard and Andrieux [228, 229, 230, 231] exploiting the decomposition of such a network in cycles as introduced by Schnakenberg [189] and using concepts from large deviation theory [232].”
Conformal field theory out of equilibrium: a review (2016, background): “rises in CFT, have exponent = 1). Note that the function F(), for 2R, is always convex: aF( 1) + (1 a)F( 2) F(a 1 + (1 a) 2) for all a2[0;1] (as a consequence of Holder’s inequality, see e.g. [70]). The aim of the following is to derive an exact formula for F() for non-equilibrium homogeneous CFTs, ”
Thermodynamics (2021, positive): “5 results obtained us-ing the purification algorithm for a Heisenberg chain with 30 sites and compare the results to QMC results for the energy and the free energy, where the latter is obtained by the respective integration of the former.”
The physics of climate variability and climate change (2019, methodology): “approach allows one to infer higher or lower instability of these regimes and, hence, their lower or higher predictability (Faranda et al., 2017; Hochman et al., 2019). Large Deviation Theory (e.g., Touchette, 2009; Varadhan, 1984) has been rst used in the context of geophysical uid dynamics for stu”
Introduction to dynamical large deviations of Markov processes (2017, background): “• Applications of large deviations in statistical physics: [12, 16].”
Entropy production in field theories without time reversal symmetry: Quantifying the non-equilibrium character of active matter (2016, background): “[61] Hugo Touchette, “The large deviation approach to statistical mechanics,” Physics Reports 478, 1–69 (2009).”