The large deviation approach to statistical mechanics

Touchette, Hugo

doi:10.1016/j.physrep.2009.05.002

Cited by 1,727 publications

(3,136 citation statements)

References 268 publications

(610 reference statements)

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“…It nevertheless gives rise to the same symmetry for the rate function as ours (in the special case of homogeneous dynamics) via the Legendre-like identity I W (z) = sup λ∈R {e(λ) − λz} [24]. For an in depth discussion of the relationship between the physics and mathematical notions of entropy and free energy, see [38].…”

Section: Theorem 37 the Free Energy C B W (λ) Exists Is Continuouslmentioning

confidence: 90%

“…Strict convexity of I W (z) can be deduced by contradiction. If the convex envelope of I W (z) (the supremum of all convex functions minorizing it) contained an interval of strict linearity with slope λ 0 , then the free energy's derivative would jump at λ 0 by Legendre duality, violating its continuity (see [38] for an intuitive discussion).…”

Section: Corollary 35 the Time-averaged Action Functional W (0 T)/tmentioning

confidence: 99%

“…The reason is one of convention. The scaled cumulant generating function e(λ) = lim t→∞ − 1 t log e −λW (t) in [24] satisfying the symmetry e(λ) = e(1 − λ) is the canonical free energy function [38], familiar from statistical mechanics when t is the particle number, W (t) the Hamiltonian of a configuration and λ the inverse temperature, but distinct from our free energy function (3) by the minus signs. It nevertheless gives rise to the same symmetry for the rate function as ours (in the special case of homogeneous dynamics) via the Legendre-like identity I W (z) = sup λ∈R {e(λ) − λz} [24].…”

Section: Theorem 37 the Free Energy C B W (λ) Exists Is Continuouslmentioning

confidence: 99%

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Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Shargel¹,

Chou

2009

J Stat Phys

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Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.

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Section: Theorem 37 the Free Energy C B W (λ) Exists Is Continuouslmentioning

confidence: 90%

Section: Corollary 35 the Time-averaged Action Functional W (0 T)/tmentioning

confidence: 99%

Section: Theorem 37 the Free Energy C B W (λ) Exists Is Continuouslmentioning

confidence: 99%

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Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Shargel¹,

Chou

2009

J Stat Phys

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“…A rigorous derivation of such macrostate entropy requires the use of large deviation theory; see e.g. Touchette (2009) for an introduction to those tools. A key difficulty in deriving rigorously this macrostate entropy from the usual Boltzmann entropy is that the microstates are continuous fields which contain an infinite number of degrees of freedom, and which are constrained by an infinite number of dynamical invariants.…”

Section: Microscopic Configurations Macroscopic Description and Varimentioning

confidence: 99%

A statistical mechanics approach to mixing in stratified fluids

2016

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Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. The huge number of degrees of freedom involved in these processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding a prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile. Assuming random evolution through turbulent stirring, the theory predicts that the inviscid, adiabatic dynamics is attracted irreversibly towards an equilibrium state characterised by a smooth, stable buoyancy profile at a coarse-grained level, upon which are fine-scale fluctuations of velocity and buoyancy. The convergence towards a coarse-grained buoyancy profile different from the initial one corresponds to an irreversible increase of potential energy, and the efficiency of mixing is quantified as the ratio of this potential energy increase to the total energy injected into the system. The remaining part of the energy is literally lost into small-scale fluctuations. We show that for sufficiently large Richardson number, there is equipartition between potential and kinetic energy, provided that the background buoyancy profile is strictly monotonic. This yields a mixing efficiency of 0.25, which provides statistical mechanics support for previous predictions based on phenomenological kinematics arguments. In the general case, the cumulative, global mixing efficiency predicted by the equilibrium theory can be computed using an algorithm based on a maximum entropy production principle. It is shown in particular that the variation of mixing efficiency with the Richardson number strongly depends on the background buoyancy profile. This approach could be useful to the understanding of mixing in stratified turbulence in the limit of large Reynolds and Péclet numbers.

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“…Note that trying to get the rate function directly, using say Gartner Ellis theorem [26], is quite complicated as s i /N is not a sum of N independent random variables with respect to the probability measure in Eq. 2.…”

Section: Rate Function For a Random Field P-spin Modelmentioning

confidence: 99%

Effect of random field disorder on the first order transition in p-spin interaction model

Sumedha

Singh

2016

Physica A: Statistical Mechanics and its Applications

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We study the random field p-spin model with Ising spins on a fully connected graph using the theory of large deviations in this paper. This is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state. For p = 2, the phase-diagram of the model, for bimodal distribution of the random field, has been well studied and is known to undergo a continuous transition for lower values of the random field (h) and a first order transition beyond a threshold, h tp (≈ 0.439). We find the phase diagram of the model, for all p ≥ 2, with bimodal random field distribution, using large deviation techniques. We also look at the fluctuations in the system by calculating the magnetic susceptibility. For p = 2, beyond the tricritical point in the regime of first order transition, we find that for h tp < h < 0.447, magnetic susceptibility increases rapidly (even though it never diverges) as one approaches the transition from the high temperature side. On the other hand, for 0.447 < h ≤ 0.5, the high temperature behaviour is well described by the Curie-Weiss law. For all p ≥ 2, we find that for larger magnitudes of the random field (h > h o = 1/p!), the system does not show ferromagnetic order even at zero temperature. We find that the magnetic susceptibility for p ≥ 3 is discontinuous at the transition point for h < h o .

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The large deviation approach to statistical mechanics

Cited by 1,727 publications

References 268 publications

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

A statistical mechanics approach to mixing in stratified fluids

Effect of random field disorder on the first order transition in p-spin interaction model

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