Models of Stochastic Behavior in Non-Equilibrium Steady States

Graham, R.

doi:10.1007/978-1-4684-4061-4_26

Cited by 9 publications

(10 citation statements)

References 31 publications

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“…The linear model of Onsager and Machlup serves as a template for constructing and studying other models of fluctuation dynamics, including models of nonequilibrium steady states (see, e.g., [133,257,258]), and for ultimately building a general theory of nonequilibrium processes (see, e.g., [16,95,122,223,227,252]). In going beyond this model, one may replace the linear force b(m) = −am by nonlinear forces, consider noise processes with a non-zero mean or noise processes that are correlated in time, in addition to studying several (possibly coupled) macrostates rather than just one as we did in the previous example.…”

Section: B Phenomenological Models Of Fluctuationsmentioning

confidence: 99%

The large deviation approach to statistical mechanics

2009

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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.PACS numbers: 65.40.Gr, Typeset by REVT E X arXiv:0804.0327v2 [cond-mat.stat-mech]

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Section: B Phenomenological Models Of Fluctuationsmentioning

confidence: 99%

The large deviation approach to statistical mechanics

2009

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“…An interesting example is the Bénard system, in which the particle flux is replaced by a heat flux maintained by reservoirs at different temperatures, with the hotter reservoir below the system, and the typical patterns change abruptly from uniform to rolls to hexagonal cells as the fluid is driven away from equilibrium [10]. Open systems of this type have been discussed extensively in the literature from both a microscopic and macroscopic point of view; see [11,12] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…The deviations of macroscopic systems from typical behavior, e.g., from the profileρ(x,t) governed by the diffusion equation, is a central issue of statistical mechanics [5,6,7,8,11,12]. The time dependent problem was first studied by Onsager and Machlup [25], who considered space-time fluctuations for Hamiltonian systems in equilibrium (for whichρ is constant).…”

Section: Introductionmentioning

confidence: 99%

Untitled

Derrida¹,

Lebowitz

Speer

2002

Journal of Statistical Physics

190

150

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We consider an open one dimensional lattice gas on sites i = 1, . . . , N , with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N → ∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x = i/N , has the asymptotic form exp[−N F({ρ})]; F is the large deviation functional. In contrast to equilibrium systems, for which F eq ({ρ}) is just the integral of the appropriately normalized local free energy density, the F we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of F in general SNS, where the long range correlations have been observed experimentally.

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“…where P(n) = ZN P(n,N) and P ( N ) = 2, P(n,N). Doing this, we find P ( n ) = k(n -l ) ( N l n -1 ) + k ( n + l)NoP(n + 1) -k ( n + 1 ) ( N J n + 1) + k3(n + 1)P(n + 1 ) -knNoP(n) -k3nP(n) (4) and P ( N ) = k ( N + l ) ( n l N + 1)…”

Section: Loss Processesmentioning

confidence: 92%

A stochastic master equation description of laserlike systems

Englund¹,

Schieve²,

Gragg³

2009

Int. J. Quantum Chem.

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We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model.Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P ( n ) . the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of ( n ) and log[ ( n 2 ) / ( n ) 2] versus the pumping parameter show close'agreement with the microscopic results of the Casagrande-Lugiato model. A similar discussion of the fluctuationscan also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.

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Models of Stochastic Behavior in Non-Equilibrium Steady States

Cited by 9 publications

References 31 publications

The large deviation approach to statistical mechanics

The large deviation approach to statistical mechanics

Untitled

A stochastic master equation description of laserlike systems

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