Large Deviations and Applications

Varadhan, S. R. S.

doi:10.1137/1.9781611970241

Cited by 651 publications

(443 citation statements)

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“…The Central Limit Theorem says that the average of the dissipation processes converges to a Gaussian but there also exist large excursion or fluctuations in the mean. The effects of these fluctuations are frequently captured by the Large Deviation Principle [39]. If these excursions are completely random then they can, for example, be modeled by a Poisson process with the rate λ.…”

Section: The Deterministic Navier-stokes Equationmentioning

confidence: 99%

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The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

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In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

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Section: The Deterministic Navier-stokes Equationmentioning

confidence: 99%

“…θ is the angle between the vectors [u(x + s, ·) − u(x, ·)] and r, and the absolute value of the former is the reason why the angle derivatives wash out in (39). The PDF is symmetric in the transversal direction, then β = µ = 0.…”

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confidence: 99%

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

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“…We observe the same violation of the FT in our QHE. Further, the long time PDF is evaluated by invoking the use of large deviation theory [40,41] and steadystate (SS) FTs are derived from large deviation results [33,42,43]. We find that the large deviation theory cannot be used to determine the PDF in presence of PBp contributions.…”

Section: Introductionmentioning

confidence: 98%

Geometric phaselike effects in a quantum heat engine

Giri

Goswami

2017

Phys. Rev. E

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By periodically driving the temperature of reservoirs in a quantum heat engine, geometric or Pancharatnam-Berry phase-like (PBp) effects in the thermodynamics can be observed. The PBp can be identified from a generating function (GF) method within an adiabatic quantum Markovian master equation formalism. The GF is shown not to lead to a standard open quantum system's fluctuation theorem in presence of phase-different modulations with an inapplicability in the use of large deviation theory. Effect of quantum coherences in optimizing the flux is nullified due to PBp contributions. The linear coefficient, 1/2, which is universal in the expansion of the efficiency at maximum power in terms of Carnot efficiency no longer holds true in presence of PBp effects.

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“…A proof of this Theorem can be adapted from [1,5,9] In order to apply this Theorem we arrange the sum in (3.26) as sums over disjoint blocks and then take advantage of the fact that the local Gibbs measures are product measures:…”

Section: Then For Any Bounded Continuous Functionmentioning

confidence: 99%

Hydrodynamical limit for a Hamiltonian system with weak noise

1993

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We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is open and with the following adiabatic boundary conditions: it is attached to a wall on the left and there is a force (tension) τ acting on the right. In order to provide the system of the good ergodic properties, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distribution of the density of particles, momentum and energy converge to the solution of the Euler equations, in the smooth regime of them.

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Large Deviations and Applications

Cited by 651 publications

References 0 publications

The Kolmogorov–Obukhov Statistical Theory of Turbulence

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Geometric phaselike effects in a quantum heat engine

Hydrodynamical limit for a Hamiltonian system with weak noise

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