Macroscopic potentials, bifurcations and noise in dissipative systems

Graham, R.

doi:10.1007/3-540-17206-8_1

Cited by 49 publications

(70 citation statements)

References 43 publications

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“…This scaling is also expected from the FreidlinWentzell-Graham (FWG) theory of large deviations in the low-noise limit [13,44] and implies the following large deviation principle for the current distribution:…”

Section: Low-noise Limitsupporting

confidence: 55%

“…This is particularly useful for understanding the low-noise limit of large deviations, studied within the so-called Freidlin-Wentzell theory [13] (see also [44]) or the macroscopic fluctuation theory [45][46][47] in terms of most probable paths or instantons minimizing a given stochastic action. We show here how to use the deterministic limit of the auxiliary process as an alternative way to recover these instantons.…”

Section: Introductionmentioning

confidence: 99%

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Large deviations of the current for driven periodic diffusions

Nyawo

Touchette

2016

Phys. Rev. E

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We study the large deviations of the time-integrated current for a driven diffusion on the circle, often used as a model of nonequilibrium systems. We obtain the large deviation functions describing the current fluctuations using a Fourier-Bloch decomposition of the so-called tilted generator and also construct from this decomposition the effective (biased, auxiliary or driven) Markov process describing the diffusion as current fluctuations are observed in time. This effective process provides a clear physical explanation of the various fluctuation regimes observed. It is used here to obtain an upper bound on the current large deviation function, which we compare to a recently-derived entropic bound, and to study the low-noise limit of large deviations.

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Section: Low-noise Limitsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Large deviations of the current for driven periodic diffusions

Nyawo

Touchette

2016

Phys. Rev. E

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“…What is the relation between the transition observed here and those observed in the weak-noise limit [2][3][4][5], which have been re-discovered recently under the name "Lagrangian phase transitions" [11]? This should be answered by comparing the variational representations of the SCGF and the rate function obtained with and without the low-noise limit [59].…”

mentioning

confidence: 75%

“…Such transitions are known to arise in many physical systems and scaling limits, including the low-noise limit of diffusion equations modeling noise-perturbed dynamical systems [1][2][3][4][5][6], thermodynamic-like limits of chaotic systems [7], and the hydrodynamic limit of interacting particles systems, which corresponds, via the macroscopic fluctuation theory [8][9][10], to a low-noise limit [11][12][13][14][15].…”

mentioning

confidence: 99%

A minimal model of dynamical phase transition

Nyawo

Touchette

2016

EPL

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We calculate the large deviation functions characterizing the long-time fluctuations of the occupation of drifted Brownian motion and show that these functions have non-analytic points. This provides the first example of dynamical phase transition that appears in a simple, homogeneous Markov process without an additional low-noise, large-volume or hydrodynamic scaling limit.Keywords: Brownian motion, large deviations, dynamical phase transitions Dynamical phase transitions are phase transitions in the fluctuations of physical observables that give rise, similarly to equilibrium phase transitions, to non-analytic points in generalized potentials or large deviation functions characterizing the likelihood of fluctuations. Such transitions are known to arise in many physical systems and scaling limits, including the low-noise limit of diffusion equations modeling noise-perturbed dynamical systems [1-6], thermodynamic-like limits of chaotic systems [7], and the hydrodynamic limit of interacting particles systems, which corresponds, via the macroscopic fluctuation theory [8][9][10], to a low-noise limit [11][12][13][14][15].Similar transitions also appear in the long-time fluctuations of time-integrated quantities, such as Lyapunov exponents [16][17][18], dynamical activities [19][20][21][22][23][24][25][26][27][28][29][30], currents [31][32][33][34][35][36][37][38][39], and the entropy production [40][41][42], which now play a central role in studies of nonequilibrium processes. In this case, the large deviation functions are found to be smooth in the long-time limit; singularities start to appear only when a low-noise or a scaling (hydrodynamic, particle or mean-field) limit is taken in addition to the long-time limit [43], leading many to believe and claim that these additional limits are necessary for dynamical phase transitions to appear in Markov processes.The study in [44] of non-homogeneous random walks that are reset in time has just shown, by a mapping to DNA models, that this is not always true -a dynamical phase transition can occur in the long-time limit without a low-noise or scaling limit. Here, we present a simple, minimal model based on a one-dimensional and homogeneous diffusion process that confirms this. The process has no reset and so also shows that random resetting is not needed for such transitions to arise.The process that we consider is the drifted Brownian motion, defined aswhere µ is the drift, σ is the noise power, and W t is the simple, one-dimensional Brownian motion (BM) started at W 0 = 0. Physically, X t may represent the position of a small particle evolving in a fluid moving at slow, constant velocity or in a static fluid but with additional forces (created, e.g., by laser tweezers [45] or an AC trap [46]) that pull the particle at constant velocity. By analogy with electrical circuits perturbed by Nyquist noise [47], X t can also represent the charge dissipated in a resistor upon the application of a ramped voltage. In both cases, we are interested in the fluctuations of the fracti...

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“…One of the longstanding unsolved problems in the theory of fluctuations is that of noise-induced escape from a chaotic attractor [1][2][3]. Chaotic systems are widespread in nature, and the study of their dynamics in the presence of noise is a topic of broad interdisciplinary interest whose potential applications include e.g.…”

Section: Introductionmentioning

confidence: 99%

Noise-Induced Escape From the Lorenz Attractor

2001

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Noise-induced escape from a quasi-hyperbolic attractor in the Lorenz system is investigated via an analysis of the distributions of both the escape trajectories and the corresponding realizations of the random force. It is shown that a unique escape path exists, and that it consists of three parts with noise playing a different role in each. It is found that the mechanism of the escape from a quasi-hyperbolic attractor differs from that of escape from a non-hyperbolic attractor. The possibility of calculating the escape probability is discussed.

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Macroscopic potentials, bifurcations and noise in dissipative systems

Cited by 49 publications

References 43 publications

Large deviations of the current for driven periodic diffusions

Large deviations of the current for driven periodic diffusions

A minimal model of dynamical phase transition

Noise-Induced Escape From the Lorenz Attractor

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