Ergodicity and large deviations in physical systems with stochastic dynamics

Jack, Robert L.

doi:10.1140/epjb/e2020-100605-3

Cited by 122 publications

(156 citation statements)

References 155 publications

(450 reference statements)

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“…To investigate the emergence of ordered dynamical phases and of strong spatio-temporal correlations it is necessary to investigate the full probability of timeintegrated observables [19][20][21][22]. We represent observables as vectors,…”

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

“…where P(ω t ) is the probability of trajectory ω t to occur. For long times this probability distribution is assumed to obey a large deviation (LD) principle [19][20][21][22] π t (O) e −tϕ(O/t) .…”

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

“…Interesting phase behaviours in trajectory space are encoded in the analytic properties of the SCGF and its singularities are indicative of dynamical phase transitions. The rate function ϕ(o) and the SCGF θ(s) are related by a Legendre-Fenchel transform [19][20][21][22]…”

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

“…The SCGF θ(s) can be obtained by means of tilted operators techniques [19][20][21][22]. In particular, it is the largest eigenvalue associated with a deformation of the original stochastic generator W, which for observables of the form (6) reads,…”

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

“…More recently, the notion of phase transition has been extended to include also critical phenomena taking place in large fluctuations of nonequilibrium processes, e.g. [6][7][8][9][10][11][12][13][14][15][16][17][18] (see [19][20][21][22] for reviews). In this scenario, sudden changes in the spatio-temporal structures of the system (trajectories) are witnessed by a non-analytic behaviour in free-energy or entropy-like functionals describing the statistics of an appropriate time-integrated observable.…”

Section: Introductionmentioning

confidence: 99%

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Trajectory phase transitions in noninteracting spin systems

et al. 2020

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We show that a collection of independent Ising spins evolving stochastically can display surprisingly large fluctuations towards ordered behaviour, as quantified by certain types of time-integrated plaquette observables, despite the underlying dynamics being non-interacting. In the large deviation (LD) regime of long times and large system size, this can give rise to a phase transition in trajectory space. As a non-interacting system we consider a collection of spins undergoing single spin-flip dynamics at infinite-temperature. For the dynamical observables we study, the associated tilted generators have an exact and explicit spin-plaquette duality. Such setup suggests the existence of a transition (in the large size limit) at the self-dual point of the tilted generator. The nature of the LD transition depends on the observable. We consider explicitly two situations: (i) for a pairwise bond observable the LD transition is continuous, and equivalent to that of the transverse field Ising model; (ii) for a higher order plaquette observable, in contrast, the LD transition is first order. Case (i) is easy to prove analytically, while we confirm case (ii) numerically via an efficient trajectory sampling scheme that exploits the non-interacting nature of the original dynamics. :1911.11739v1 [cond-mat.stat-mech] arXiv

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Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

Section: Stochastic Dynamics and Large Deviationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Trajectory phase transitions in noninteracting spin systems

et al. 2020

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Multi-species Neutron Transport Equation

et al. 2019

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The Neutron Transport Equation (NTE) describes the flux of neutrons through inhomogeneous fissile medium. Whilst well treated in the nuclear physics literature (cf. [8, 28]), the NTE has had a somewhat scattered treatment in mathematical literature with a variety of different approaches (cf. [7,26]). Within a probabilistic framework it has somewhat undeservingly received little attention in recent years; nonetheless, probabilistic treatments can be found see for example [18,27,23,30,4,3]. In this article our aim is threefold. First we want to introduce a slightly more general setting for the NTE, which gives a more complete picture of the different species of particle and radioactive fluxes that are involved in fission. Second we consolidate the classical c 0 -semigroup approach to solving the NTE with the method of stochastic representation which involves expectation semigroups. Third we provide the leading asymptotic of our multi-species NTE, which will turn out to be crucial for further stochastic analysis of the NTE in forthcoming work [14,12,5]. The methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c 0 -semigroup theory from functional analysis. In this respect, our presentation is thus part review of existing theory and part presentation of new research results based on generalisation of existing results.1. Introduction. The neutron transport equation (NTE) describes the flux of neutrons across a directional planar cross-section in an inhomogeneous fissile medium (typically measured is number of neutrons per cm 2 per second). As such, flux is described as a function of time, t, Euclidian location, r ∈ R 3 , direction of travel, Ω ∈ S 2 , speed c > 0 (and hence velocity υ = cΩ), and neutron energy, E ∈ R. It is not uncommon in the physics literature, as indeed we shall do here, to assume that energy is a function of velocity (E = m|υ| 2 /2), thereby reducing the number of variables by one. This allows us to describe the dependency of flux more simply in terms of time and, what we call, the configuration variables (r, υ) ∈ D × V where D ⊆ R 3 is a smooth, open, connected and bounded domain of concern such that ∂D has zero Lebesgue measure and V is the velocity space, which can now be taken to be V = {v ∈ R 3 : υ min < |v| < υ max }, where 0 < υ min < υ max < ∞.

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Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

2021

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We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.

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Ergodicity and large deviations in physical systems with stochastic dynamics

Cited by 122 publications

References 155 publications

Trajectory phase transitions in noninteracting spin systems

Trajectory phase transitions in noninteracting spin systems

Multi-species Neutron Transport Equation

Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

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