Duality and Hidden Symmetries in Interacting Particle Systems

Giardinà, Cristian; Kurchan, Jorge; Redig, Frank; Vafayi, Kiamars

doi:10.1007/s10955-009-9716-2

Cited by 143 publications

(316 citation statements)

References 21 publications

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“…Such underlying structure is usually provided by a Lie algebra naturally associated to the generator of the process. The first result in this direction was given in [24] for the symmetric process, while a systematic and general approach has been described in [6,12]. When passing from symmetric to asymmetric processes, one has to change from the original Lie algebra to the corresponding deformed quantum Lie algebra, where the deformation parameter is related to the asymmetry.…”

Section: Informal Description Of the Resultsmentioning

confidence: 99%

“…In this mapping the spins are represented by 2 × 2 matrices satisfying the sl 2 algebra. By considering higher values of the spins, represented by (2 j + 1)-dimensional matrices with j ∈ N/2, one obtains the generalized Symmetric Simple Exclusion Process with up to 2 j particles per site (SSEP(2 j) for short), sometimes also called "partial exclusion" [5,12,24]. Namely, denoting by η i ∈ {0, 1, .…”

Section: Previous Extensions Of the Asepmentioning

confidence: 99%

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A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

Carinci

Giardinà

Redig

et al. 2015

Probab. Theory Relat. Fields

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134

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We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ∈ (0, 1) and where at most 2 j ∈ N particles per site are allowed. The process is constructed from a (2 j + 1)-dimensional representation of a quantum Hamiltonian with U q (sl 2 ) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure. Mathematics Subject Classification

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Section: Informal Description Of the Resultsmentioning

confidence: 99%

Section: Previous Extensions Of the Asepmentioning

confidence: 99%

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

Carinci

Giardinà

Redig

et al. 2015

Probab. Theory Relat. Fields

Self Cite

134

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“…Generalisations of the KMP kernel (2.1) have been obtained as instantaneous thermalisation limits of so-called Brownian energy processes [5]. Here we consider the kernels…”

Section: Parameter-dependent Kmp Modelsmentioning

confidence: 99%

“…The notion that the r-point correlation functions in the nonequilibrium steady state can be obtained from absorption probabilities of r dual particles has in the recent years led to a number of other fruitful applications of duality in the context of interacting particle systems [4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Heat conduction and the nonequilibrium stationary states of stochastic energy exchange processes

Gilbert¹

2017

J. Stat. Mech.

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Abstract. I revisit the exactly solvable Kipnis-Marchioro-Presutti model of heat conduction [J. Stat. Phys. 27 65 (1982)] and describe, for one-dimensional systems of arbitrary sizes whose ends are in contact with thermal baths at different temperatures, a systematic characterisation of their non-equilibrium stationary states. These arguments avoid resorting to the analysis of a dual process and yield a straightforward derivation of Fourier's law, as well as higher-order static correlations, such as the covariant matrix. The transposition of these results to families of gradient models generalising the KMP model is established and specific cases are examined.

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“…20, the KMP process is related to the process BEP(2), in the following way. Choose any i and consider the single-pair generator L BEP(m) i,i+1 .…”

Section: The Kipnis-marchioro-presutti Processmentioning

confidence: 99%

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Peletier

Redig

Vafayi

2014

Journal of Mathematical Physics

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We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the GBEP(a). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a).We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form − log ρ; they involve dissipation or mobility terms of order ρ 2 for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.

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Duality and Hidden Symmetries in Interacting Particle Systems

Cited by 143 publications

References 21 publications

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

Heat conduction and the nonequilibrium stationary states of stochastic energy exchange processes

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

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