Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems

Schütz, Gunter M.; Sandow, Sven

doi:10.1103/physreve.49.2726

Cited by 133 publications

(163 citation statements)

References 25 publications

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“…It is interesting to recall that in the symmetric case D R = D L the conditional probability P (x 1 , x 2 ; t|y 1 , y 2 ; 0) determines not only the behaviour of the two-particle system, but also the behaviour of various time-dependent density-density correlation functions in N-particle systems, viz. the equal-time two-point correlator n x 1 (t)n x 2 (t) for an arbitrary initial state [3], the two-time correlator n x 1 (t 1 )n x 2 (t 2 ) for an arbitrary initial state [23] and the (time-translationally invariant) fourpoint correlator n x 1 (t 1 )n x 2 (t 2 )n x 3 (t 3 )n x 4 (t 4 ) averaged over the stationary distribution [23]. The contribution of the bound state to pair diffusion was discussed in [24].…”

Section: Discussionmentioning

confidence: 99%

Exact solution of the master equation for the asymmetric exclusion process

1997

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Section: Discussionmentioning

confidence: 99%

Exact solution of the master equation for the asymmetric exclusion process

1997

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“…The results of the section on exclusion processes partly overlap with [13]. We study indeed the same generalized version of exclusion processes with partial exclusion, i.e.…”

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confidence: 96%

“…In that case, one can describe the same physical situation as the transport of another quantity (another element of the group), and in some cases this makes the problem simpler. In the physics literature Sandow and Schutz [13] realized that this is case for the SEP process, whose SU(2) symmetry they made explicit by writing the evolution operator in quantum spin notation. In this paper we study in full generality the relation between duality and symmetries.…”

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confidence: 99%

Duality and Hidden Symmetries in Interacting Particle Systems

Kurchan²,

et al. 2009

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In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the "hidden" symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the KipnisMarchioro-Presutti (KMP) model for which we unveil its SU(1, 1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1, 1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.

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“…One may relax the exclusion constraint to allow for up to m particles on each lattice site. This gives rise to the partial exclusion process [22,23] with m + 1 states per site. Also particle systems with nonconserved internal degrees of freedom such as velocities in traffic flow models [24,25] have more than one possible state per site, but still obey a single continuity equation of the form (1.2).…”

Section: Basic Modelsmentioning

confidence: 99%

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

Schütz

2003

J. Phys. A: Math. Gen.

137

136

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Recent work on stochastic interacting particle systems with two particle species (or single-species systems with kinematic constraints) has demonstrated the existence of spontaneous symmetry breaking, long-range order and phase coexistence in nonequilibrium steady states, even if translational invariance is not broken by defects or open boundaries. If both particle species are conserved, the temporal behaviour is largely unexplored, but first results of current work on the transition from the microscopic to the macroscopic scale yield exact coupled nonlinear hydrodynamic equations and indicate the emergence of novel types of shock waves which are collective excitations stabilized by the flow of microscopic fluctuations. We review the basic stationary and dynamic properties of these systems, highlighting the role of conservation laws and kinetic constraints for the hydrodynamic behaviour, the microscopic origin of domain wall (shock) stability and the coarsening dynamics of domains during phase separation.

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Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems

Cited by 133 publications

References 25 publications

Exact solution of the master equation for the asymmetric exclusion process

Exact solution of the master equation for the asymmetric exclusion process

Duality and Hidden Symmetries in Interacting Particle Systems

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

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