Asymmetric Stochastic Transport Models with $${\mathscr {U}}_q(\mathfrak {su}(1,1))$$ U q ( su ( 1 , 1 ) ) Symmetry

Abstract

By using the algebraic construction outlined in Carinci et al. (arXiv:1407.3367, 2014, we introduce several Markov processes related to the U q (su(1, 1)) quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the and which turns out to have the Symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

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

“…Recently it has been applied to algebras with higher rank, such as U q (gl(3)) [2,15] or U q (sp (4)) [15], yielding two-component asymmetric exclusion process with multiple conserved species of particles. In [7] we also applied the method to non-compact algebras such as U q (su (1, 1)), finding new diffusion processes of heat transport with duality.…”

Local Equilibrium in Inhomogeneous Stochastic Models of Heat Transport

“…Some of these results were announced in [LNY15]. The systems we consider are somewhat similar to the ones recently introduced in [CGRS15], but are not the same.…”

“…Then we study the macroscopic energy propagation and the emergence of local equilibrium. The special case of constant 2 degrees of freedom is [KMP82], while systems with (arbitrary) constant degrees of freedom have been studied in [CGRS15]. A chain of alternating billiard particles (2 degrees of freedom) and pistons (1 degree of freedom), has been proposed in [BGNSzT15].…”

Orthogonal Dualities of Markov Processes and Unitary Symmetries

“…For the reader convenience we recall [3,10,18,20,25,31] for some applications to non-equilibrium statistical physics, [5,24] for duality in population models and [2,8,9] for the study of singular stochastic PDE via duality. We also mention the algebraic approach to duality that shows that several Markov processes dualities in turn derive from algebraic structures, see for instance [6,7,15,22,28].…”