Duality in quantum transport models

Duality for stochastic models of transport

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particles densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneous…

Duality and hidden equilibrium in transport models

A large family of diffusive models of transport that has been considered in the past years admits a transformation into the same model in contact with an equilibrium bath. This mapping holds at the full dynamical level, and is independent of dimension or topology. It provides a good opportunity to discuss questions of time reversal in out of equilibrium contexts. In particular, thanks to the mapping one may define the free-energy in the non-equilibrium states very naturally as the (usual) free energy of the mapped system.

Duality and phase diagram of one dimensional transport

The observation of duality by Mukherji and Mishra in one dimensional transport problems has been used to develop a general approach to classify and characterize the steady state phase diagrams. The phase diagrams are determined by the zeros of a set of coarse-grained functions without the need of detailed knowledge of microscopic dynamics. In the process, a new class of nonequilibrium multicritical points has been identified.

Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport

Let $X$ be a finite set and $Ω=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $σ$ the shift map acting on $Ω$. For a fixed probability $μ$ on $X$ with supp($μ$)$=X$, define $Π(μ,σ)$ as the set of all Borel probabilities $π\in P(X\times Ω)$ such that the $x$-marginal of $π$ is $μ$ and the $y$-marginal of $π$ is $σ$-invariant. We consider a fixed Lipschitz cost function $c: X \times Ω\to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times Ω$. Moreover, we define entropy, pressure and equilibrium plans. The study …

最优网络中的热力学与勒让德对偶性

Thermodynamics and Legendre Duality in Optimal Networks

本文建立非平衡输运网络最优性与广义热力学形式（含勒让德对偶）的联系，量化偏离工作点的距离，统一广义耗散与熵产生最小化，并揭示成本优化导致分支输运及相变行为。

Self-duality in quantum impurity problems

We establish the existence of an exact non-perturbative self-duality in a variety of quantum impurity problems, including the Luttinger liquid or quantum wire with impurity. The former is realized in the fractional quantum Hall effect, where the duality interchanges electrons with Laughlin quasiparticles. We discuss the mathematical structure underlying this property, which bears an intriguing resemblance with the work of Seiberg and Witten on supersymmetric non-abelian gauge theory.