We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven-diffusive systems—which includes exclusion processes—focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix-product form. In addition to a gentle introduction to this matrix-product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free-energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix-product state for a given model and more complicated variants of the matrix-product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

Nonequilibrium steady states of matrix-product form: a solver's guide

A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

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Quantum graphs: I. Some basic structures

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.

https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-von-oppen/papers/nlin_CD_0605028.pdf

Quantum graphs: Applications to quantum chaos and universal spectral statistics

This paper reviews a recently discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

The ODE/IM correspondence

A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Graph models for waves in thin structures

The inverse scattering problem on branching graphs is studied. The definition of the Schrodinger operator on such graphs is discussed. The operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices. It is shown that in general the scattering matrix does not determine the topology of the graph, the potentials on the edges and the boundary conditions uniquely.

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Fredrik Stenberg

Papers

On the inverse scattering problem on branching graphs