No Arabic abstract
The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumptions of their continuity at the interfaces are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. We indicate also how to extend the solution method to the setting of one finite-sized domain surrounded on both sides by semi-infinite domains, and on that of three finite-sized domains.
We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $alphain [0,1]$, and the linearinteraction with the reservoirs by $(1-alpha)$, we prove that for all $alpha$ close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is emph{unique}, and there are no spatially non-uniform NESS with a spatial density $rho$ belonging to $L^p$ for any $p>1$. We also show that for all $alphain [0,1]$, the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.
We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix S, which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on S and is explicitly constructed. In this context we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current-current correlation function is derived and the zero-frequency noise power is determined.
We present a general construction of matrix product states for stationary density matrices of one-dimensional quantum spin systems kept out of equilibrium through boundary Lindblad dynamics. As an application we review the isotropic Heisenberg quantum spin chain which is closely related to the generator of the simple symmetric exclusion process. Exact and heuristic results as well as numerical evidence suggest a local quantum equilibrium and long-range correlations reminiscent of similar large-scale properties in classical stochastic interacting particle systems that can be understood in terms of fluctuating hydrodynamics.
This paper studies steady-state traffic flow on a ring road with up- and down- slopes using a semi-discrete model. By exploiting the relations between the semi-discrete and the continuum models, a steady-state solution is uniquely determined for a given total number of vehicles on the ring road. The solution is exact and always stable with respect to the first-order continuum model, whereas it is a good approximation with respect to the semi-discrete model provided that the involved equilibrium constant states are linearly stable. In an otherwise case, the instability of one or more equilibria could trigger stop-and-go waves propagating in certain road sections or throughout the ring road. The indicated results are reasonable and thus physically significant for a better understanding of real traffic flow on an inhomogeneous road.
Electric resistance in conducting media is related to heat (or entropy) production in presence of electric fields. In this paper, by using Arakis relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the non-autonomous C*-dynamical system obtained from the fermionic second quantization of a discrete Schrodinger operator with bounded static potential in presence of an electric field that is time- and space-dependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context, and is studied the present paper. We use tree-decay bounds of the $n$-point, $nin 2mathbb{N}$, correlations of the many-fermion system to analyze this regime. We verify below the 1st law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among other things, technically convenient. We initially focus our study on non-interacting (or free) fermions, but our approach will be later applied to weakly interacting fermions.