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Register a new userUniqueness of the Non-Equilibrium Steady State for a $1$d BGK model in kinetic theory
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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.
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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.
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.
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 prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $a in L^2(partialOmega)$ satisfies ${1over 2pi}|int_{partialOmega}acdot |<1$. If $Omega$ is of class $C^{1,1}$, we can assume $ain W^{-1/4,4}(partialOmega)$. Moreover, we show that for every $D$--solution $(u,p)$ of the Navier--Stokes equations it holds $ abla p = o(r^{-1}), abla_k p = O(r^{epsilon-3/2}), abla_ku = O(r^{epsilon-3/4})$, for all $kin{Bbb N}setminus{1}$ and for all positive $epsilon$, and if the flux of $u$ through a circumference surrounding $complementOmega$ is zero, then there is a constant vector $u_0$ such that $u=u_0+o(1)$.
We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.
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