Long time asymptotics of a degenerate linear kinetic transport equation
Tomasz Komorowski
Kinetic & Related Models 2014, 7(1): 79-108 doi: 10.3934/krm.2014.7.79
In the present article we prove an algebraic rate of decay towards the equilibrium for the solution of a non-homogeneous, linear kinetic transport equation. The estimate is of the form $C(1+t)^{-a}$ for some $a>0$. The total scattering cross-section $R(k)$ is allowed to degenerate but we assume that $R^{-a}(k)$ is integrable with respect to the invariant measure.
keywords: Kinetic transport equation scattering kernel. Fredholm determinant
Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential
Tomasz Komorowski Lenya Ryzhik
Discrete & Continuous Dynamical Systems - B 2012, 17(3): 871-914 doi: 10.3934/dcdsb.2012.17.871
We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenbeck random potential when the initial data is spatially localized. The limit of the fluctuations of the Wigner transform satisfies a kinetic equation with random initial data. This result generalizes that of [12] where the random potential was assumed to be white noise in time.
keywords: Wigner transform kinetic equation radiative transport equation.
Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation
Tomasz Komorowski Łukasz Stȩpień
Kinetic & Related Models 2018, 11(2): 239-278 doi: 10.3934/krm.2018013

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

keywords: Harmonic oscillators kinetic limit Ornstein-Uhlenbeck process Wigner functions correctors
Kinetic limits for waves in a random medium
Guillaume Bal Tomasz Komorowski Lenya Ryzhik
Kinetic & Related Models 2010, 3(4): 529-644 doi: 10.3934/krm.2010.3.529
keywords: Waves in random media Kinetic equation Scintillation function Time reversal. Wigner transform Wave-wave correlation Radiative transport equation Self-averaging Particles in random flow Fokker-Planck equation
Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities
Tomasz Komorowski Stefano Olla Marielle Simon
Kinetic & Related Models 2018, 11(3): 615-645 doi: 10.3934/krm.2018026

We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

keywords: Hydrodynamic limit chain of oscillators heat diffusion Wigner distribution thermalization

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