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Register a new userApproach to equilibrium for the phonon Boltzmann equation
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We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.
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We use a probabilistic approach to study the rate of convergence to equilibrium for a collisionless (Knudsen) gas in dimension equal to or larger than 2. The use of a coupling between two stochastic processes allows us to extend and refine, in total variation distance, the polynomial rate of convergence given in [AG11] and [KLT13]. This is, to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 that does not require any symmetry of the domain, nor a monokinetic regime. Our study is also more general in terms of reflection at the boundary: we allow for rather general diffusive reflections and for a specular reflection component.
A derivation of the Dirac equation in `3+1 dimensions is presented based on a master equation approach originally developed for the `1+1 problem by McKeon and Ord. The method of derivation presented here suggests a mechanism by which the work of Knuth and Bahrenyi on causal sets may be extended to a derivation of the Dirac equation in the context of an inference problem.
Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows to write down the master equation based on the type of of the kinetic equations and assumptions about the nature of the process. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. Leaving in the expansion terms up to the second order we can get the Fokker-Planck equation, and thus the Langevin equation. It should be clearly understood that these equations are approximate recording of the master equation. However, this does not eliminate the need for the study of the master equation. Moreover, the power series produced during the master equation decomposition may be divergent (for example, in spatial models). This makes it impossible to apply the classical perturbation theory. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. As an example the Verhulst model is used because of its simplicity and clarity (the first order equation is independent of the spatial variables, however, contains non-linearity). We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.
In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad assumption on transition probability rate, that comprises hard potentials for both the relative speed and internal energy with the rate in the interval $(0,2]$, which is multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ODE theory in Banach spaces, for an initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite $k_*$ polynomial moment, with $k_*$ depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated to the solution are both generated and propagated uniformly in time.
We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In cite{pomeau1987shock}, a self-similarity approach was proposed for infinite total cross section resulting from a power law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau, Bobylev and Cercignani started the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy cite{bobylev2002self,bobylev2003eternal}. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.
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