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Register a new userEquilibration in the Kac Model using the GTW Metric $d_2$
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Authors
Hagop Tossounian
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We use the Fourier based Gabetta-Toscani-Wennberg (GTW) metric $d_2$ to study the rate of convergence to equilibrium for the Kac model in $1$ dimension. We take the initial velocity distribution of the particles to be a Borel probability measure $mu$ on $mathbb{R}^n$ that is symmetric in all its variables, has mean $vec{0}$ and finite second moment. Let $mu_t(dv)$ denote the Kac-evolved distribution at time $t$, and let $R_mu$ be the angular average of $mu$. We give an upper bound to $d_2(mu_t, R_mu)$ of the form $min{ B e^{-frac{4 lambda_1}{n+3}t}, d_2(mu,R_mu)}$, where $lambda_1 = frac{n+2}{2(n-1)}$ is the gap of the Kac model in $L^2$ and $B$ depends only on the second moment of $mu$. We also construct a family of Schwartz probability densities ${f_0^{(n)}: mathbb{R}^nrightarrow mathbb{R}}$ with finite second moments that shows practically no decrease in $d_2(f_0(t), R_{f_0})$ for time at least $frac{1}{2lambda}$ with $lambda$ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in [14].
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We study a model of random colliding particles interacting with an infinite reservoir at fixed temperature and chemical potential. Interaction between the particles is modeled via a Kac master equation cite{kac}. Moreover, particles can leave the system toward the reservoir or enter the system from the reservoir. The system admits a unique steady state given by the Grand Canonical Ensemble at temperature $T=beta^{-1}$ and chemical potential $chi$. We show that any initial state converges exponentially to equilibrium by computing the spectral gap of the generator in a suitable $L^2$ space and by showing exponential decrease of the relative entropy with respect to the steady state. We also show propagation of chaos and thus the validity of a Boltzmann-Kac type equation for the particle density in the infinite system limit.
We introduce a global thermostat on Kacs 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation. We prove ergodicity for this equation, and show that its equilibrium distribution has a density that, depending on the parameters of the model, can exhibit heavy tails, and whose behaviour at the origin can range from being analytic, to being $C^k$, and even to blowing-up. Finally, we prove propagation of chaos for the associated $N$-particle system, with a uniform-in-time rate of order $N^{-eta}$ in the squared $2$-Wasserstein metric, for an explicit $eta in (0, 1/3]$.
We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $Mle N$ particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelsons hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.
This article reviews recent work on the Kac master equation and its low dimensional counterpart, the Kac equation.
We consider a system of $M$ particles in contact with a heat reservoir of $Ngg M$ particles. The evolution in the system and the reservoir, together with their interaction, are modeled via the Kacs Master Equation. We chose the initial distribution with total energy $N+M$ and show that if the reservoir is initially in equilibrium, that is if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decay exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in [2].
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