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Register a new userOn a thermostated Kac model with rescaling
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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 $eta in (0, 1/3]$.
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We study a system of $N$ particles interacting through the Kac collision, with $m$ of them interacting, in addition, with a Maxwellian thermostat at temperature $frac{1}{beta}$. We use two indicators to understand the approach to the equilibrium Gaussian state. We prove that i) the spectral gap of the evolution operator behaves as $frac{m}{N}$ for large $N$ ii) the relative entropy approaches its equilibrium value (at least) at an eventually exponential rate $sim frac{m}{N^2}$ for large $N$. The question of having non-zero entropy production at time $0$ remains open. A relationship between the Maxwellian thermostat and the thermostat used in Bonetto, Loss, Vaidyanathan (J. Stat. Phys. 156(4):647-667, 2014) is established through a van Hove limit.
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 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].
We consider Kacs 1D N-particle system coupled to an ideal thermostat at temperature T, introduced by Bonetto, Loss, and Vaidyanathan in 2014. We obtain a propagation of chaos result for this system, with explicit and uniform-in-time rates of order N^(-1/3) in the 2-Wasserstein metric. We also show well-posedness and equilibration for the limit kinetic equation in the space of probability measures. The proofs use a coupling argument previously introduced by Cortez and Fontbona in 2016.
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.
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