Kac Master Equation Research Articles

Grand Canonical Evolution for the Kac Model

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 fast 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.

Decay of Information for the Kac Evolution

We consider a system of $M$ particles in contact with a heat reservoir of $N\gg M$ particles. The evolution in the system and the reservoir, together with their interaction, are modeled via the Kac's 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].

Chaos, ergodicity, and equilibria in a quantum Kac model

We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup PN,t, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the N particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study of the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.

Entropy Decay for the Kac Evolution

We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $M\le N$ 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 Nelson's 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.

Perfect and ε-perfect simulation methods for the one-dimensional Kac equation

Abstract We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel “ε-perfect sampling” algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.

Propagation of Chaos for the Thermostatted Kac Master Equation

The Kac model is a simplified model of an $N$-particle system in which the collisions of a real particle system are modeled by random jumps of pairs of particle velocities. Kac proved propagation of chaos for this model, and hence provided a rigorous validation of the corresponding Boltzmann equation. Starting with the same model we consider an $N$-particle system in which the particles are accelerated between the jumps by a constant uniform force field which conserves the total energy of the system. We show propagation of chaos for this model.

The Kac Model Coupled to a Thermostat

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $\beta$. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.

Determination of the spectral gap for Kac's master equation and related stochastic evolution

Determination of the spectral gap for Kac's master equation and related stochastic evolution

Bounds for Kac's Master Equation

Mark Kac considered a Markov Chain on the n-sphere based on random rotations in randomly chosen coordinate planes. This same walk was used by Hastings on the orthogonal group. We show that the walk has spectral gap bounded below by c/n 3. This and curvature information are used to bound the rate of convergence to stationarity.

Generalized scheme of the no-time-counter scheme for the DSMC in rarefied gas flow analysis

Abstract The generalized scheme of the no-time-counter (NTC) scheme for the direct simulation Monte Carlo method in the analysis of rarefied gas flow is proposed. The generalized scheme is theoretically derived from Kac's master equation. This scheme gives the theoretical basis of the NTC scheme, in that the NTC scheme can be derived approximately from the generalized scheme. The applicability of the present scheme is verified through an application to the relaxation problem in which the non-equilibrium distribution function relaxes to an equilibrium one. The present scheme and the NTC scheme show good agreement with respect to the averaged quantities, such as distribution function and collision rate. From the point of view of application, the NTC scheme is sufficient for obtaining a result for the averaged quantities at least.

On the use of the modified Nanbu direct simulation scheme

The application of direct simulation Monte Carlo methods to complex rarefied gas problems is well established. Several such methods now exist and these may be put into one of two categories, i.e., those which model the Boltzmann equation and those which model the Kac master equation. Previously, the latter methods have been in the ascendancy mainly due to their numerical efficiency. However, recent advances in using the Boltzmann equation as the kinetic model mean that the application of this solution technique warrants investigation. Our calculations show that the new DSMC method derived from the Boltzmann equation can achieve a processing time comparable with the more popular Monte Carlo schemes based on the Kac formulation.

Fluctuations in a Markovian system of pairwise interacting particles

Analogously to the Kac's master equation approach to the spatially homogeneous Boltzmann equation we introduce a system of Markov processes of many particles moving on a countable set with pairwise interaction, and investigate the fluctuation around McKean's non-linear limit process. Our model possibly admits simultaneous jumps of two particles, which make impossible both such characterizations of the fluctuation in the limit and techniques (based on the Cameron-Martin formula) as has previously been obtained or used for diffusion models and for McKean's two-speed gas model. We obtain a new description of the variance functional for the fluctuation, and, by applying it in the case of no simultaneous jumps, give a new derivation of a formula of H. Tanaka [7].

Derivation from Kac's Master Equation of the Stochastic Laws for Simulating Molecular Collisions

Belotserkovskiy and Yanitskiy proposed stochastic laws for simulating molecular collisions in gas; these laws are consistent with the assumptions on which Kac's master equation is built up. Firstly, it is shown that the laws can be derived directly from Kac's master equation by use of the stochastic solution method recently proposed by the author. Secondly, by use of the laws the solution of Kac's master equation is obtained and compared with the known exact solution of the Boltzmann equation. If the number of simulated molecules greater than about 40 is used, the obtained solution agrees with the exact solution.

Interrelations between Various Direct Simulation Methods for Solving the Boltzmann Equation

Interrelations between the five direct simulation methods of Nanbu, Belotserkovskiy and Yanitskiy, Koura, Deshpande, and Bird are examined by means of probability consideration. These methods are divided into two groups; one based on the Boltzmann equation and the other based on the Kac master equation. Nanbu's method belongs to the first group while all other methods belong to the second group. The methods of Koura, Deshpande, and Bird are modifications of that of Belotserkovskiy and Yanitskiy. Although the probability law for simulating molecular collisions is different from method to method, the mean rate of collision is essentially the same for all methods. This implies that there is no significant difference at least among macroscopic properties resulting from all methods.