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Register a new userPartially Thermostated Kac Model
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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.
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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 $eta in (0, 1/3]$.
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.
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 review the exact results on the various critical regimes of the antiferromagnetic $Q$-state Potts model. We focus on the Bethe Ansatz approach for generic $Q$, and describe in each case the effective degrees of freedom appearing in the continuum limit.
Kinetic energy equipartition is a premise for many deterministic and stochastic molecular dynamics methods that aim at sampling a canonical ensemble. While this is expected for real systems, discretization errors introduced by the numerical integration may distort such assumption. Fortunately, backward error analysis allows us to identify the quantity that is actually subject to equipartition. This is related to a shadow Hamiltonian, which coincides with the specified Hamiltonian only when the time-step size approaches zero. This paper deals with discretization effects in a straightforward way. With a small computational overhead, we obtain refine
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