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Register a new userSpectral Gaps for Reversible Markov Processes with Chaotic Invariant Measures: The Kac Process with Hard Sphere Collisions in Three Dimensions
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We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.
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Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a ``thermodynamic theory can also be developed based on Tsallis generalized entropy $S^{(q)}=sum_{i=1}^N(p_i-p_i^q)/[q(q-1)]$ and Shiinos generalized free energy $F^{(q)}=[sum_{i=1}^Np_i(p_i/pi_i)^{q-1}-1]/[q(q-1)]$, where $pi_i$ is the stationary distribution. $dF^{(q)}/dt=-f_d^{(q)}le 0$ and it is zero iff the system is in its stationary state. $dS^{(q)}/dt-Q_{ex}^{(q)} = f_d^{(q)}$ where $Q_{ex}^{(q)}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d^{(q)}$ is the product of Onsagers thermodynamic flux and force. However, it is discovered that the Onsagers force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis statistics.
Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation.
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.
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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