A complex of issues related to the problem proving Boltzmann formula for probability density (distribution) w(η) values a certain set parameters η an equilibrium system is investigated. The classical expresses this distribution through non-equilibrium free energy F (η) , which depends on mentioned parameters. This possible because after occurrence fluctuations, finds itself in state, evolves fu...

Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field Abstract We consider the long time evolution of a quantum particle weakly interacting with a phonon field. We show that in the weak coupling limit the Wigner distribution of the electron density matrix converges to the solution of the linear Boltzmann equation globally in time. The collision kern...

describes the evolution of a density of particles in a medium in which the particles don’t interact among themselves. The motion of each particle is described by a jump process: The speed of a particle is constant (equal to one), and also the direction is constant in exponentially distributed time intervals. At the end of such an interval, the direction jumps according to a law that corresponds...

We present a new learning algorithm for Mean Field Boltzmann Machines based on the contrastive divergence optimization criterion. In addition to minimizing the divergence between the data distribution and the equilibrium distribution that the network believes in, we maximize the divergence between one-step reconstructions of the data and the equilibrium distribution. This eliminates the need to...

In 1872 Ludwig Boltzmann published a lengthy memoir containing two fundamental results: an integro-differential equation describing the time evolution of an ideal gas (Boltzmann equation) and a mathematical argument proving that such evolution must reach, sooner or later, the state of equilibrium described by the Maxwell distribution. The latter achievement, later called the H-theorem, expresse...

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite v...

We develop a numerical framework for the quantum analogue of the “classical” lattice Boltzmann method (LBM), with the Maxwell-Boltzmann distribution replaced by the Fermi-Dirac function. To accommodate the spin density matrix, the distribution functions become 2× 2-matrix valued. We show that the efficient, commonly used BGK approximation of the collision operator is valid in the present settin...

Two-dimensional decaying turbulence is simulated using the lattice Boltzmann method with the Bhatnagar-Gross-Krook collision operator. The simulation results are analyzed in the spectral space and the auto power spectral density functions of the one-particle distribution functions are presented. It is shown that both the equilibrium and the non-equilibrium parts of the distribution functions ha...

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to...

We know that the velocity distribution of a gas of classical particles in equilibrium is the Maxwellian distribution. This is a very well experimentally confirmed fact. The approach in kinetic theory that gives the time evolution of the velocity distribution of a gas of particles is the Boltzmann equation. Hence, the Boltzmann equation should have the Maxwellian distribution as an attractive fi...