One considers the motion of a test particle in an homogeneous fluid in equilibrium at temperature T , undergoing dissipative collisions with the fluid particles. It is shown that the corresponding linear Boltzmann equation still posseses a stationary Maxwellian velocity distribution, with an effective temperature smaller than T . This effective temperature is explicitly given in terms of the restitution parameter and the masses.
The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the non-linear Boltzmann equation for dimensions larger than one and provides expressions for the exponents describing the decay of the particle density n(t) ∝ t −ξ and the root mean-square velocity v ∝ t −γ in term of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents ξ and γ are obtained in arbitrary dimension as a function of the parameter µ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d = 1. For the two dimensional case, we implement Monte-Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.
We present a study of the Casimir effect in an imperfect (mean-field) Bose gas contained between two infinite parallel plane walls. The derivation of the Casimir force follows from the calculation of the excess grand-canonical free energy density under periodic, Dirichlet, and Neumann boundary conditions with the use of the steepest descent method. In the one-phase region, the force decays exponentially fast when distance D between the walls tends to infinity. When the Bose-Einstein condensation point is approached, the decay length in the exponential law diverges with critical exponent ν(IMP) = 1, which differs from the perfect gas case where ν(P) = 1/2. In the two-phase region, the Casimir force is long range and decays following the power law D(-3), with the same amplitude as in the perfect gas.
The adhesive dynamics of a one-dimensional aggregating gas of point particles is rigorously described. The infinite hierarchy of kinetic equations for the distributions of clusters of nearest neighbours is shown to be equivalent to a system of two coupled equations for a large class of initial conditions. The solution to these nonlinear equations is found by a direct construction of the relevant probability distributions in the limit of a continuous initial mass distribution. We show that those limiting distributions are identical to those of the statistics of shocks in the Burgers turbulence. The analysis relies on a mapping on a Brownian motion problem with parabolic constraints.
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