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Register a new userQuasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases
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The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behavior at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochastic driving is regular and for infinitesimal dissipation, has a stretched exponential tail, with an unusual stretching exponent $b_{QE} = 2b$, twice as large as the standard one for the corresponding $d$-dimensional system at finite dissipation. For deterministic driving the behavior is more subtle and displays singularities, such as multi-peaked velocity distribution functions. We classify the corresponding velocity distributions according to the nature and scaling behavior of such singularities.
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Combining analytical and numerical methods, we study within the framework of the homogeneous non-linear Boltzmann equation, a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the new method presented in a previous paper [J. Stat. Phys. 124, 549 (2006)] and extend our results to a different heating mechanism, namely a deterministic non-linear friction force. We derive analytically the high energy tail of the velocity distribution and compare the theoretical predictions with high precision numerical simulations. Stretched exponential forms are obtained when the non-equilibrium steady state is stable. We derive sub-leading corrections and emphasize their relevance. In marginal stability cases, power-law behaviors arise, with exponents obtained as the roots of transcendental equations. We also consider some simple BGK (Bhatnagar, Gross, Krook) models, driven by similar heating devices, to test the robustness of our predictions.
The algorithm for Dissipative Particle Dynamics (DPD), as modified by Espagnol and Warren, is used as a starting point for proving an H-theorem for the free energy and deriving hydrodynamic equations. Equilibrium and transport properties of the DPD fluid are explicitly calculated in terms of the system parameters for the continuous time version of the model.
We connect two different generalizations of Boltzmanns kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given.
The physics of highly excited Rydberg atoms is governed by blockade or exclusion interactions that hinder the excitation of atoms in the proximity of a previously excited one. This leads to cooperative effects and a relaxation dynamics displaying space-time heterogeneity similar to what is observed in the relaxation of glass-forming systems. Here we establish theoretically the existence of a glassy dynamical regime in an open Rydberg gas, associated with phase coexistence at a first-order transition in dynamical large deviation functions. This transition occurs between an active phase of low density in which dynamical processes take place on short timescales, and an inactive phase in which excited atoms are dense and the dynamics is highly arrested. We perform a numerically exact study and develop a mean-field approach that allows to understand the mechanics of this phase transition. We show that radiative decay --- which becomes experimentally relevant for long times --- moves the system away from dynamical phase coexistence. Nevertheless, the dynamical phase transition persists and causes strong fluctuations in the observed dynamics.
We study the kinetics of nonlinear irreversible fragmentation. Here fragmentation is induced by interactions/collisions between pairs of particles, and modelled by general classes of interaction kernels, and for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the non-vanishing mass flux criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters.
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