Exact steady state solution of the Boltzmann equation: A driven 1-D inelastic Maxwell gas

The exact nonequilibrium steady state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of the distribution function $f(c)$. In this paper we have inverted the Fourier transform to express $f(c)$ in the form of an infinite series of exponentially decaying terms. The dominant high energy tail is exponential, $f(c)\simeq A_0\exp(-a|c|)$, where $a\equiv 2/\sqrt{1-\alpha^2}$ and the amplitude $A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($\alpha\to 0$) and in the quasi-elastic limit ($\alpha\to 1$). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high energy tail exists for velocities $|c-c_0|\sim 1/\sqrt{q}$ around a crossover velocity $c_0\simeq \ln q^{-1}/\sqrt{q}$, where $q\equiv (1-\alpha)/2\ll 1$. In this crossover region the distribution function is extremely small, $\ln f(c_0)\simeq q^{-1}\ln q$.Comment: 11 pages, 4 figures; a table and a few references added; to be published in PR

Apparent Rate Constant for Diffusion-Controlled Three molecular (catalytic) reaction

We present simple explicit estimates for the apparent reaction rate constant for three molecular reactions, which are important in catalysis. For small concentrations and $d> 1$, the apparent reaction rate constant depends only on the diffusion coefficients and sizes of the particles. For small concentrations and $d\le 1$, it is also time -- dependent. For large concentrations, it gains the dependence on concentrations.Comment: 12 pages, LaTeX, Revised: missing ref. for important paper by G. Oshanin and A. Blumen was added and minor misprints correcte