Particle Production in Strong Time-dependent Fields

In these lecture notes we give an introduction to the kinetic equation approach to pair production form the vacuum in strong, time-dependent external fields (dynamical Schwinger process). We first give a derivation of the kinetic equation with the source term for the case of fermions starting from the Dirac equation and for bosons from the Klein-Gordon equation. In a second part we discuss the application of the approach to the situation of external field pulses as single-sheeted functions of time (like the Sauter-pulse) and as multi- sheeted functions approximating the situation in the focal point of counter-propagating laser beams. Special emphasis is on the discussion of the time evolution of the system that exhibits the characteristics of a field-induced phase transition for which we discuss the behaviour of the entropy and particle density of the system. We give an outlook to applications of the approach in describing particle production in strong fields formed in particle and nuclear collisions.Comment: 23 pages, 7 figures, Lecture Notes based on arXiv:hep-ph/9809227 and arxiv:1607.08775; to appear in Proceedings of the Helmholtz International Summer School on "Quantum Field Theory at the Limits: From Strong Fields to Heavy Quarks", July 18-30, 2016, Dubna, Russi

Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions

We study the high-energy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hard-sphere Boltzmann equations, which imply that the velocity distribution functions $f(v)$ behave in a certain sense as $C\exp(-r|v|^s)$ for $|v|$ large. The values of $s$, which we call {\em the orders of tails}, range from $s=1$ to $s=2$, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of the Povzner-type inequalities.Comment: 22 page