Abstract:
Although kinetic equations for various scattering processes have proven their practical significance, the question about their area of applicability and derivation from the initial dynamic equations is satisfactorily solved only for the case of gases. This question remains relevant for kinetic equations describing both nonlinear wave processes and transport phenomena in semiconductors. Using the simplest example of elastic scattering, we show that the validity of the kinetic equation for the distribution function $n_{k}$ implies the existence of an internal characteristic time scale $\tau_{\rm s}$ much smaller than standard scattering times such as the transport time $\tau_{\rm tr}$. The time $\tau_{\rm s}$ corresponds to the rapid stochastization of the phases of the waves (quasiparticles). As show our estimates, this time satisfies the inequalities $1 \ll \omega_k\tau_{\rm s} \ll \omega_k\tau_{\rm tr}$, where $\omega_k = \varepsilon_k/\hbar$ is the characteristic frequency of the quasiparticles. This means that over time $\tau_{\rm s}$ the density matrix $\rho_{\rm k, k'}$ relaxes to the diagonal form $n_{\rm k} \delta_{\rm k- k'}$, so that its contribution to transport phenomena is small in the parameter $\tau_{\rm s}/\tau_{\rm tr}$ compared to the contribution of $n_{k}$.
Keywords:kinetic equations, scattering processes, density matrix, stochastization time, transport time
PACS:05.20.Dd
Received:June 17, 2025 Revised:August 7, 2025 Accepted: August 29, 2025
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