Relaxation of a single-particle excitation in a Fermi system within the diffusion approximation of kinetic theory

TL;DR

This paper investigates the relaxation dynamics of a single-particle excitation in a Fermi system using the diffusion approximation of kinetic theory. It solves a nonlinear diffusion equation with constant coefficients $D_{p,0}$ and $K_{p,0}$ and decomposes the distribution into a step-like core and a single-particle peak to extract separate relaxation times. The results show both the total relaxation time and the single-particle relaxation time are of order $\tau \sim 10^{-24}$ s, with finite-size effects causing $\tau$ to decrease with excitation energy $E_{ex}$ while $\tau_1$ increases with $E_{ex}$ and decreases with mass number $A$. A parametric scaling of the kinetic coefficients reveals that significantly reducing $D_{p,0}$ and $K_{p,0}$ is required to attain longer times, indicating a discrepancy with prior coefficient estimates and prompting refinement of the diffusion model and consideration of quantum effects.

Abstract

The time evolution of the Wigner distribution function for a single-particle excitation in a Fermi system was studied within the framework of the diffusion approximation of kinetic theory by numerically solving a nonlinear diffusion equation with constant kinetic coefficients. A method was proposed to separate the dissipative processes into contributions from the relaxation of the single-particle excitation and from the relaxation of the nuclear core, with a distinct relaxation time introduced for each process. The influence of the diffusion and drift coefficients on the characteristic relaxation time scale was analyzed. It was found that the resulting relaxation times exhibit a discrepancy relative to the kinetic coefficient estimates known from previous studies.

Figures (6)

• Figure 1: (Color online) Profiles of the distribution function $f_0(p,t)$ at the time points indicated in the captions. The blue line represents the initial step-like distribution function (\ref{['fin0']}), while the red curve shows the equilibrium Fermi distribution function (\ref{['feq']}), corresponding to $t = 3.0 \times 10^{-22}$ s.
• Figure 2: (Color online) Profiles of the distribution function $f(p,t)$ at different time moments, as indicated in the captions. The blue line corresponds to the initial distribution function (\ref{['fin1p']}), while the red curve represents the equilibrium Fermi distribution function (\ref{['feq']}).
• Figure 3: (Color online) Relaxation time of the nucleus, $\tau$, as a function of the single-nucleon excitation energy $E_\text{ex}$. Calculations are shown for mass numbers $A = 50$, $150$, and $250$.
• Figure 4: (Color online) Profiles of the deviation of the distribution function $\delta f_1(p,t)$ at the time moments indicated in the captions. The blue line represents the initial difference $f_\text{in}(p)-f_{\text{in},0}(p)$, while the red curve corresponds to the equilibrium Fermi distribution (\ref{['feq']}).
• Figure 5: (Color online) Relaxation time of a single-nucleon excitation, $\tau_1$, as a function of the excitation energy $E_\text{ex}$. Calculations are shown for mass numbers $A = 50$, $150$, and $250$.