Non-equilibrium distribution function in ultra-fast processes
K. S. Glavatskiy
TL;DR
The paper tackles fast and ultra-fast non-equilibrium processes by allowing the distribution function to depend on the time derivatives of equilibrium integrals of motion, yielding a transient (non-equilibrium) Gibbs relation with shift-time terms. A threshold between slow and fast processes is defined via a characteristic time $\tau$ such that $t \gg \tau$ is quasi-equilibrium and $t \ll \tau$ is transient; the non-equilibrium framework also introduces two shift times $\tau_E$ and $\tau_N$. Applied to inertial heat conduction, the theory reproduces the Maxwell-Cattaneo-Vernotte equation $J + \tau_E\dot{J} = -K\,(T_1 - T_2)$ and relates $\tau_E$ to a finite propagation speed via $\tau_E = \kappa/(c_v\,u^2)$. The approach also yields a non-equilibrium expression for work and reveals two distinct pressures, external $p$ and internal/thermodynamic $P$, along with a transient non-equilibrium Gibbs relation, offering a rigorous statistical-mechanical basis for ultra-fast thermodynamics.
Abstract
A simple expression for the non-equilibrium distribution function in ultra-fast transient processes is proposed. Postulating its dependence on temporal derivatives of the equilibrium integrals of motion, non-equilibrium analogues of the thermodynamic relationships are derived and the conditions that maximize the non-equilibrium entropy are identified. A rigorous threshold between ``slow" and ``fast" processes is suggested, identifying the range of applicability of classical quasi-equilibrium description. The proposed theory is validated by deriving the known law of inertial heat conduction, which accounts for finite speed of thermal propagation. Finally, a new expression for the non-equilibrium work is derived, revealing two kinds of pressure that emerge in fast non-equilibrium.