Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state

Abstract

This paper presents a study of the nonequilibrium relaxation process of chemically reactive systems using steepest-entropy-ascent quantum thermodynamics (SEAQT). The trajectory of the chemical reaction, i.e., the accessible intermediate states, is predicted and discussed. The prediction is made using a thermodynamic-ensemble approach, which does not require detailed information about the particle mechanics involved (e.g., the collision of particles). Instead, modeling the kinetics and dynamics of the relaxation process is based on the principle of steepest-entropy ascent (SEA) or maximum-entropy production, which suggests a constrained gradient dynamics in state space. The SEAQT framework is based on general definitions for energy and entropy and at least theoretically enables the prediction of the nonequilibrium relaxation of system state at all temporal and spatial scales. However, to make this not just theoretically but computationally possible, the concept of density of states is introduced to simplify the application of the relaxation model, which in effect extends the application of the SEAQT framework even to infinite energy eigenlevel systems. The energy eigenstructure of the reactive system considered here consists of an extremely large number of such levels (on the order of 10^{130}) and yields to the quasicontinuous assumption. The principle of SEA results in a unique trajectory of system thermodynamic state evolution in Hilbert space in the nonequilibrium realm, even far from equilibrium. To describe this trajectory, the concepts of subsystem hypoequilibrium state and temperature are introduced and used to characterize each system-level, nonequilibrium state. This definition of temperature is fundamental rather than phenomenological and is a generalization of the temperature defined at stable equilibrium. In addition, to deal with the large number of energy eigenlevels, the equation of motion is formulated on the basis of the density of states and a set of associated degeneracies. Their significance for the nonequilibrium evolution of system state is discussed. For the application presented, the numerical method used is described and is based on the density of states, which is specifically developed to solve the SEAQT equation of motion. Results for different kinds of initial nonequilibrium conditions, i.e., those for gamma and Maxwellian distributions, are studied. The advantage of the concept of hypoequilibrium state in studying nonequilibrium trajectories is discussed.

Generalized thermodynamic relations for a system experiencing heat and mass diffusion in the far-from-equilibrium realm based on steepest entropy ascent

“…The state representation and the equation of motion can be simplified by combining degenerate energy eigenlevels [16]. The system is defined by a group of different energy eigenlevels {ǫ i , i = 1, 2, ...} and their degeneracy {n i , i = 1, 2, ...}.…”

“…More discussion is presented in reference [16], and an example is provided below. The energy eigenlevels of the system {ǫ i , i = 1, 2, 3...} with degeneracy {n i , i = 1, 2, 3...} can be divided into to M sets {ǫ…”

“…Given a way to divide the energy eigenlevels, if the system probability distributions in the M subspaces are all canonical distribution, the state of the system is called a M th-order hypoequilibrium state [16], which can be described uniquely by the total probability in each subspace ({p…”

“…The macroscopic properties of entropy, energy, and particle number, which are well defined for any state of any system [15], are used to develop the governing equation and describe system state evolution. Recently, this description has been simplified via the concept of hypoequilibrium state [16], which captures the global features of the microscopic description for the relaxation process. In addition, the concept of nonequilibrium intensive properties introduced in [16] based on the concept of hypoequilibrium state enables a complete description of the nonequilibrium evolution of state when combined with the set of nonequilibrium extensive properties.…”

“…Recently, this description has been simplified via the concept of hypoequilibrium state [16], which captures the global features of the microscopic description for the relaxation process. In addition, the concept of nonequilibrium intensive properties introduced in [16] based on the concept of hypoequilibrium state enables a complete description of the nonequilibrium evolution of state when combined with the set of nonequilibrium extensive properties. Unlike the intensive property definitions of other nonequilibrium thermodynamic approaches (definitions which require local-equilibrium, near-equilibrium, or steady state assumption or a phenomenological basis), the definitions in the SEAQT framework are fundamental and available to all nonequilibrium states and are especially suitable for the description of the evolution in state of relaxation processes.…”