Minimum Stabilizing Energy Release for Mixing Processes

TL;DR

This paper investigates the limits of energy release in mixing processes, identifying bounds on the energy of stable states and providing methods to compute these bounds for both continuous and discrete systems.

Contribution

It introduces the concept of the minimum stabilizing energy release, linking it to the upper bound of accessible ground states, and develops computational techniques for discrete systems.

Findings

The upper bound of ground state energy corresponds to the quasilinear plateau.

Methods to compute bounds are extended from continuous to three-state discrete systems.

The upper bound can be calculated directly for small discrete systems.

Abstract

Diffusive operations, which mix the populations of different elements of phase space, can irreversibly transform a given initial state into any of a spectrum of different states from which no further energy can be extracted through diffusive operations. We call these ground states. The lower bound of accessible ground state energies represents the maximal possible release of energy. This lower bound, sometimes called the diffusively accessible free energy, is of interest in theories of instabilities and wave-particle interactions. On the other hand, the upper bound of accessible ground state energies has escaped identification as a problem of interest. Yet, as demonstrated here, in the case of a continuous system, it is precisely this upper bound that corresponds to the paradigmatic "quasilinear plateau" ground state of the bump-on-tail distribution. Although for general discrete systems the complexity of calculating the upper bound grows rapidly with the number of states, using techniques adapted from treatments of the lower bound, the upper bound can in fact be computed directly for the three-state discrete system.