Recovering Gardner Restacking with Purely Diffusive Operations

TL;DR

This paper demonstrates that the Gardner free energy in plasma physics can be recovered from the diffusive free energy in the continuous limit, showing that phase-space conservation is possible through entropy-increasing operations.

Contribution

It reveals that the Gardner free energy can be obtained from the diffusive free energy in the continuous limit, bridging a gap between two fundamental free energy concepts.

Findings

Gardner free energy can be recovered from diffusive free energy in the continuous limit.

Macroscopic phase-space conservation can result from microscopic diffusive operations.

Diffusive free energy is generally less than Gardner free energy for finite discretizations.

Abstract

The maximum particle kinetic energy that can be extracted from an initial six-dimensional phase space distribution motivates the concept of free or available energy. The free energy depends on the allowed operations that can be performed. A key concept underlying the theoretical treatment of plasmas is the Gardner free energy, where the exchange of the contents of equal phase volumes is allowed. A second free energy concept is the diffusive free energy, in which the contents of volumes are instead averaged. For any finite discretization of phase space, the diffusive free energy is known to be less than the Gardner free energy. However, despite the apparent fundamental differences between these free energies, it is demonstrated here that the Gardner free energy may be recovered from the continuous limit of the diffusive free energy, leading to the surprise that macroscopic phase-space conservation can be achieved by ostensibly entropy-producing microscopic operations.