Upper and Lower Bounds on Phase-Space Rearrangements*

Theories of free or available energy quantify the energy that could be released by phase-space rearrangements of a given distribution. Different theories correspond to different rules for which rearrangement operations are allowed. The Gardner free energy is defined by Gardner restacking, which allows any operation that conserves phase space volumes [1]. The diffusively accessible free energy instead allows pairwise averaging operations [2-6]. Different rearrangements can map a given initial state to different ground states – that is, states from which no further energy can be extracted. Gardner retacking maps any initial state to a unique ground state; diffusive exchange operations can map an initial state into any of a spectrum of final states. The lowest-energy diffusively accessible state turns out to be identical to the Gardner ground state [5]. The highest-energy diffusively accessible ground state recovers certain intuitions in, for example, bump-on-tail distributions [6]. This presentation will focus on the characterization of these accessible ground states, and on their applications. Applications include instabilities [6-8] and phase-space engineering, such as the transfer of energy from fusion products to fuel ions in alpha-channeling [9].

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[5] E. J. Kolmes and N. J. Fisch, Phys. Rev. E 102, 063209 (2020).

[6] E. J. Kolmes and N. J. Fisch, Phys. Rev. E 106, 055209 (2022).

[7] P. Helander, J. Plasma Phys. 83, 715830401 (2017).

[8] P. Helander, J. Plasma Phys. 86, 905860201 (2020).

[9] N. J. Fisch and J.-M. Rax, Phys. Rev. Lett. 69, 612 (1992).