A collision operator for describing dissipation in noncanonical phase space

Journal Article · Mon May 13 00:00:00 EDT 2024 · Fundamental Plasma Physics

DOI:https://doi.org/10.1016/j.fpp.2024.100054· OSTI ID:2349563

Sato, Naoki ^[1]; Morrison, Philip J. ^[2]

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1. National Inst. for Fusion Science (Japan)
2. Univ. of Texas, Austin, TX (United States)

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

• Research Organization: Univ. of Texas, Austin, TX (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: FG02-04ER54742; FG05-80ET53088
• OSTI ID: 2349563
• Alternate ID(s): OSTI ID: 2997888
• Journal Information: Fundamental Plasma Physics, Journal Name: Fundamental Plasma Physics Vol. 10; ISSN 2772-8285
• Publisher: Elsevier BV
• Country of Publication: United States
• Language: English

Related Subjects

Boltzmann equation
Collision operator
Collisionless relaxation
Noncanonical Hamiltonian mechanics
Self-organization