A Collision Operator for Describing Dissipation in Noncanonical Phase Space

Abstract

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 an 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.

arXiv comment34 pages

References in corpus (21)

• Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas
• On Foundation of the Generalized Nambu Mechanics
• Thermal disequilibration of ions and electrons by collisionless plasma turbulence
• Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski
• Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions
• Nonadditive Entropies Yield Probability Distributions with Biases not Warranted by the Data
• A General Metriplectic Framework with Application to Dissipative Extended Magnetohydrodynamics
• Metriplectic Framework for Dissipative Magneto-Hydrodynamics
• Self-organization in foliated phase space: Construction of a scale hierarchy by adiabatic invariants of magnetized particles
• Energetically-consistent collisional gyrokinetics
• Collisionless relaxation of a Lynden-Bell plasma
• Diffusion with finite-helicity field tensor: a new mechanism of generating heterogeneity
• Up-Hill Diffusion Creating Density Gradient - What is the Proper Entropy?
• Statistical mechanics of two-dimensional turbulence
• Conservative dissipation: How important is the Jacobi identity in the dynamics?
• Differential formulation of the gyrokinetic Landau operator
• Calculation of large-aspect-ratio tokamak and toroidally-averaged stellarator equilibria of high-beta reduced magnetohydrodynamics via simulated annealing
• Dissipative Brackets for the Fokker-Planck Equation in Hamiltonian Systems and Characterization of Metriplectic Manifolds
• Charged Particle Diffusion in a Magnetic Dipole Trap
• Maximum entropy states of collisionless positron-electron plasma in a dipole magnetic field
• Generalization of Hamiltonian Mechanics to a Three Dimensional Phase Space

Cited by in corpus (4)

• Scattering Theory in Noncanonical Phase Space: A Drift-Kinetic Collision Operator for Weakly Collisional Plasmas
• A Collision Operator for Field-Mediated Interactions in General Relativistic Kinetic Theory
• Revealing Noncanonical Hamiltonian Structures in Relativistic Fluid Dynamics
• Axisymmetric Dynamos Sustained by a Modified Ohm's Law in a Toroidal Volume