[Paper Review] Landau equation for self-gravitating classical and quantum particles: Application to dark matter

Pierre-Henri Chavanis|arXiv (Cornell University)|Dec 22, 2020

Advanced Thermodynamics and Statistical MechanicsPhysics and Astronomy102 references39 citations

TL;DR

The paper develops the kinetic theory of classical and quantum self-gravitating particles, derives the quantum Landau equation, and discusses applications to dark matter including Bose–Einstein condensation and soliton formation.

ABSTRACT

We develop the kinetic theory of classical and quantum particles (fermions and bosons) in gravitational interaction. The kinetic theory of quantum particles may have applications in the context of dark matter. For simplicity, we consider an infinite and spatially homogeneous system (or make a local approximation) and neglect collective effects. This leads to the quantum Landau equation derived heuristically in [Chavanis, Physica A 332, 89 (2004)]. We establish its main properties: conservation laws, $H$-theorem, equilibrium state, relaxation time, quantum diffusion and friction coefficients, quantum Rosenbluth potentials, self-consistent evolution, (thermal) bath approximation, quantum Fokker-Planck equation, quantum King model... For bosonic particles, the Landau equation can describe the process of Bose-Einstein condensation. We discuss the relation of our study with the works of [Levkov et al., Phys. Rev. Lett. 121, 151301 (2018); Bar-Or et al., Astrophys. J. 871, 28 (2019)] on fuzzy dark matter halos and the formation of Bose stars and solitons.

Motivation & Objective

• Develop a kinetic theory for self-gravitating classical and quantum particles (fermions and bosons).
• Derive and analyze the quantum Landau equation in a weak-deflection, local approximation.
• Establish conservation laws, H-theorem, equilibria, and relaxation properties for classical and quantum cases.
• Explore implications for dark matter, including Bose-Einstein condensation, fermionic degeneracy, and soliton/core-halo structures.
• Connect the theory to existing frameworks (King models, thermal bath approximation) and discuss relevance to fuzzy dark matter.

Proposed method

• Start from the Boltzmann equation and take the weak-deflection (Landau) limit for gravitational interactions.
• Generalize to a multispecies quantum Landau equation including f and 1±f factors for fermions/bosons.
• Derive diffusion and friction coefficients, quantum Rosenbluth potentials, and the quantum Fokker-Planck form.
• Discuss conservation laws, H-theorem, and approach to equilibrium (Boltzmann, Fermi-Dirac, Bose-Einstein distributions).
• Analyze special cases: thermal bath approximation, bosonic condensation below Tc, and fermionic blocking effects.

Experimental results

Research questions

• RQ1 What is the appropriate multispecies quantum Landau framework for self-gravitating systems?
• RQ2 How do quantum statistics (f(1±f)) modify relaxation, diffusion, and condensation in gravitational encounters?
• RQ3 What are the equilibrium states and relaxation times for classical, fermionic, and bosonic self-gravitating systems?
• RQ4 How does the theory describe Bose-Einstein condensation and the formation of solitons/core-halo structures in dark matter?

Key findings

• The Landau equation conserves mass and energy and increases the Boltzmann entropy, relaxing toward Boltzmann, Fermi-Dirac, or Bose-Einstein distributions depending on statistics.
• A quantum Landau framework yields quantum diffusion/friction coefficients and Rosenbluth potentials for self-gravitating particles.
• Bosons above the condensation temperature behave essentially as collisionless under gravity, while below Tc condensation and Bose stimulation drive condensation dynamics.
• Fermionic degeneracy (Pauli blocking) tends to lengthen relaxation times, making fermionic dark matter effectively collisionless on astrophysical timescales.
• The work connects to dark matter phenomenology, including fuzzy dark matter halos and Bose stars, and discusses core-halo structures and potential resolutions of cusp-core problems.
• It provides a basis for kinetic descriptions of self-gravitating quantum systems relevant to dark matter scenarios.