Statistical mechanics of self-gravitating systems: Mixing as a criterion for indistinguishability

Abstract

We propose an association between the phase-space mixing level of a self-gravitating system and the indistinguishability of its constituents (stars or dark matter particles). This represents a refinement in the study of systems exhibiting incomplete violent relaxation. Within a combinatorial analysis similar to that of Lynden-Bell, we make use of this association to obtain a distribution function that deviates from the Maxwell-Boltzmann distribution, increasing its slope for high energies. Considering the smallness of the occupation numbers for large distances from the center of the system, we apply a correction to Stirling's approximation which increases the distribution slope also for low energies. The distribution function thus obtained presents some resemblance to the "S" shape of distributions associated with cuspy density profiles (as compared to the distribution function obtained from the Einasto profile), although it is not quite able to produce sharp cusps. We also argue how the association between mixing level and indistinguishability can provide a physical meaning to the assumption of particle-permutation symmetry in the N-particle distribution function, when it is used to derive the one-particle Vlasov equation, which raises doubts about the validity of this equation during violent relaxation.

The phase-space structure of tidally stripped haloes

“…There are also several proposed models for the 'true' DF of dark matter halos, based on maximizing entropy under various constraints, or other arguments (e.g. Hjorth & Williams 2010;Pontzen & Governato 2013;Beraldo e Silva et al 2014;see Halle et al 2017 for a discussion). Our focus here is not on which of these theoretical models is correct, but on the generic effects of tidal mass loss on any DF.…”

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

“…Moreover, Jaynes (1971) argues that the fact the "H" function is not monotonic can be associated to the non-validity of the hypothesis of molecular chaos, which is of fundamental importance in the derivation of the Vlasov-Poisson equation through the BBGKY hierarchy, a possibility also suggested by Beraldo e Silva et al (2014).…”

“…(3) refers to the coordinates of one single typical particle while it is also intended to describe the evolution of the system as whole, and thus is clearly based on mechanical plus statistical considerations, what is made explicit in the construction of the BBGKY hierarchy: as pointed out e.g. by Beraldo e Silva et al (2014), to reduce the full hierarchy of N -body evolution equations to an (effective) onebody problem one needs to assume the molecular-chaos hypothesis, i.e. that f (N ) ( r 1 , v 1 , ..., r N , v N ) can be written as a N -fold product of one-particle distribution functions.…”

Propriedades dinâmicas da matéria escura

“…No modelo discutido neste capítulo, baseado no trabalho publicado recentemente (Beraldo e Silva, Lima, Sodré & Perez, 2014), propomos uma associação entre o nível de mistura e o conceito de indistinguibilidade, e apresentamos as possíveis conseqüências para o estado estacionário gerado pelo processo de relaxação violenta. Por "mistura", não nos referimosà "mistura de fases" (phase mixing), queé um processo associado á orbitas determinísticas em um potencial integrável.…”

“…Longe de ser apenas uma estratégia de cálculo, estas hipóteses guardam um significado estatístico profundo, e sem elas nãoé possível obter a equação de Vlasov. A simetria de f (N )é comumente tratada como uma conseqüência direta da hipótese de partículas idênticas (ver Binney & Tremaine, 2008;Saslaw, 1987 Silva, Lima, Sodré & Perez (2014). Também pretendemos testar diferentes perfis de densidades, em particular aquele associado a este modelo e o DARKexp, utilizando um novo conjunto de dados produzido recentemente por Umetsu, Medezinski, Nonino et al (2014).…”