On the coarse-grained evolution of collisionless stellar systems

Abstract

Abstract.We describe the dynamical evolution of collisionless stellar systems on a coarse-grained scale. We first discuss the statistical theory of violent relaxation, following the seminal paper of Lynden-Bell (1967). Consistently with this statistical approach, we present kinetic equations for the coarse-grained distribution function f (r, u, t) based on a Maximum Entropy Production Principle or on a quasi-linear theory of the Vlasov-Poisson system. Then, we develop a deterministic approach where the coarse-grained distribution function is defined as a convolution of the fine-grained distribution function f (r, u, t) by a Gaussian window. We derive the dynamical equation satisfied by f (r, u, t) and show that its stationary states are different from those predicted by the statistical theory of violent relaxation. This implies that the notion of coarse-graining must be defined with care. We apply these results to the HMF (Hamiltonian Mean Field) model and find that the spatial density is similar to a Tsallis q-distribution where the q parameter is related to the resolution length.

Vlasov–Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory

“…One common way to try predicting the structure of the system after relaxation consists in using a statistical approach combined with entropy maximization, such as in the theory of Lynden-Bell (1967) and its numerous extensions (see, e.g., the recent investigations by Pontzen & Governato 2013;Carron & Szapudi 2013, for the case of dark matter halos, but this list is far from exhaustive). However, this approach requires some level of coarse graining of the distribution function, an operation which is not unique nor free of biases (see, e.g., Chavanis & Bouchet 2005;Arad & Lynden-Bell 2005). Furthermore, the concept of entropy is not necessarily well defined in the continuous limit (see, e.g., Tremaine, Henon, & Lynden-Bell 1986;Chavanis 2006, and references therein).…”

Consequences of entropy bifurcation in non-Maxwellian astrophysical environments

“…Remarkably, we have to add to this diversity scale invariant power-law distributions relying on self organized criticality (SOC) (Bak et al, 1988;Bak, 1996;Chapman et al, 1998;Watkins et al, 2001;Chapman and Watkins, 2001) as well as gravitationally bound astrophysical stellar systems (Nakamichi et al, 2002;Chavanis and Bouchet, 2005). Furthermore, Leubner (2005) developed in this conjunction recently a nonextensive theory representing accurately the hot plasma and dark matter (DM) density profiles in galaxies and clusters in the context of scale invariant power-law distributions.…”

Metastability and Anomalous Behavior in the HMF Model: Connections to Nonextensive Thermodynamics and Glassy Dynamics

“…In this work we will discuss such features in the context of the so called Hamiltonian Mean Field (HMF) model, a system of inertial spins with long-range interaction. The model, originally introduced by Antoni and Ruffo [2], has been thoroughly investigated and generalized in the last years for its anomalous dynamical behavior [3,4,5,6,7,8,9,10,11,12,13]. With respect to systems with short-range interactions, the dynamics and the thermodynamics of many-body systems of particles interacting with longrange forces, as the HMF, are particularly rich and interesting.…”

“…Thus, it represents a very useful "laboratory" for exploring metastability and glassy dynamics in systems with long-range interactions. The model can be considered as a minimal and pedagogical model for a large class of complex systems, among which one can surely include self-gravitating systems [13] and glassy systems [10], but also systems apparently belonging to different fields as biology or sociology. In fact, we recently found similar features also in the context of the Kuramoto Model [19], one of the simplest models for synchronization in biological systems [20].…”