A Maximum Entropy Principle for inferring the Distribution of 3D Plasmoids [HEAP]


The Principle of Maximum Entropy, a powerful and general method for inferring the distribution function given a set of constraints, is applied to deduce the overall distribution of plasmoids (flux ropes/tubes). The analysis is undertaken for the general 3D case, with mass, total flux and (3D) velocity serving as the variables of interest, on account of their physical and observational relevance. The distribution functions for the mass, width, total flux and helicity exhibit a power-law behavior with exponents of $-4/3$, $-2$, $-3$ and $-2$ respectively for small values, whilst all of them display an exponential falloff for large values. In contrast, the velocity distribution, as a function of $v = |{\bf v}|$, is shown to be flat for $v \rightarrow 0$, and becomes a power law with an exponent of $-7/3$ for $v \rightarrow \infty$. Most of these results exhibit a high degree of universality, as they are nearly independent of the free parameters. A preliminary comparison of our results with the observational evidence is presented, and some of the ensuing space and astrophysical implications are discussed.

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M. Lingam, L. Comisso and A. Bhattacharjee
Tue, 21 Feb 17

Comments: 15 pages, 6 figures

Statistical mechanics of gravitating gas like galaxy [GA]


The most probable state of an infinite self-gravitating gas in the dynamical equilibrium is defined by “gravitational haziness”, a parameter representing many-body effects and like the temperature in the case of thermal equilibrium. A kinetic equation for the distribution function of gas particles in the phase space is derived from a concept of statistical equipartition of the virial among subsystems. Its solution, an analog of the Maxwell-Boltzmann weight, is found in the limit of thick “gravitational haziness” (the high-temperature expansion) where the gravitational potential follows the Lane-Emden equation. A more general equation for arbitrary “gravitational haziness” is conjectured as a special property of the kinetic equation. The first law of a “hazydynamics” (thermodynamics) states that the total mass of an astronomical stellar collection is the sum of the Archimedes displaced mass and an excess “gobbled” mass determined by the “gravitational haziness” and history.

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A. Kashuba
Mon, 20 Feb 17

Comments: 5 pages

Kinetic theory of fermions in curved spacetime [CL]


We build a statistical description of fermions, taking into account the spin degree of freedom in addition to the momentum of particles, and we detail its use in the context of the kinetic theory of gases of fermions particles. We show that the one-particle distribution function needed to write a Liouville equation is a spinor valued operator. The degrees of freedom of this function are covariantly described by an intensity function and by a polarisation vector which are parallel transported by free streaming. Collisions are described on the microscopic level and lead to a Boltzmann equation for this operator. We apply our formalism to the case of weak interactions, which at low energies can be considered as a contact interaction between fermions, allowing us to discuss the structure of the collision term for a few typical weak-interaction mediated reactions. In particular we find for massive particles that a dipolar distribution of velocities in the interacting species is necessary to generate linear polarisation, as opposed to the case of photons for which linear polarisation is generated from the quadrupolar distribution of velocities.

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C. Fidler and C. Pitrou
Wed, 1 Feb 17

Comments: 47 pages, 1 figure

Isotropic-Nematic Phase Transitions in Gravitational Systems [GA]


We examine dense self-gravitating stellar systems dominated by a central potential, such as nuclear star clusters hosting a central supermassive black hole. Different dynamical properties of these systems evolve on vastly different timescales. In particular, the orbital-plane orientations are typically driven into internal thermodynamic equilibrium by vector resonant relaxation before the orbital eccentricities or semimajor axes relax. We show that the statistical mechanics of such systems exhibit a striking resemblance to liquid crystals, with analogous ordered-nematic and disordered-isotropic phases. The ordered phase consists of bodies orbiting in a disk in both directions, with the disk thickness depending on temperature, while the disordered phase corresponds to a nearly isotropic distribution of the orbit normals. We show that below a critical value of the total angular momentum, the system undergoes a first-order phase transition between the ordered and disordered phases. At the critical point the phase transition becomes second-order while for higher angular momenta there is a smooth crossover. We also find metastable equilibria containing two identical disks with mutual inclinations between $90^{\circ}$ and $180^\circ$.

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Z. Roupas, B. Kocsis and S. Tremaine
Fri, 13 Jan 17

Comments: 33 pages, 23 figures

Formation and relaxation of quasi-stationary states in particle systems with power law interactions [CL]


We explore the formation and relaxation of so-called quasi-stationary states (QSS) for particle distributions in three dimensions interacting via an attractive radial pair potential $V(r \rightarrow \infty) \sim 1/r^\gamma$ with $\gamma > 0$, and either a soft-core or hard-core regularization at small $r$. In the first part of the paper we generalize, for any spatial dimension $d \geq 2$, Chandrasekhar’s approach for the case of gravity to obtain analytic estimates of the rate of collisional relaxation due to two body collisions. The resultant relaxation rates indicate an essential qualitative difference depending on the integrability of the pair force at large distances: for $\gamma >d-1$ the rate diverges in the large particle number $N$ (mean field) limit, unless a sufficiently large soft core is present; for $\gamma < d-1$, on the other hand, the rate vanishes in the same limit even in the absence of any regularization. In the second part of the paper we compare our analytical predictions with the results of extensive parallel numerical simulations in $d=3$, for a range of different exponents $\gamma$ and soft cores leading to the formation of QSS. We find, just as for the previously well studied case of gravity (which we also revisit), excellent agreement between the parametric dependence of the observed relaxation times and our analytic predictions. Further, as in the case of gravity, we find that the results indicate that, when large impact factors dominate, the appropriate cut-off is the size of the system (rather than, for example, the mean inter-particle distance). Our results provide strong evidence that the existence of QSS is robust only for long-range interactions with a large distance behavior $\gamma < d-1$; for $\gamma \geq d-1$ the existence of such states will be conditioned strongly on the short range properties of the interaction.

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B. Marcos, A. Gabrielli and M. Joyce
Tue, 10 Jan 17

Comments: 21 pages, 11 figures, submitted to PRE

Dynamical system modeling fermionic limit [CL]


The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

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D. Bors and R. Stanczy
Mon, 19 Dec 16

Comments: 12 pages, 2 figures

Thermodynamics of gravitational clustering phenomena: $N$-body self-gravitating gas on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$ [CL]


This work is devoted to the thermodynamics of gravitational clustering, a collective phenomenon with a great relevance in the $N$-body cosmological problem. We study a classical self-gravitating gas of identical non-relativistic particles defined on the sphere $\mathbb{S}^{3}\subset \mathbb{R}^{4}$ by considering gravitational interaction that corresponds to this geometric space. The analysis is performed within microcanonical description of an isolated Hamiltonian system by combining continuum approximation and steepest descend method. According to numerical solution of resulting equations, the gravitational clustering can be associated with two microcanonical phase transitions. A first phase transition with a continuous character is associated with breakdown of $SO(4)$ symmetry of this model. The second one is the gravitational collapse, whose continuous or discontinuous character crucially depends on the regularization of short-range divergence of gravitation potential. We also derive the thermodynamic limit of this model system, the astrophysical counterpart of Gibbs-Duhem relation, the order parameters that characterize its phase transitions and the equation of state. Other interesting behavior is the existence of states with negative heat capacities, which appear when the effects of gravitation turn dominant for energies sufficiently low. Finally, we comment the relevance of some of these results in the study of astrophysical and cosmological situations. Special interest deserves the gravitational modification of the equation of state due to the local inhomogeneities of matter distribution. Although this feature is systematically neglected in studies about Universe expansion, the same one is able to mimic an effect that is attributed to the dark energy: a negative pressure.

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F. Tello-Ortiz and L. Velazquez
Mon, 10 Oct 16

Comments: To appear in J. Stat. Mech.: Theo. Exp