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Register a new userThermodynamics of gravitational clustering phenomena: $N$-body self-gravitating gas on the sphere $mathbb{S}^{3}subsetmathbb{R}^{4}$
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Added by
Luisberis Velazquez-Abad
Publication date
2016
fields
Physics
and research's language is
English
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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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The thermodynamic behaviour of self-gravitating $N$-body systems has been worked out by borrowing a standard method from Molecular Dynamics: the time averages of suitable quantities are numerically computed along the dynamical trajectories to yield thermodynamic observables. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. The dynamics of self-gravitating $N$-body systems has been computed using two different kinds of regularization of the newtonian interaction: the usual softening and a truncation of the Fourier expansion series of the two-body potential. $N$ particles of equal masses are constrained in a finite three dimensional volume. Through the computation of basic thermodynamic observables and of the equation of state in the $P - V$ plane, new evidence is given of the existence of a second order phase transition from a homogeneous phase to a clustered phase. This corresponds to a crossover from a polytrope of index $n=3$, i.e. $p=K V^{-4/3}$, to a perfect gas law $p=K V^{-1}$, as is shown by the isoenergetic curves on the $P - V$ plane. The dynamical-microcanonical averages are compared to their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. A major disagreement is found, because the canonical ensemble seems to have completely lost any information about the phase transition. The microcanonical ensemble appears as the only reliable statistical framework to tackle self-gravitating systems. Finally, our results -- obtained in a ``microscopic framework -- are compared with some existing theoretical predictions -- obtained in a ``macroscopic (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.
For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension $d leq 3$ the configuration must be central, in dimension $d geq 4$ new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of $S$-balanced configurations, containing central ones, which yield simple solutions of the $n$-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear $S$-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.
We study the statistical mechanics of binary systems under gravitational interaction of the Modified Newtonian Dynamics (MOND) in three-dimensional space. Considering the binary systems, in the microcanonical and canonical ensembles, we show that in the microcanonical systems, unlike the Newtonian gravity, there is a sharp phase transition, with a high-temperature homogeneous phase and a low temperature clumped binary one. Defining an order parameter in the canonical systems, we find a smoother phase transition and identify the corresponding critical temperature in terms of physical parameters of the binary system.
We report on the results of a study of the motion of a four particle non-relativistic one-dimensional self-gravitating system. We show that the system can be visualized in terms of a single particle moving within a potential whose equipotential surfaces are shaped like a box of pyramid-shaped sides. As such this is the largest $N$-body system that can be visualized in this way. We describe how to classify possible states of motion in terms of Braid Group operators, generalizing this to $N$ bodies. We find that the structure of the phasetextcolor{black}{{} space of each of these systems yields a large variety of interesting dynamics, containing regions of quasiperiodicity and chaos. Lyapunov exponents are calculated for many trajectories to measure stochasticity and previously unseen phenomena in the Lyapunov graphs are observed.
We study confining strings in ${cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $mathbb{R}^{1,2} times mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which are lines in $mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c ge 5$, the two domain walls confining unit $N$-ality quarks attract and form non-BPS bound states, collapsing to a single flux line. We determine the $N$-ality dependence of the string tensions for $2 le N_c le 10$. Compared to known scaling laws, we find a weaker, almost flat $N$-ality dependence, which is qualitatively explained by the properties of BPS domain walls. We also quantitatively study the behavior of confining strings upon increasing the $mathbb{S}^1$ size by including the effect of virtual $W$-bosons and show that the qualitative features of double-string confinement persist.
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