Negative specific heat in self-gravitatingN-body systems enclosed in a spherical container with reflecting walls

Abstract

Gravity-dominated systems have a negative specific heat. We investigate the negative specific heat of self-gravitating systems enclosed in a spherical container with reflecting walls by means of N-body simulations. To simulate nonequilibrium processes, a particle reflected at a nonadiabatic wall is cooled to mimic energy loss by reflecting walls, while an adiabatic wall is employed for microcanonical ensembles. We show that a negative specific heat occurs not only in the microcanonical ensemble but also in certain nonequilibrium processes with the nonadiabatic wall. With increasing cooling rates, the dependence of temperature T on energy , i.e., the − T curve, gradually deviates from the microcanonical ensemble and approaches a certain common curve at a low-energy region. The common curve agrees with an − T curve for stellar polytropes, especially for the polytrope index of n ϳ 5. We show that the stellar polytrope should be related to the present nonequilibrium process appearing in the self-gravitating system with the nonadiabatic wall. In the nonequilibrium process, a rapid change in velocity at the nonadiabatic wall significantly affects the velocity and density profiles. In particular, the greater the cooling rate, the greater the local velocity gradient at a low-energy region.

Transition of velocity distributions in collapsing self-gravitatingN-body systems

“…In this study, the units of time t and velocity v are R 3 /(Gm) and √ Gm/R, respectively [29]. The units are set to be G = R = m = 1, to ensure generality of the system.…”

“…In our simulations, r 0 for the Plummer softened potential is set to be 0.005R [29,30], which is sufficiently smaller than the critical value 0.021R, above which the collapse-explosion transition is replaced by a normal first-order phase transition [34,47]. The collapse energy for the system with an adiabatic wall is ε coll ≈ −0.339 [34], which means that if ε of a uniform state becomes lower than ε coll , the system should undergo a collapse to a core-halo state.…”

“…For simulating an N -body system, the set of equations of motion is integrated using Verlet's algorithm, as for our previous studies [29,30]. To maintain the accuracy of our simulations, a time step of t = 10 −5 or t = 0.5 × 10 −5 is selected, based on a simulation with several different time steps [29,30,58]. (Since Verlet's algorithm is a third-order symplectic integrator, higher-order symplectic integrators are 021132-2 required to select a larger time step.)…”

“…As shown in Fig. 1(a), we first consider a "cold collapse process" under a restriction of constant mass and energy using adiabatic walls, because a system enclosed in a spherical container is suitable for examining the fundamental characteristics of the thermodynamic properties [15,29,34]. (To simulate the cold collapse process, we employ a typical small N -body system which has been analyzed in detail [34,48], since the well-studied system is an important benchmark system for examining a collapse theoretically and numerically.)…”

“…Self-gravitating or long-range attractive interacting systems exhibit several peculiar features [1][2][3][4], such as gravothermal catastrophe [5,6], negative specific heat [7][8][9], violent relaxation [10], nonequilibrium thermodynamics, and nonextensive statistical mechanics [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In these systems, the velocity distributions are non-Gaussian [31][32][33], especially in quasiequilibrium states and metastable states.…”