April 15, 2002
We investigate the thermodynamic properties of stellar self-gravitating system arising from the Tsallis generalized entropy. In particular, physical interpretation of the thermodynamic instability, as has been revealed by previous paper(Taruya & Sakagami, cond-mat/0107494, Physica A 307, 185 (2002)), is discussed in detail based on the non-extensive thermostatistics. Examining the Clausius relation in a quasi-static experiment, we obtain the standard result of thermodynamic relation that the physical temperature of the equilibrium non-extensive system is identified with the inverse of the Lagrange multiplier, $T_{phys}=1/\beta$. Using this relation, the specific heat of total system is computed, and confirm the common feature of self-gravitating system that the presence of negative specific heat leads to the thermodynamic instability. In addition to the gravothermal instability discovered previously, the specific heat shows the curious divergent behavior at the polytrope index $n>3$, suggesting another type of thermodynamic instability. Evaluating the second variation of free energy, we find that the marginal stability condition indicated from the specific heat can be exactly recovered from the second variation of free energy. Thus, the stellar polytropic system is consistently characterized by the non-extensive thermostatistics as a plausible thermal equilibrium state. We also clarify the non-trivial scaling behavior appeared in specific heat and address the origin of non-extensive nature in stellar polytrope.
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November 15, 2002
We revisit the issues on the thermodynamic property of stellar self-gravitating system arising from Tsallis' non-extensive entropy. Previous papers (Taruya & Sakagami, Physica A 307 (2002) 185 (cond-mat/0107494); ibid. (2002) in press (cond-mat/0204315)) have revealed that the extremum-state of Tsallis entropy characterized by the so-called stellar polytrope has consistent thermodynamic structure, which predicts the thermodynamic instability due to the negative specific heat....
October 4, 2003
After introducing the fundamental properties of self-gravitating systems, we present an application of Tsallis' generalized entropy to the analysis of their thermodynamic nature. By extremizing the Tsallis entropy, we obtain an equation of state known as the stellar polytrope. For a self-gravitating stellar system confined within a perfectly reflecting wall, we discuss the thermodynamic instability caused by its negative specific heat. The role of the extremum as a quasi-equi...
July 24, 2001
We present a first physical application of Tsallis' generalized entropy to the thermodynamics of self-gravitating systems. The stellar system confined in a spherical cavity of radius $r_e$ exhibits an instability, so-called gravothermal catastrophe, which has been originally investigated by Antonov (1962) and Lynden-Bell & Wood (1968) on the basis of the maximum entropy principle for the phase-space distribution function. In contrast to previous analyses using the Boltzmann-G...
March 18, 2003
With particular attention to the recently postulated introduction of a non-extensive generalization of Boltzmann-Gibbs statistics, we study the long-term stellar dynamical evolution of self-gravitating systems on timescales much longer than the two-body relaxation time. In a self-gravitating N-body system confined in an adiabatic wall, we show that the quasi-equilibrium sequence arising from the Tsallis entropy, so-called stellar polytropes, plays an important role in charact...
November 11, 2014
The pure self-gravitating system in this paper refers to a multi-body gaseous system where the self-gravity plays a dominant role and the intermolecular interactions can be neglected. Therefore its total mass must be much more than a limit mass, the minimum mass of the system exhibiting long-range nature. Thee method to estimate the limit mass is then proposed. The nonequlibrium stationary state in the system is identical to the Tsallis equilibrium state, at which the Tsallis...
September 22, 2004
We discuss the meaning of Tsallis functional in astrophysics. The energy functional of a polytropic star is similar to Tsallis free energy and the H-function associated with a stellar polytrope is similar to Tsallis entropy. More generally, the energy functional of a barotropic star is similar to a generalized free energy and the H-function associated with a spherical stellar system is similar to a generalized entropy. Their optimization under appropriate constraints determin...
April 23, 2008
We present a stability analysis of the classical ideal gas in a new theory of nonextensive statistics and use the theory to understand the phenomena of negative specific heat in some self-gravitating systems. The stability analysis is made on the basis of the second variation of Tsallis entropy. It is shown that the system is thermodynamically unstable if the nonextensive parameter is q>5/3, which is exactly equivalent to the condition of appearance of the negative specific h...
September 22, 2017
In this letter, we study the limit behavior of the evolution of Tsallis entropy in self-gravitating systems. The study is carried out under two different situations, drawing the same conclusion. No matter in the energy transfer process or in the mass transfer process inside the system, when nonextensive parameter q is more than unity, the total entropy is bounded; on the contrary, when this parameter is less than unity, the total entropy is unbounded. There are proofs in both...
February 11, 2004
We use physical constrains imposed from the H-Theorem and from the negative nature of the heat capacity of self-gravitating thermodynamically isolated systems to investigate some possible limits on the stellar polytrope index $n$ within the domain of a classical non-extensive kinetic theory.
March 6, 2018
The statistical mechanics of a cloud of particles interacting via their gravitational potentials is an old problem which encounters some issues when the traditional Boltzmann-Gibbs statistics is applied. In this article, we consider the generalized statistics of Tsallis and analyze the statistical and thermodynamical implications for a self-gravitating gas, obtaining analytical and convergent expressions for the equation of state and specific heat in the canonical as well as ...