The nonextensive parameter as a stability criterion of convection in a fluid

Abstract

We apply the nonextensive statistical theory given by Landau and Lifshitz to study the stability condition of convection in a fluid. When introducing Du's equation (Du J. L., Europhys. Lett., 67 (2004) 893), a relationship between the temperature gradient and the potential field of a nonequilibrium complex system by the nonextensive parameter, into the stability condition, we find that the nonextensive parameter can be used as the stability criterion of the convection taking place in a fluid. With the data of the standard solar models for BP2000 and BS2005, we apply this criterion to the solar interior and show a stability range from the center of the Sun to 0.74 R ,which is in excellent agreement with the helioseismological measurements inside the Sun. With the same data, the lower limit of the adiabatic index, when stability is needed, has been calculated.

The stationary state and gravitational temperature in a pure self-gravitating system

“…According to the relationship (24), Q>0 actually reveals the characteristic of the distribution of the Lagrange temperature in the self-gravitating system. In fact, we need to give more restricts to the nonextensive parameter q.…”

“…For example, the dark matter density distribution in the galactic halo is one example of the power-law forms [3]. The calculation using the data of standard solar model further restricts the range of Q to be 0<Q<1 [24].…”

“…(21) and (25) don't hold at the same time. In the evolution, the system can be at hydrostatic equilibrium and satisfies the equation (24). Apart from the Tsallis non-equilbirum states, (25) also does not hold in some other situations, such as in the first kind and third kind of self-gravitating systems [15].…”

“…This is the gravitational thermal capacity of the pure self-gravitational system (i.e., the second kind of self-gravitational system [15] What here we should mention is that the evolution exists in the self-gravitating system whose total energy is definite, which leads to the change of Q according to the relationship (24). On the basis of the classical statistics, the isolated self-gravitating system will evolve to the so-called gravothermal catastrophe.…”

The nonextensive parameter for the rotating astrophysical systems with power-law distributions

“…Eq. (2) has been applied to test the nonextensivity in the helioseismological measurements of the Sun [44], to constitute the astrophysical convection instability criterion [45,46], to analyze the thermodynamic stability of self-gravitating system [47], and to introduce the concept of the gravitational heat conduction [48] and the gravitational temperature [49]. Eq.…”