The Nonextensive Parameters Of Two Types Of The Nonequilibrium System

The nonextensive q-parameter is crucially important in Tsallis statistics. With this parameter, the Tsallis entropy is essentially different from the older Boltzmann-Gibbs entropy. The deviation of q from unity represents the degree of nonextensivity of the system under consideration, so it is undoubtedly very important to correctly understand the physical meaning of this parameter. Only in this way can we ascertain what physical systems Tsallis statistics applies to. The long-range interacting systems have offered a well framework for in-depth studies of the Tsallis statistics. For plasma systems and self-gravitating systems, the main interactions are respectively electromagnetic force and universal gravitation, both of which are long-distance force, i.e., these two types of systems are typically dominated by the long-range interactions. We take the generalized Boltzmann equation as starting point to obtain expressions of the nonextensive parameter q for these two types of systems and study the corresponding physical meaning.We exactly obtain an equation of the q-parameter for nonequilibrium plasmas in magnetic field, based on the generalized Boltzmann equation, the q-H theorem and the power-law velocity q-distribution. It is shown that the q-parameter different from unity is closely related to the temperature gradient, the electric field strength, the magnetic induction as well as the overall bulk velocity of the plasma. The effect of the magnetic field on the q-parameter depends on the overall bulk velocity of the plasma. If magnetic induction is parallel to the overall bulk velocity of the system or the overall bulk velocity is zero, the magnetic field will not influence q. For the magnetic-field-free case, the equation of q exactly reduces to the counterpart without magnetic field after the magnetic field term is removed. For the rotating self-gravitating systems, we follow a similar approach to derive an expression of the q-parameter. We show that the q-parameter different from unity is not only related to the temperature gradient and the gravitational acceleration, but also depends on the inertial centrifugal acceleration. The expression of q can reduce to the counterpart of non-rotation if we take no account of the rotation effect. In addition, we define a relative nonextensive parameter of the rotating self-gravitating system, and take the Sun, Earth, Jupiter and Saturn as examples to illustrate the nonextensive effect introduced by the rotations at the equator surfaces.We respectively adopt two methods, the linearized collision-less Boltzmann equation and the fluid equation, to study the effect of Coriolis force of rotating self-gravitating systems on the Jeans’ criterion for gravitational instability in the framework of nonextensive statistics and reach the same conclusions. It is shown that the formula of the generalized Jeans’ length remains unaffected apart from the direction perpendicular to rotation axis, where the gravitational instability is divided into two cases according to relations between rotate speed and density. The case of taking account of inertial centrifugal force is relatively complicated, and needs further research.We show the equivalence of three versions of the quantum theory with position-dependent mass. We correct the first and third versions of this theory and find that although the stationary state energy eigenfunctions are different, the energy eigenvalues are identical for all three versions. In addition, we show that both the probability density and the probability flux in the second version are consistent with those in the third version. For this account, the two versions of the quantum theory are proved to be exactly equivalent. As an application example of the theory, we reanalyze the transmission and tunneling probability for a particle subjected to a potential barrier, and reach the same conclusion as before.