The Basic Problems And Stochastic Dynamical Foundation Of Nonextensive Statistical Mechanics

In order to solve the limitations and difficulties in traditional statistical mechanics, Tsallis proposed the nonextensive entropy and founded the nonextensive statistical mechanics in 1988. Although this new theory has lots of applications in physics, it is still under development and improvement. In this thesis, we mainly discuss some basic problems and the stochastic dynamical foundation of nonextensive statistical mechanics, including the following content:First, the energy fluctuation in nonextensive statistical mechanics is studied. Under the framework of nonextensive statistical mechanics, we derive the energy distribution and the energy fluctuation in canonical ensemble. Then, we apply the results to classical ideal gas, and analyze its energy fluctuation and the ensemble equivalence by employing thermodynamic limit. In the end, combining the studies of Plastino et al, we compare our results with Liu’s results, and discuss a possible meaning of the nonextensive parameter q.Second, the static linear response theory under isothermal and adiabatic condition is investigated in nonextensive statistical mechanics. We analyze the difference between the three energy constraints at first. Then, using the third generation of the energy constraint, we revise the isothermal static linear response function and then derive the static linear response function under the adiabatic condition. We present the relationship between the isothermal and adiabatic linear response functions. At last, by analyzing an example, we illustrate the influence by the different definitions of temperature.Third, we study the stochastic dynamical system governed by the two-variable Langevin equation, namely the Brownian motion in an inhomogeneous medium. We find that under the different rules of stochastic integrals, the system always have the power-law stationary distributions if it follows the generalized fluctuation-dissipation relation. These stationary distributions can be presented in a unified form of power-law function. In the end, we use numerical method to examine the correction of our derivation.Fourth, we examine whether the principle of detailed balance holds for the power-law distributions generated from the Langevin equations in the generalized fluctuation- dissipation relation. Then, under both conditions of detailed balance and generalized fluctuation-dissipation relation, we derive analytically the power-law distribution stationary solutions of Fokker-Planck equations. At last, we give the conditions if the principle of detailed balance holds for the power-law distributions, and discuss the relation between detailed balance and equilibrium.Fifth, we study the time behavior of the Fokker-Planck equation in Zwanzig’s rule. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation-dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker-Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. At last, we apply our results to Ornstein-Uhlenbeck process, and compare with the results in the traditional theory.