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Register a new userConstraining nonextensive statistics with plasma oscillation data
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R. Silva
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We discuss experimental constraints on the free parameter of the nonextensive kinetic theory from measurements of the thermal dispersion relation in a collisionless plasma. For electrostatic plane-wave propagation, we show through a statistical analysis that a good agreement between theory and experiment is possible if the allowed values of the $q$-parameter are restricted by $q=0.77 pm 0.03$ at 95% confidence level (or equivalently, $2-q = 1.23$, in the largely adopted convention for the entropy index $q$). Such a result rules out (by a large statistical margin) the standard Bohm-Gross dispersion relation which is derived assuming that the stationary Maxwellian distribution ($q=1$) is the unperturbed solution.
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We study the thermodynamic properties of solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation. First we study the nonextensive grand canonical distribution function and the nonextensive quantum statistics with a parameter transformation. Then we derive the generalized Boson distribution and Fermi distribution in the nonextensive quantum statistics. Further we study the thermodynamic properties of solid and metal electrons in the nonextensive quantum system, including the generalized Debye models, the generalized internal energies, the generalized capacities and chemical potential. We derive new expressions of these thermodynamic quantities, and we show that they all depend significantly on the nonextensive parameter and in the limit they recover to the forms in the classical quantum statistics. These new expressions may be applied to study the new characteristics in some nonextensive quantum systems where the long-range interactions and/or long-range correlations play a role.
The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand thermodynamic potential in nonextensive thermodynamics. In order to establish appropriate nonextensive thermodynamic formalism, the dual thermodynamic interpretations are necessary for thermodynamic relations and thermodynamic quantities. By using a new technique of parameter transformation, the single-particle distribution can be deduced from the power-law Gibbs distribution. This technique produces a link between the statistical ensemble and the quasi-independent system with two kinds of nonextensive parameter having quite different physical explanations. Furthermore, the technique is used to construct nonextensive quantum statistics and effectively to avoid the factorization difficulty in the power-law grand canonical distribution.
In this paper, we study the probability distribution of the observable $s = (1/N)sum_{i=N-N+1}^N x_i$, with $1 leq N leq N$ and $x_1<x_2<cdots< x_N$ representing the ordered positions of $N$ particles in a $1d$ one-component plasma, i.e., $N$ harmonically confined charges on a line, with pairwise repulsive $1d$ Coulomb interaction $|x_i-x_j|$. This observable represents an example of a truncated linear statistics -- here the center of mass of the $N = kappa , N$ (with $0 < kappa leq 1$) rightmost particles. It interpolates between the position of the rightmost particle (in the limit $kappa to 0$) and the full center of mass (in the limit $kappa to 1$). We show that, for large $N$, $s$ fluctuates around its mean $langle s rangle$ and the typical fluctuations are Gaussian, of width $O(N^{-3/2})$. The atypical large fluctuations of $s$, for fixed $kappa$, are instead described by a large deviation form ${cal P}_{N, kappa}(s)simeq exp{left[-N^3 phi_kappa(s)right]}$, where the rate function $phi_kappa(s)$ is computed analytically. We show that $phi_{kappa}(s)$ takes different functional forms in five distinct regions in the $(kappa,s)$ plane separated by phase boundaries, thus leading to a rich phase diagram in the $(kappa,s)$ plane. Across all the phase boundaries the rate function $phi(kappa,s)$ undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.
We investigate the dispersion relation and Landau damping of ion acoustic waves in the collisionless magnetic-field-free plasma if it is described by the nonextensive q-distributions of Tsallis statistics. We show that the increased numbers of superthermal particles and low velocity particles can explain the strengthened and weakened modes of Landau damping, respectively, with the q-distribution. When the ion temperature is equal to the electron temperature, the weakly damped waves are found to be the distributions with small values of q.
Recent work has suggested that in highly correlated systems, such as sandpiles, turbulent fluids, ignited trees in forest fires and magnetization in a ferromagnet close to a critical point, the probability distribution of a global quantity (i.e. total energy dissipation, magnetization and so forth) that has been normalized to the first two moments follows a specific non Gaussian curve. This curve follows a form suggested by extremum statistics, which is specified by a single parameter a (a=1 corresponds to the Fisher-Tippett Type I (Gumbel) distribution.) Here, we present a framework for testing for extremal statistics in a global observable. In any given system, we wish to obtain a in order to distinguish between the different Fisher-Tippett asymptotes, and to compare with the above work. The normalizations of the extremal curves are obtained as a function of a. We find that for realistic ranges of data, the various extremal distributions when normalized to the first two moments are difficult to distinguish. In addition, the convergence to the limiting extremal distributions for finite datasets is both slow and varies with the asymptote. However, when the third moment is expressed as a function of a this is found to be a more sensitive method.
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