Tsallis statistics

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

Monte Carlo Methods for Pure Tsallis Statistics... 

We might then assume a Maxwell distribution of momenta so that the overall phase space distribution is that of a Maxwell-Tsallis statistics. 

As in the case of pure Tsallis statistics and Eq. (33), in the limit that q 1 the standard transition state theory result... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

I Andricioaei, JE Straub. On Monte Carlo and molecular dynamics methods inspired by Tsallis statistics Methodology, optimization, and application to atomic clusters. J Chem Phys 107 9117-9124, 1997. 

Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

Straub, J.E. Andricioaei, I., Computational methods inspired by Tsallis statistics Monte Carlo and molecular dynamics algorithms for the simulation of classical and quantum systems, Braz. J. Phys. 1999, 29, 179-186... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

The aims of this study are to establish the connection between the Tsallis statistics, i.e., the statistical mechanics based on the Tsallis statistical entropy, and the equilibrium thermodynamics and to prove the zero law of thermodynamics. 

The structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section. 

The Tsallis statistical mechanics is based on the generalized entropy which is a function of the entropic parameter q and probing probabilities pt [3, 4] ... 

To prove the homogeneity properties of the thermodynamic quantities and the Euler theorem for the Tsallis statistics in the canonical ensemble, we will consider, as an example, the exact analytical results for the nonrelativistic ideal gas. 

It is convenient to obtain the exact results for the ideal gas in the Tsallis statistics by means of the integral representation for the Gamma function (see [9] and reference therein) ... 

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics in the microcanonical ensemble. Equation (140) proves the zero law of thermodynamics for the microcanonical ensemble [6]. 

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable E of the microcanonical ensemble, we obtain Eq. (96) for the variable E of the canonical ensemble. Thus, for the Tsallis statistics, the canonical and microcanonical ensembles are equivalent in the thermodynamic limit when the entropic parameter z is considered to be an extensive variable of state. 

Parvan A S. Microcanonical ensemble extensive thermodynamics of Tsallis statistics. Phys. Lett. A. 2006 350(5-6) 331-338. 

Annealing Algorithms using Tsallis Statistics Application to Conformational Optimization of a Tetrapeptide. 

Andricioaei and Straub have recently employed a similar acceptance probability where the trial step is sampled from a distribution function of a form proposed by Tsallis. In Tsallis statistics , the standard Gibbs entropy S = —k / dx p(x) In p(x) is modified to the form Sq = k J dx(l — Pq xf)j q — 1) which is equal to the Gibbs entropy formula in the limit that q =. The equilibrium probability distribution functions take the form... 

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