Tsallis distributions

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]... 

Also note that the definition of the effective potential V in (8.14) enables one to conceive a constant-temperature molecular dynamics method (instead of MC) to generate the Tsallis distributions. Given this effective potential, it is possible to define a constant-temperature molecular dynamics algorithm such that the distribution Pq(x) is sampled in the trajectory. The equation of motion takes on the simple and suggestive form... 

When a system has long-time correlation, for which we expect fractional power scaling of excess heat, our assumption of the Boltzmann equilibrium distribution may always not be valid. Actually some power distributions such as the Tsallis distribution [14] have been reported at the edge of chaos [15]. A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. Beck and Cohen [13] considered many types of distributions for the inverse of temperature. For example, they chose Gaussian, uniform, gamma, log-normal, and others. In particular, the Tsallis distribution is realized for gamma distribution. We will show that excess heat can be written as a superposition of correlation functions... 

Before we close this section, we will comment on the area of the hysteresis loop. When the decay constant of a correlation functions depends on the inverse of temperature, we can expect various behavior for the area of the hysteresis loop. In the case of the Tsallis distribution, the inverse of temperature is distributed as a gamma distribution. If the decay constant is proportional to the temperature, the area of the histeresis loop decays as a modified Bessel function for the large period of external transformation. On the other hand, if the decay constant is proportional to the inverse of temperature, we can expect the fractional power scaling. 

Gaussian while the Tsallis distribution function q = 2) is a Cauchy-Lorentz distribution. 

Third, generalized equations of motion have been proposed to sample arbitrary (i.e., not necessarily canonical) probability distributions [134, 135, 136, 137]. Such methods can be used, e.g., to optimize the efficiency of conformational searches [134, 135, 137] or for generating Tsallis distributions of microstates [136]. 

The Tsallis probability distribution is equivalent to the classical Boltzmann distribution in the limit q 1. As shown numerically elsewhere/ for q > 1 the Tsallis distribution broadens, overcoming energy barriers, and like high-temperature Boltzmann particle densities, the Tsallis distribution has maxima at the coordinates of potential minima. The Tsallis distribution can be used like high-temperature Boltzmann distributions if Tsallis configurations are accepted or rejected with probability... 

Successful applications of jumps to Tsallis distributions have included cluster... 

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]... 

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

As one of the possible ways to alter the sampling distribution in a manner that is conducive to enhanced sampling, we present a strategy based on probability distributions that arise in a generalization of statistical mechanics proposed by Tsallis [31]. In this... 

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. 

Notice that this information approach to Levy statistics is even more direct than the nonextensive thermodynamic approach. As shown in Ref. 52, the adoption of the method of entropy maximization, with the Shannon entropy replaced by the Tsallis entropy [53], does not yield directly the Levy distribution, but a probability density function n(x) whereby reiterated application of the convolution generates the stable Levy distribution. 

In recent times, the term superstatistics has been coined [153] to denote an approach to non-Poisson statistics, of any form, not only the Nutting (Tsallis) form, as in the original work of Beck [154]. We note that Cohen points out explicitly [153] that the time scale to change from a Poisson distribution to... 

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. 

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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