Maximum path information and the principle of least action for chaotic system

A path information is defined in connection with the different possible paths of chaotic system moving in its phase space between two cells. On the basis of the assumption that the paths are differentiated by their actions, we show that the maximum path information leads to a path probability distribution as a function of action from which the well known transition probability of Brownian motion can be easily derived. An interesting result is that the most probable paths are just the paths of least action. This suggests that the principle of least action, in a probabilistic situation, is equivalent to the principle of maximization of information or uncertainty associated with the probability distribution.Comment: 12 pages, LaTeX, 1 eps figure, Chaos, Solitons & Fractals (2004), in pres

Incomplete information and fractal phase space

The incomplete statistics for complex systems is characterized by a so called incompleteness parameter $\omega$ which equals unity when information is completely accessible to our treatment. This paper is devoted to the discussion of the incompleteness of accessible information and of the physical signification of $\omega$ on the basis of fractal phase space. $\omega$ is shown to be proportional to the fractal dimension of the phase space and can be linked to the phase volume expansion and information growth during the scale refining process.Comment: 12 pages, 2 ps figure, Te

Unnormalized nonextensive expectation value and zeroth law of thermodynamics

We show an attempt to establish the zeroth law of thermodynamics within the framework of nonextensive statistical mechanics based on the classic normalization $\texttt{Tr}\hat{\rho}=1$ and the unnormalized expectation $x=\texttt{Tr}\hat{\rho}^q\hat{x}$. The first law of thermodynamics and the definition of heat and work in this formalism are discussed.Comment: 6 pages, no figure, RevTeX. To appear in Chaos, Solitons & Fractals (2002

Extensive generalization of statistical mechanics based on incomplete information theory

Statistical mechanics is generalized on the basis of an additive information theory for incomplete probability distributions. The incomplete normalization $\sum_{i=1}^wp_i^q=1$ is used to obtain generalized entropy $S=-k\sum_{i=1}^wp_i^q\ln p_i$. The concomitant incomplete statistical mechanics is applied to some physical systems in order to show the effect of the incompleteness of information. It is shown that this extensive generalized statistics can be useful for the correlated electron systems in weak coupling regime.Comment: 15 pages, 3 eps figures, Te

Non quantum uncertainty relations of stochastic dynamics

First we describe briefly an information-action method for the study of stochastic dynamics of hamiltonian systems perturbed by thermal noise and chaotic instability. It is shown that, for the ensemble of possible paths between two configuration points, the action principle acquires a statistical form $=0$. The main objective of this paper is to prove that, via this information-action description, some quantum like uncertainty relations such as $\geq\frac{1}{\sqrt{2}\eta}$ for action, $<\Delta P>\geq\frac{1}{\eta}$ for position and momentum, and $<\Delta t>\geq\frac{1}{\sqrt{2}\eta}$ for hamiltonian and time, can arise for stochastic dynamics of classical hamiltonian systems. A corresponding commutation relation can also be found. These relations describe, through action or its conjugate variables, the fluctuation of stochastic dynamics due to random perturbation characterized by the parameter $\eta$

Nonextensive statistics and incomplete information

We comment on some open questions and theoretical peculiarities in Tsallis nonextensive statistical mechanics. It is shown that the theoretical basis of the successful Tsallis' generalized exponential distribution shows some worrying properties with the conventional normalization and the escort probability. These theoretical difficulties may be avoided by introducing an so called incomplete normalization allowing to deduce Tsallis' generalized distribution in a more convincing and consistent way.Comment: 21 pages, RevTeX, no figures, published version to appear in Euro. J. Phys. B (2002