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The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics.

ABSTRACT: As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.

SUBMITTER: Tirnakli U

PROVIDER: S-EPMC4804329 | biostudies-other | 2016 Mar

REPOSITORIES: biostudies-other

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Project description:The relationship among the standard reaction Gibbs free energy ΔG°, the standard reduction potential E°, and the atomic structure parameters of radius, nuclear charge, and isoelectronic orbitals nl is accomplished through the attraction electric force F elec. In relationship with E°, it was necessary to define two new reference scales: E 0 ° with a final state of E° in the element, which allowed to have a parabolic trend of ΔG° versus F elec, and E °,0 whose final state is the ion with a more negative charge (e.g., -1, -2, -3). The relationship with ΔG° is related to the concept of chemical stability, and the relationship with E °,0 is more related to the concept of electronegativity. In relationship with ΔG°, it was necessary to predict the values of possible new cations and noncommon cations in order to find a better trend of ΔG° versus F elec, whose stability is analyzed by Frost diagrams of the isoelectronic series. This dependence of ΔG° on F elec is split into two terms. The first term indicates the behavior of the minimum of ΔG° for each isoelectronic orbital nl, while the second term deals with the parabolic trend of this orbital. For the minima of the configuration np6, a hysteresis behavior of the minima of ΔG° is found: an exponential behavior from periods 1 and 2 and a sigmoidal behavior from periods 5 and 4 to interpolate period 3. It is also found that the proximity of unfilled np or (n + 1)s orbitals induces instability of the ion in configurations ns2/nd2/4f2 and nd10/nd8(n + 1)s2, respectively. On the contrary, the stability of the orbitals np6 does not depend on the neighboring empty (n + 1)s0 orbitals. Both phenomena can be explained by the stability of the configuration of noble gas np6 and the nd10(n + 1)s2 configuration. We have also found that it is possible to increase the reduction potential E °,0 (macroscopic electronegativity), although the electric force F elec decreases because the orbital overlap influences the electronegativity.

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The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics.

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