We introduce a new universality class of one-dimensional iteration model giving rise to self-similar motion, in which the Feigenbaum constants are generalized as self-similar rates and can be predetermined. The curves of the mean-square displacement versus time generated here show that the motion is a kind of anomalous diffusion with the diffusion coefficient depending on the self-similar rates. In addition, it is found that the distribution of displacement agrees to a reliable precision with the q-Gaussian type distribution in some cases and bimodal distribution in some other cases. The results obtained show that the self-similar motion may be used to describe the anomalous diffusion and nonextensive statistical distributions.

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In order to overcome the limitations of the original expression of the probability distribution appearing in literature of Incomplete Statistics, a new expression of the probability distribution is derived, where the Lagrange multiplier β introduced here is proved to be identical with that introduced in the second and third choices for the internal energy constraint in Tsallis' statistics and to be just equal to the physical inverse temperature. It is expounded that the probability distribution described by the new expression is invariant through uniform translation of the energy spectrum. Moreover, several fundamental thermodynamic relations are given and the relationship between the new and the original expressions of the probability distribution is discussed.

The cycle model of a general micro-scaled regenerative quantum refrigerator working with an ideal Bose or Fermi gas is established. The combined effects of quantum boundary and degeneracy on the performance of the cycle are investigated based on the thermodynamic properties of a confined ideal Bose or Fermi gas. The inherent regenerative losses of the cycle are analyzed and calculated. Expressions for several important performance parameters, such as the refrigeration load, work input, and coefficient of performance (COP), are derived under the cases of the gas degeneracy, weak gas degeneracy, high temperature limit, and thermodynamic limit. The curves of the refrigeration load and coefficient of performance versus the volume and surface area ratios of the cycle and the refrigeration load versus the coefficient of performance are represented. The effects of the size effect on the refrigeration load and coefficient of performance are discussed. The general performance characteristics of the cycle are revealed. It is found that both the refrigeration load and coefficient of performance of the micro-scaled quantum Stirling refrigeration cycle depend on the surface area of the cyclic system besides the temperature of the heat reservoirs, the volume of cyclic system, and other parameters, while those of the macro-scaled refrigerator are independent of the surface area of a cyclic system. The results obtained here are more general and significant than those in the current literature. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4754420]National Natural Science Foundation [11175148]; Natural Science Foundation of Fujian Province [2011J01012]; Science Research Fund by Huaqiao University, People's Republic of China [09BS510

By using a fonctionelle of probability distributions, several different statistical physics including extensive and nonextensive statistics are unified in a general method. The essential equivalence between the MaxEnt process of the most probable probility distribution in these statistics and the famous thermodynamical relation dU TdS = is strictly proved without any additional assumption. Moreover, itis expounded that all the conclusions of these different statistics can be directly derived from the equivalent relation.