On the basis of the entropy of incomplete statistics （IS） and the joint probability factorization condition, two controversial problems existing in IS are investigated： one is what expression of the internal energy is reasonable for a composite system and the other is whether the traditional zeroth law of thermodynamics is suitable for IS. Some new equivalent expressions of the internal energy of a composite system are derived through accurate mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it cannot be proven theoretically. Finally, it is pointed out that the generalized zeroth law of thermodynamics for incomplete nonextensive statistics is unnecessary and the nonextensive assumptions for the composite internal energy will lead to mathematical contradiction.
The performance of a two-state quantum engine under different conditions is analyzed. It is shown that the efficiency of the quantum engine can be enhanced by superposing the eigenstates at the beginning of the cycle. By employing the finite-time movement of the potential wall, the power output of the quantum engine as well as the efficiency at the maximum power out- put （EMP） can be obtained. A generalized potential is adopted to describe a class of two-level quantum engines in a unified way. The results obtained show clearly that the performances of these engines depend on the external potential, the geometric configuration of the quantum engines, and the superposition effect. Moreow^r, it is found that the superposition effect will en- large the optimally operating region of quantum engines.
Tsallis entropy and incomplete entropy are proven to have equivalent mathematical structure except for one nonextensive factor q through variable replacements on the basis of their forms. However, employing the Lagrange multiplier method, it is judged that neither yields the q-exponential distributions that have been observed for many physical systems. Consequently, two generalized entropies under complete and incomplete probability normalization conditions are proposed to meet the experimental observations. These two entropic forms are Lesche stable, which means that both vary continuously with probability distribution functions and are thus physically meaningful.
Based on the q-exponential distribution which has been observed in more and more physical systems, the uncertainty measure of such an abnormal distribution can be derived by employing a variational relationship which can be traced from the first and second thermodynamic laws. The uncertainty measure obtained here can be considered as the entropic form for the abnormal physical systems having observable q-exponential distribution. This entropy will tend to the Boltzmann-Gibbs entropy when the nonextensive parameter tends to unity. It is very important to find that this entropic form is always concave and the systemic entropy is maximizable.
OU CongJieEL KAABOUCHI AzizWANG QiuPing AlexandreCHEN JinCan
