Tsallis Constantino, Souza Andre M C
Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 2):026106. doi: 10.1103/PhysRevE.67.026106. Epub 2003 Feb 6.
The basic aspects of both Boltzmann-Gibbs (BG) and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional (S(BG)=-k Sigma(i)p(i)ln p(i) for the BG formalism) with the appropriate constraints (Sigma(i)p(i)=1 and Sigma(i)p(i)E(i)=U for the BG canonical ensemble). Second, through optimization, the equilibrium or stationary-state distribution (p(i)=e(-betaE(i))/Z(BG) with Z(BG)= Sigma(j)e(-betaE(j)) for BG). Third, the connection to thermodynamics (e.g., F(BG)=-(1/beta)ln Z(BG) and U(BG)=-(partial differential/partial differential beta)ln Z(BG)). Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor B(E)= integral (infinity)(0)dbetaf(beta)e(-betaE). This corresponds to the second stage described above. In this paper, we solve the corresponding first stage, i.e., we present an entropic functional and its associated constraints which lead precisely to B(E). We illustrate with all six admissible examples given by Beck and Cohen.
玻尔兹曼-吉布斯（BG）统计力学和非广延统计力学的基本方面都可以通过三个不同阶段来理解。首先，提出一个具有适当约束条件（对于BG正则系综，(\sum_{i}p_{i}=1)且(\sum_{i}p_{i}E_{i}=U)）的熵泛函（对于BG形式体系，(S(BG)=-k\sum_{i}p_{i}\ln p_{i})）。其次，通过优化得到平衡态或稳态分布（对于BG，(p_{i}=e^{-\beta E_{i}}/Z(BG))，其中(Z(BG)=\sum_{j}e^{-\beta E_{j}})）。第三，建立与热力学的联系（例如，(F(BG)=-(1/\beta)\ln Z(BG))且(U(BG)=-(\partial/\partial\beta)\ln Z(BG))）。假设存在温度涨落，贝克和科恩最近提出了一个广义玻尔兹曼因子(B(E)=\int_{0}^{\infty}d\beta f(\beta)e^{-\beta E})。这对应于上述的第二阶段。在本文中，我们解决了相应的第一阶段问题，即我们提出了一个熵泛函及其相关约束条件，这些条件恰好能导出(B(E))。我们用贝克和科恩给出的所有六个可允许的例子进行了说明。
