A first-principles derivation is presented of canonical distributions for a finite thermostat taking into account nonextensive energy.Parameterizing this energy by , we derive an explicit form for the distribution functions by regulating , and then explore the nontrivial relationship between these functions and energy nonextensivity, as well other system parameters such as system size. A variational entropy function is also derived from these distribution functions.. finite system, energy nonextensivity, first-principles method, canonical distribution function
Citation:Zheng L, Li W. Thermoequilibrium statistics for a finite system with energy nonextensivity. Chinese Sci Bull, 2011Bull, , 56: 3666-3670, doi: 10.1007 Studies into the family of small or finite-size systems reveal that these systems show complex statistical characteristics very different from large systems, and expose failings in our understanding of thermostatistics for finite systems and their nonequilibrium dynamics [1][2][3][4]. One of these characteristics is nonextensivity (refer to [2] and references therein for examples), which means that macroscopic quantities of such systems may not be proportional to system size. A thermodynamic quantity of a nonextensive system is called nonadditive if this quantity is not the sum of that of its subsystems. Nonextensivity might arise from surface effects or interactions between subsystems. Thus, extensive energy and entropy may become inappropriate in treating finite systems. There has already been much discussion on the properties of finite systems that has raised many questions and controversies [5][6][7]. To describe the behaviors of these systems, one point worth noting is that the Boltzmann-Gibbs statistical mechanics, which is based on the concept of extensive entropy, is not applicable [1,8,9]. Consequently, it is questionable to assume an exponential distribution for finite systems. Therefore, one of the first things to work out for a statistical description of finite systems is to find the appropriate probability distribution. Some published results obtained by first principles (see [8] for example) have shown that a small system, in equilibrium with a finite reservoir, may follow a q-exponential distribution of nonextensive statistical mechanics (NSM), as proposed by Tsallis [9]. This NSM has been used widely in many areas, and considered effective in solving many physical problems [10][11][12][13]. Results from mathematical proofs are able to demonstrate a connection between the system's finiteness and nonextensivity of the theory. However, most proofs have relied on the additivity of energy. Clearly, for large systems, this assumption is acceptable and provides a helpful approximation in obtaining the statistics in the thermodynamic limit. The assumption is, however, questionable when establishing statistics for finite systems. Related problems arising from this additive energy assumption can be found in [14][15][16][17].Work in [18] represents a first attempt to build statistics for fin...