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摘要翻译：
我们考虑了一个具有特定分形维数d$的自相似相空间，它以谱函数F(d)$分布。结果表明，相关的热统计是由非扩展统计的Tsallis形式所支配的，其中不可加性参数等于${bar\tau}(q)\equiv1/\tau(q)>1$,多重分形函数$\tau(q)=qd_q-f(d_q)$是用[1,infty)$中的多重分形参数$q\确定的比热。这样，均分定律就被证明发生了。通过对多重分形谱函数f(d)$的优化，导出了统计权重与系统复杂度之间的关系。结果表明，统计权重指数$\tau(q)$可以用Tsallis指数和Kaniadakis指数变形的双曲切线来描述任意多重分形相空间。证明了谱函数$f(d)$从最小值$f=-1$at$d=0$单调增加到最大值1$f=1$at$d=1$。同时，单分形的数量随着相空间体积的增加而增加，在小维D$时，单分形的数量下降到1$。
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英文标题：
《Multifractal spectrum of phase space related to generalized
thermostatistics》
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作者：
A. I. Olemskoi, V. O. Kharchenko, V. N. Borisyuk
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
We consider a self-similar phase space with specific fractal dimension $d$ being distributed with spectrum function $f(d)$. Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter is equal to ${\bar\tau}(q)\equiv 1/\tau(q)>1$, and the multifractal function $\tau(q)= qd_q-f(d_q)$ is the specific heat determined with multifractal parameter $q\in [1,\infty)$. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum function $f(d)$ derives the relation between the statistical weight and the system complexity. It is shown the statistical weight exponent $\tau(q)$ can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials to describe arbitrary multifractal phase space explicitly. The spectrum function $f(d)$ is proved to increase monotonically from minimum value $f=-1$ at $d=0$ to maximum one $f=1$ at $d=1$. At the same time, the number of monofractals increases with growth of the phase space volume at small dimensions $d$ and falls down in the limit $d\to 1$.
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PDF链接：
https://arxiv.org/pdf/708.187
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