Nonconcave Entropies in Multifractals and the Thermodynamic Formalism
Nonconcave Entropies in Multifractals and the Thermodynamic Formalism
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多重分形中的非凹熵和热力学形式主义
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
C. Beck
中科院分区：
文献类型：
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作者：
H. Touchette;C. Beck
我们讨论了一个微妙的参与计算的多重分形谱时，这些表示为勒让德-Fenchel变换的功能类似于自由能函数。我们表明，勒让德Fenchel变换的自由能函数产生正确的多重分形谱时，后者是全凹的。如果谱没有确定的包络，那么变换产生的是谱的凹包络，而不是谱本身。一些数学和物理的例子来说明这一结果，这是在微正则和正则系综的不等价的根源。在一个更积极的注意，我们还表明，通过勒让德-芬切尔变换的自由能表示非凹多重分形谱的不可能性可以规避的帮助下，一个广义的自由能函数，这涉及到最近推出的广义正则系综。最后讨论了与大偏差理论中速率函数计算的类比。
We discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as Legendre-Fenchel transforms of functions analogous to free energy functions. We show that the Legendre-Fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave. If the spectrum has no definite concavity, then the transform yields the concave envelope of the spectrum rather than the spectrum itself. Some mathematical and physical examples are given to illustrate this result, which lies at the root of the nonequivalence of the microcanonical and canonical ensembles. On a more positive note, we also show that the impossibility of expressing nonconcave multifractal spectra through Legendre-Fenchel transforms of free energies can be circumvented with the help of a generalized free energy function, which relates to a recently introduced generalized canonical ensemble. Analogies with the calculation of rate functions in large deviation theory are finally discussed.