Non-commutative geometry and irreversibility

Abstract:We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear "spontaneously" in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.

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“…In Ref. [140], a kinetics is built upon the q-calculus of discrete dilatations and is shown to describe diffusion on a hierarchical lattice.…”

Section: Multilacunarity and Quasi-log-periodicitymentioning

confidence: 99%

Discrete-scale invariance and complex dimensions

1998

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“…An example of generalized calculus including the q-derivative and the q-integral was proposed long ago by Jackson [17]. This q-calculus arises in the description of systems with discrete dilatation symmetries [18]. Of course, it appears as a convenient tool for the study of the thermodynamics of a q-Bose gas.…”

Section: Introductionmentioning

confidence: 99%

Elements of µ-calculus and Thermodynamics of µ-Bose Gas Model

Rebesh¹,

Gavrilik²,

Качурик³

2013

Ukr. J. Phys.

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We review on and give some further details about the thermodynamical properties of the µ-Bose gas model (arXiv:1309(arXiv: .1363) introduced by us recently. This model was elaborated in connection with µ-deformed oscillators. Here, we present the necessary concepts and tools from the so-called µ-calculus. For the high temperatures, we obtain the virial expansion of the equation of state, as well as five virial coefficients. In the regime of low temperatures, the critical temperature of condensation is inferred. We also obtain the specific heat, internal energy, and entropy for a µ-Bose gas for both low and high temperatures. All thermodynamical functions depend on the deformation parameter µ. The dependences of the entropy and the specific heat on the deformation parameter are visualized. K e y w o r d s: deformed oscillators, deformed analogs of the Bose gas model, µ-calculus, equation of state, virial coefficients, critical temperature, specific heat, entropy.

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“…But it does seem possible to "sneak" fractals into TD via entropy ideas and fractional calculus. For more on this theme we sketch from [22,23] Thus the purpose in [22] is to point out that q-derivatives are naturally suited for describing systems with discrete dilatation symmetries such as fractal and multi-fractal sets, where the limit q → 1 corresponds to continuous scale change. Thus the q-derivative can be defined via (5.1)…”

Section: Td and Fractalsmentioning

confidence: 99%

Some topics in thermodynamics and quantum mechanics

Carroll

2012

Preprint

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Non-commutative geometry and irreversibility

Cited by 5 publications

References 21 publications

Discrete-scale invariance and complex dimensions

Discrete-scale invariance and complex dimensions

Elements of µ-calculus and Thermodynamics of µ-Bose Gas Model

Some topics in thermodynamics and quantum mechanics

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