Phase transitions on strange irrational sets

Artuso, Roberto; Cvitanović, Predrag; Kenny, Brian G.

doi:10.1103/physreva.39.268

Cited by 51 publications

(49 citation statements)

References 33 publications

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“…This maybe be calculated analytically for negative half integer β (see Table 1) and computationally when beta is positive for finite order Farey tree levels. See [1] who conjecture that asymptotically, F (β) converges to β = 1 so that F (β) for the Farey tree partition is C 1 at β = 1. These different partitions lead to different thermodynamic free energy F (β) which have only two points in common.…”

Section: Resultsmentioning

confidence: 99%

“…P and Q are coprime integers; k and δ are constants (we assume δ > 2 ). In this model we know the widths of the mode locked intervals, however we do not know precisely where the left and right hand ends of the interval are located on [0,1]. This means we do not know exactly where the holes are located.…”

Section: Model For Interval Widthsmentioning

confidence: 99%

“…For the purpose of illustration, the value δ = 2 has been chosen. For negative half integer β, F (β) is known analytically for the latter partition [1]. See Table 2.…”

Section: Comparison Of Free Energies For Different Partitionsmentioning

confidence: 99%

“…In fact such a choice has been used [1] for evaluating the free energy (or its equivalent) and the Hausdorff dimension in the case of the real mode locking critical circle map. This is a different partition of the rationals which also utilized Farey series.…”

Section: Free Energy For Farey Series Partitionmentioning

confidence: 99%

“…The Farey tree level partition of the rationals has been frequently used in discussions of the circle map [1,3] and one may readily calculate the free energy using this scheme. Since the number of rationals doubles from one level to the next, the free energy may be defined through (10) when comparing level n of the Farey tree with level (n + 1).…”

Section: Free Energy For Farey Tree Partitionmentioning

confidence: 99%

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Ambiguity in the Determination of the Free Energy for a Model of the Circle Map

Kenny¹,

Dixon²

2009

ANZIAMJ

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We consider a simple model to describe the widths of the mode locked intervals for the critical circle map. Using two different partitions of the rational numbers, based on Farey series and Farey tree levels respectively, we calculate the free energy analytically at selected points for each partition. It is found that the result of the calculation depends on the method of partition. An implication of this is that the generalized dimensions Dq are different for each partition except when q = 0, i.e. only the Hausdorff dimension is the same in each case.

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Section: Resultsmentioning

confidence: 99%

Section: Model For Interval Widthsmentioning

confidence: 99%

“…For the purpose of illustration, the value δ = 2 has been chosen. For negative half integer β, F (β) is known analytically for the latter partition [1]. See Table 2.…”

Section: Comparison Of Free Energies For Different Partitionsmentioning

confidence: 99%

Section: Free Energy For Farey Series Partitionmentioning

confidence: 99%

Section: Free Energy For Farey Tree Partitionmentioning

confidence: 99%

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Ambiguity in the Determination of the Free Energy for a Model of the Circle Map

Kenny¹,

Dixon²

2009

ANZIAMJ

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Phase Transitions on Strange Sets

Cvitanović¹

1988

Nonlinear Evolution and Chaotic Phenomena

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Abstract:The "thermodynamic" formalism provides a very general division of strange (Cantor, fractal) sets into two classes; those which do exhibit phase transitions, and those which do not.The "thermodynamic formalism,,1,2,3,4) is based on the observation that the sum used in determining the Hausdorff dimension of a strange (fractal, Cantor) set resembles a partition sum over "configurations" i, with T playing the role of "temperature", and covering interval sizes playing the role of "Boltzmann weights".The "thermodynamic" functions extracted from the above partition sum exhibit a phenomenon that might have gone unnoticed were it not for the thermodynamic formalism. They can undergo "phase transitions". These phase transitions can be visualized in the following way: the exponent T acts as a "magnifying lens" which blows up some of the covering intervals, and (relatively), shrinks the others. For negative T, the fat intervals are expanded, and the thin ones are made even thinner. For positive T, the thin intervals are (relatively) blown up. Below the phase transition, the sum is dominated by a few fat intervals. Visually, the cover consists of a few + Carlsberg Fellow

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Experimental Study of the Multifractal Structure of the Quasiperiodic Set

Barkley

Cumming

1989

NATO ASI Series

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Phase transitions on strange irrational sets

Cited by 51 publications

References 33 publications

Ambiguity in the Determination of the Free Energy for a Model of the Circle Map

Ambiguity in the Determination of the Free Energy for a Model of the Circle Map

Phase Transitions on Strange Sets

Experimental Study of the Multifractal Structure of the Quasiperiodic Set

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