Statistical-Thermodynamics Formalism of Self-Similarity

Fujisaka, Hirokazu; Inoue, Masayoshi

doi:10.1143/ptp.77.1334

Cited by 77 publications

(32 citation statements)

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“…Equivalently, one can consider the dynamics of the local time average of a time series α t = 1 t t j=1 ln B(x j ) [14]. Map (1) can be treated by putting…”

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confidence: 99%

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Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map

Rajagopalan¹,

Sabir²

2001

Phys. Rev. E

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We analyse the chaotic motion and its shape dependence in a piecewise linear map using Fujisaka's characteristic function method. The map is a generalization of the one introduced by R. Artuso. Exact expressions for diffusion coefficient are obtained giving previously obtained results as special cases. Fluctuation spectrum relating to probability density function is obtained in a parametric form. We also give limiting forms of the above quantities. Dependence of diffusion coefficient and probability density function on the shape of the map is examined.

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“…Equivalently, one can consider the dynamics of the local time average of a time series α t = 1 t t j=1 ln B(x j ) [14]. Map (1) can be treated by putting…”

mentioning

confidence: 99%

“…In this brief report, we apply the characteristic function formalism [13][14] to analyse the chaotic motion in a generalized piecewise linear (GPL) map with a variable shape. It is a generalization of the exactly solvable model in ref.…”

mentioning

confidence: 99%

Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map

Rajagopalan¹,

Sabir²

2001

Phys. Rev. E

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“…In this case, the master equation for any e can be treated as a finite-dimensional linear ordinary differential equation. The more detailed analysis of the time fluctuation based on the large deviation theory [13,14] would enhance the usefulness of the e-approximation. This is now under ongoing study and will be presented elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Investigating the Gap between Discrete and Continuous Models of Chemically Reacting Systems

Haruna

2010

J. Comput. Chem. Jpn.

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Molecular discreteness would be important in intracellular chemical reactions since the number of copies of molecules included in the reactions is small compared to in vitro situations [1][2][3][4][5]. Recently an experimental evidence for the significance of the molecular discreteness was found [6]. For a theoretical study, Togashi and Kaneko [3] reported a new transition phenomenon in a small autocatalytic system by a discrete stochastic simulation of the system. It cannot be observed in a model based on the classical reaction rate equation. However, Ohkubo et al. [7] found that the transition phenomenon reported However, we use CLE as a reference model in order to investigate the discrete nature of chemically reacting systems J. Comput. Chem. Jpn., Vol. 9, No. 3, pp.135-142 (2010) © The aim of this paper is to investigate the discrete nature of chemically reacting systems. In order to achieve our purpose we propose a systematic method to compare the discrete stochastic model of chemically reacting systems with the continuous stochastic model. We adopt the chemical master equation ( well-known idea of approximating diffusion processes by birth-death processes, we construct a family of master equations parameterized by the degree of discreteness. This family of master equations bridges CME and CFPE.With full degree of discreteness we obtain CME and as decreasing discreteness the family of master equations converges to CFPE. Our strategy is not to study CME directly but to distinguish the properties of CME by putting CME into the family of master equations bridging CME and CFPE. We examine the usefulness of our construction by two simple examples.

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“…The real system corresponds to (q, β) = (1, 0), where the summands in the partition function coincide with the probability measures of the UPO ensemble, hence we call it physical situation hereafter. Note that the partition function (6) is similar to that introduced by Fujisaka and Inoue [4], but here we explicitly consider the scaling dependence on the number of DOFs N as well as the period p in order to argue phase transitions. The relation to the space-time Gibbs measure should be also referred to.…”

Section: B Thermodynamic Formalismmentioning

confidence: 99%

“…Since one cannot follow individual trajectories in chaotic systems by any means, one of the subjects attracting interest is evaluation of dynamical averages, namely asymptotic time averages and fluctuations of some observables along typical orbits. The thermodynamic formalism [1,2], which is frequently used for multifractal analysis [2,3], is exploited for this purpose [4] and the concept of dynamical averaging has been remarkably developed by means of unstable periodic orbit expansion, trace formulae, and dynamical zeta function, which reveal the role of unstable periodic orbits (UPOs) as a skeleton of chaos [5]. On the other hand, the thermodynamic formalism is sometimes discussed in the context of phase transitions, called q-phase transitions.…”

Section: Introductionmentioning

confidence: 99%

Role of unstable periodic orbits in phase transitions of coupled map lattices

Takeuchi

Sano

2007

Phys. Rev. E

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The thermodynamic formalism for dynamical systems with many degrees of freedom is extended to deal with time averages and fluctuations of some macroscopic quantity along typical orbits, and applied to coupled map lattices exhibiting phase transitions. Thereby, it turns out that a seed of phase transition is embedded as an anomalous distribution of unstable periodic orbits, which appears as a so-called q-phase transition in the spatio-temporal configuration space. This intimate relation between phase transitions and q-phase transitions leads to one natural way of defining transitions and their order in extended chaotic systems. Furthermore, a basis is obtained on which we can treat locally introduced control parameters as macroscopic "temperature" in some cases involved with phase transitions.

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Statistical-Thermodynamics Formalism of Self-Similarity

Cited by 77 publications

References 1 publication

Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map

Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map

Investigating the Gap between Discrete and Continuous Models of Chemically Reacting Systems

Role of unstable periodic orbits in phase transitions of coupled map lattices

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