Effects of non-Gaussian noise near supercritical Hopf bifurcation

We have studied the effects of non-Gaussian colored noise in a chemical oscillation system, the well-known Brusselator model, in the parameter region close to the supercritical Hopf bifurcation. With the variation of the parameter q, which quantifies the deviation from Gaussian character, the signal-to-noise ratio of noise induced oscillation exhibits a bell-shaped change, indicating the presence of resonant activity. The cooperative effects of q and the correlation time τ on the performance of noise induced oscillation are also investigated. Interestingly, resonance-like behavior can be induced by either q or τ when the other parameter is properly fixed. Stochastic normal form theory is used to analyze these nontrivial effects and the simulation results are well reproduced. This work provides us comprehensive understanding of how non-Gaussian noise influences the dynamics in chemical oscillation systems.     

On spurious and corrupted multifractality: The effects of additive noise, short-term memory and periodic trends

We study the performance of multifractal detrended fluctuation analysis (MF-DFA) applied to long-term correlated and multifractal data records in the presence of additive white noise, short-term memory and periodicities. Such additions and disturbances that can be typically found in the observational records of various complex systems ranging from climate dynamics to physiology, network traffic, and finance. In monofractal records, we find that (i) additive white noise hardly results in spurious multifractality, but causes underestimated generalized Hurst exponents h(q) for all q values; (ii) short-range correlations lead to pronounced crossovers in the generalized fluctuation functions F_q(s) at positions that decrease with increasing moment q, thus causing significantly overestimated h(q) for small q and spurious multifractality; (iii) periodicities like seasonal trends (with standard deviations comparable with the one of the studied process) result in spurious “reversed” multifractality where h(q) increases with increasing q (except for very short time windows). We also show that in multifractal cascades moderate additions of noise, short-range memory, or periodic trends cause flawed results for h(q) with q<2, while h(q) with q>2 remains nearly unchanged.     

Stochastic resonance in an asymmetric bistable system with coloured noises and periodic rectangular signal

In this paper, we consider the phenomenon of stochastic resonance (SR) in an asymmetric bistable system with coloured noises and periodic rectangular signal. Expression of the signal-to-noise ratio (SNR) has been obtained under the adiabatic limit. We investigate the effect of any system parameter (such as p, q, r, τ₁, τ₂) on the SNR. The plot of SNR-τ₁ shows SR for some values of the additive noise self-correlation time τ₂, but not for the whole range of τ₂. The system bias r suppresses the SNR. When the intensity of additive noise q is increased, the SR phenomenon disappears in the plot of SNR-p, but the plot of SNR-q presents SR for almost all values of the multiplicative noise intensity p.     

Quantumq-white noise and aq-central limit theorem

We establish a connection between the Azéma martingales and certain quantum stochastic processes with increments satisfyingq-commutation relations. This leads to a theory ofq-white noise onq-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bialgebras. In particular, quantum Azéma noise,q-interpolations between Fermion and Boson quantum Brownian motion and unitary evolutions withq-independent multiplicative increments are studied. It follows from our results that the Azéma martingales and theq-interpolations are central limits of sums ofq-independent, identically distributed quantum random variables.     

Finite Fermi systems theory and self-consistency relations

The self-consistent theory of the finite Fermi systems is outlined. This approach is based on the same Fermi liquid theory principles as the familiar theory for finite Fermi systems (FFS) by Migdal. We show that the basic Fermi system properties can be evaluated in terms of the quasiparticle Lagrangian L_q which incorporates the energy dependency effects. This Lagrangian is defined so that the corresponding Lagrange equations should coincide with the FFS theory equations of motion of the quasiparticles. The quasiparticle energy E_q defined in the terms of t he quasiparticle Lagrangian L_q according to the usual canonical rules is shown to be equal to the binding energy E_o of the system. For a given Lagrangian L_q the particle densities in nuclei, the nuclear single-particle spectra, the low-lying collective states (LCS) properties, and the amplitude of the interquasiparticle interaction are also evaluated. The suggested approach is compared with the Hartree-Fock theory with effective forces.     

Return distributions in dog-flea model revisited

A recent study of coherent noise model for the system size independent case provides an exact relation between the exponent τ of avalanche size distribution and the q value of the appropriate q-Gaussian that fits the return distribution of the model. This relation is applied to Ehrenfest’s historical dog-flea model by treating the fluctuations around the thermal equilibrium as avalanches. We provide a clear numerical evidence that the relation between the exponent τ of fluctuation length distribution and the q value of the appropriate q-Gaussian obeys this exact relation when the system size is large enough. This allows us to determine the value of the q-parameter a priori from one of the well known exponents of such dynamical systems. Furthermore, it is shown that the return distribution in dog-flea model gradually approaches q-Gaussian as the system size increases and this tendency can be analyzed by a well defined analytical expression.     

The structure of the multi-instanton gas

We parametrize the q-instanton solutions of an SU(2) gauge theory in terms of the positions of 2q constituent particles or “instanton quarks”. Explicit computations of lowest order quantum fluctuations about the q = 1 and q = 2 solutions show that the short-distance interaction between instanton quark pairs is logarithmic. Extending these interactions to arbitrary q, we describe the multi-instanton gas as a plasma of instanton quarks.     

Wavevector dependence of the two-dimensional plasmon dispersion relationship in the (100) silicon inversion layer

The two-dimensional plasmon dispersion relationship in the inversion layer on (100) p-silicon is studied as a function of wavevector q. This is facilitated by a simple technique which allows plasmon resonances corresponding to several values of q to be excited in the same sample. Over the experimentally obtained values of q the results are in good agreement with theory.     

Representation of osp (2, 2) q (2) andS-matrix of super sine-Gordon theory

We investigate the representations of the osp (2, 2) _q ⁽²⁾ algebra, which leads to theS-matrix of super sine-Gordon theory. TheS-matrix has been derived from supersymmetric conformal field theory with some assumptions. We show that the conjecturedS-matrix can be derived from the representation theory using a correspondence between the representations of osp (1, 2) _q and those of sl(2) _q .     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

Entropic stochastic resonance in a confined asymmetrical bistable system with non-Gaussian noise

Particles moving in confined regions with bistable potential driven by weak signal and fluctuations may present a similar stochastic resonance (SR). This similar SR is called entropic stochastic resonance (ESR). The phenomenon of ESR in a confined asymmetrical bistable potential driven by a transverse periodic force is investigated when a non-Gaussian fluctuation is considered in the system. The left potential of the bistable system can be adjusted by introducing the asymmetrical ratio α. By the functional analysis and two-state theory, the approximate Signal-to-Noise ratio (SNR) is derived. It is found that there is a global maximum in the SNR curves when the asymmetrical ratio α and the noise intensity D are changed. It is also shown that the noise correlation time τ will suppress the ESR of the system and the departure q can obviously change the position of peak in the SNR. The above clues are helpful in achieving weak signal detection under irregular, small-scale systems.     

A novel approach for analyzing open Z-scan experiments

In this work, a novel method for analyzing open Z-scan experiments is presented. It is based on the calculation of the nonlinear absorption parameter q directly from the transmittance measurements. Specifically, we have found an analytic formula describing the dependence of the parameter q on the normalized transmittance. The influence of noise has been studied and the method was proved robust even under conditions of relatively high noise. Extension to the general case of an elliptic Gaussian incident beam has also been realized. The technique has been experimentally tested by analyzing transmittance measurements obtained from a recently synthesized pyrylium salt, illuminated by a femtosecond laser system. The results are in very good agreement with those obtained by the standard Z-scan analysis, especially regarding the nonlinear absorption coefficient β.     

From trees to galaxies: The Potts model on a random surface

The matrix model of random surfaces with c = ∞ has recently been solved and found to be identical to a random surface coupled to a q-states Potts model with q = ∞. The mean field-like solution exhibits a novel type of tree structure. The natural question is, down to which—if any—finite values of c and q does this behavior persist? In this work we develop, for the Potts model, an expansion in the fluctuations about the q = ∞ mean field solution. In the lowest—cubic—non-trivial order in this expansion the corrections to mean field theory can be given a nice interpretation in terms of structures (trees and “galaxies”) of spin clusters. When q drops below a finite q_c, the galaxies overwhelm the trees at all temperatures, thus suppressing mean field behavior. Thereafter the phase diagram resembles that of the Ising model, q = 2.     

Nonextensive formalism and continuous Hamiltonian systems

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function Sq while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy Sq yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q>1. Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q<1, which raises the question of its formal validity range for continuous Hamiltonian systems.     

Stochastic resonance in a bistable system with time-delayed feedback and non-Gaussian noise

The phenomenon of stochastic resonance in a bistable system with time-delayed feedback driven by non-Gaussian noise is investigated. Combining the small time delay approximation, the path-integral approach and the unified colored noise approximation, a general approximate Fokker–Planck equation of a stochastic system is obtained. The effects of the parameter q indicating the departure from the Gaussian noise, the delay time τ  , and the correlation time τ₀${\tau }_0$ of the non-Gaussian noise on the quasi-steady-state probability distribution function (SPD) and the signal-to-noise ratio (SNR) are discussed. It is found that the number of peaks in SPD and the reentrant transition between one peak and two peaks and then to one peak again in the curve of SNR depends on the parameter q, the delay time τ  , and the noise correlation time τ₀${\tau }_0$.     

A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics

We show that a connection between the generalized entropy and theory of quantum groups, recently pointed out by Tsallis [Phys. Lett. A 195 (1994) 329], can naturally be understood in the framework of q-calculus. We present a new entropy which has q↔q⁻¹ invariance and discuss its basic properties.     

Large q expansions for q-state gauge-matter Potts models in lagrangian form

We consider the lagrangian form of a q-state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in $\text{1}\text{q}{}^{\mathrm{<mtext>1</mtext><mtext>d</mtext>}}$. Extrapolating the series for the free energies and latent heats by the method of Padé approximants, we have constructed the phase diagrams for all values of q. Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.     

Effects of non-Gaussian noise on a calcium oscillation system

We investigate the effects of the non-Gaussian colored noise on a calcium oscillation system using stochastic simulation methods. It is found that the reciprocal coefficient of variance R has a maximum (R max ) with increasing noise intensity Q. The non-Gaussian noise parameter q has an important effect on the system. For some values of q (e.g., q = 0.9, q = 1.0), R has a maximum with increasing correlation time τ. Non-Gaussian noise induced spikes are more regular than Gaussian noise induced spikes when q is small and Q has large values. The R has a maximum with increasing q. Therefore, non-Gaussian noise could play more effective roles in the calcium oscillation system.     

A topological quantum field theory construction on piecewise linear manifolds

The generalisation of the topological quantum field theory construction of Turaev and Viro to arbitrary dimension is presented, and it is shown that q-deformed spin-networks, or the recoupling theory of the quantum group U_qsl(2) provide a realisation of the initial data for the construction of 2-, 3- and 4-dimensional TQFTs.     

Kinetic theory of classical liquids. I. Basic theory

A somewhat new approach to a kinetic theory of classical liquids is presented, and it is used to calculate the dynamical structure factor S(qω). It gives correctly the zeroth, second, and fourth frequency moments of S(qω), and it goes correctly over to the free particle value for large q. For small q and ω it goes over to proper hydrodynamics, including the coupling to heat diffusion. Results of numerical calculations on liquid argon are presented and they show very good agreement with available neutron scattering and molecular dynamics data.