arXiv subjects

Hideo Hasegawa

Publications and source records attributed to Hideo Hasegawa.

At least 37 records · Page 2Linked to original sources

arXiv2010

Specific heat and entropy of $N$-body nonextensive systems

We have studied finite $N$-body $D$-dimensional nonextensive ideal gases and harmonic oscillators, by using the maximum-entropy methods with the $q$- and normal averages ($q$: the entropic index). The validity range, specific heat and Tsallis entropy obtained by the two average methods are compared. Validity ranges of the $q$- and normal averages are $0 < q < q_U$ and $q > q_L$, respectively, where $q_U=1+(ηDN)^{-1}$, $q_L=1-(ηDN+1)^{-1}$ and $η=1/2$ ($η=1$) for ideal gases (harmonic oscillators). The energy and specific heat in the $q$- and normal averages coincide with those in the Boltzmann-Gibbs statistics, % independently of $q$, although this coincidence does not hold for the fluctuation of energy. The Tsallis entropy for $N |q-1| \gg 1$ obtained by the $q$-average is quite different from that derived by the normal average, despite a fairly good agreement of the two results for $|q-1 | \ll 1$. It has been pointed out that first-principles approaches previously proposed in the superstatistics yield $additive$ $N$-body entropy ($S^{(N)}= N S^{(1)}$) which is in contrast with the $nonadditive$ Tsallis entropy.

cond-mat.stat-mech↗

arXiv2010

The entropy in finite $N$-unit nonextensive systems: the ordinary average and $q$-average

We have discussed the Tsallis entropy in finite $N$-unit nonextensive systems, by using the multivariate $q$-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the $q$-average ($q$: the entropic index). The Tsallis entropy obtained by the $q$-average has an exponential $N$ dependence: $S_q^{(N)}/N \simeq \:e^{(1-q)N \:S_1^{(1)}} $ for large $N$ ($\gg \frac{1}{(1-q)} >0$). In contrast, the Tsallis entropy obtained by the normal average is given by $S_q^{(N)}/N \simeq [1/(q-1)N]$ for large $N$ ($\gg \frac{1}{(q-1)} > 0)$. $N$ dependences of the Tsallis entropy obtained by the $q$- and normal averages are generally quite different, although the both results are in fairly good agreement for $\vert q-1 \vert \ll 1.0$. The validity of the factorization approximation to PDFs which has been commonly adopted in the literature, has been examined. We have calculated correlations defined by $C_m= \langle (δx_i \:δx_j)^m \rangle -\langle (δx_i)^m \rangle\: \langle (δx_j)^m \rangle$ for $i \neq j$ where $δx_i=x_i -\langle x_i \rangle$, and the bracket $\langle \cdot \rangle$ stands for the normal and $q$-averages. The first-order correlation ($m=1$) expresses the intrinsic correlation and higher-order correlations with $m \geq 2$ include nonextensivity-induced correlation, whose physical origin is elucidated in the superstatistics.

cond-mat.stat-mech↗

arXiv2010

Synchronization induced by periodic inputs in finite $N$-unit bistable Langevin models: The augmented moment method

We have studied the synchronization induced by periodic inputs applied to the finite $N$-unit coupled bistable Langevin model which is subjected to cross-correlated additive and multiplicative noises. Effects on the synchronization of the system size ($N$), the coupling strength and the cross-correlation between additive and multiplicative noises have been investigated with the use of the semi-analytical augmented moment method (AMM) which is the second-order moment approximation for local and global variables [H. Hasegawa, Phys. Rev. E {\bf 67} (2003) 041903]. A linear analysis of the stationary solution of AMM equations shows that the stability is improved (degraded) by positive (negative) couplings. Results of the nonlinear bistable Langevin model are compared to those of the linear Langevin model.

cond-mat.dis-nn↗

arXiv2010

Nonextensive thermodynamics of the two-site Hubbard model: Canonical ensembles

Canonical ensembles consisting of $M$-unit $Hubbard$ $dimers$ have been studies within the nonextensive statistics (NES). The temperature dependences of the energy, entropy, specific heat and susceptibility have been calculated for the number of dimers, $M = 1, 2, 3$ and $\infty$. We have assumed the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2\:M$ for $M$ dimers), which was previously derived by several methods. For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have been adopted: $T=1/k_B β$ in the method A [Tsallis {\it et al.} Physica A {\bf 261}, 534 (1998)], and $T=c_q/k_B β$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281}, 126 (2001)], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, and $p_i$ the probability distribution of the $i$th state. The susceptibility and specific heat of spin dimers ({\it Heisenberg dimers}) described by the Heisenberg model have been discussed also by using the NES with the methods A and B. A comparison between the two methods suggests that the method B may be more reasonable than the method A for nonextensive systems.

cond-mat.stat-mech↗

arXiv2009

Validity of the factorization approximation and correlation induced by nonextensivity in $N$-unit independent systems

We have discussed the validity of the factorization approximation (FA) and nonextensivity-induced correlation, by using the multivariate $q$-Gaussian probability distribution function (PDF) for $N$-unit independent nonextensive systems. The Tsallis entropy is shown to be expressed by $S_q^{(N)} = S_{q,FA}^{(N)}+ ΔS_q^{(N)}$ where $q$ denotes the entropic index, $S_{q,FA}^{(N)}$ a contribution in the FA, and $ΔS_q^{(N)}$ a correction term. It is pointed out that the correction term of $ΔS_q^{(N)}$ is considerable for large $| q-1 |$ and/or large $N$ because the multivariate PDF cannot be expressed by the factorized form which is assumed in the FA. This implies that the pseudoadditivity of the Tsallis entropy, which is obtained with PDFs in the FA, does not hold although it is commonly postulated in the literatures. We have calculated correlations defined by $C_m= < (δx_i \:δx_j)^m >_q -< (δx_i)^m >_q\: < (δx_j)^m >_q$ for $i \neq j$, where $δx_i=x_i -< x_i >_q$ and $<\cdot >_q$ stands for $q$-average over the escort PDF. It has been shown that $C_1$ expresses the intrinsic correlation and that $C_m$ with $m \geq 2$ signifies correlation induced by nonextensivity whose physical origin is elucidated within the superstatistics. PDFs calculated for the classical ideal gas and harmonic oscillator are compared with the $q$-Gaussian PDF. A discussion on the $q$-product PDF is presented also.

cond-mat.stat-mech↗

arXiv2009

Generalized Fisher information matrix in nonextensive systems with spatial correlation

By using the $q$-Gaussian distribution derived by the maximum entropy method for spatially-correlated $N$-unit nonextensive systems, we have calculated the generalized Fisher information matrix of $g_{θ_n θ_m}$ for $(θ_1, θ_2, θ_3) = (μ_q, σ_q^2$, $s$), where $μ_q$, $σ_q^2$ and $s$ denote the mean, variance and degree of spatial correlation, respectively, for a given entropic index $q$. It has been shown from the Cramér-Rao theorem that (1) an accuracy of an unbiased estimate of $μ_q$ is improved (degraded) by a negative (positive) correlation $s$, (2) that of $σ_q^2$ is worsen with increasing $s$, and (3) that of $s$ is much improved for $s \simeq -1/(N-1)$ or $s \simeq 1.0$ though it is worst at $s = (N-2)/2(N-1)$. Our calculation provides a clear insight to the long-standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles. We discuss also a calculation of the $q$-Gaussian distribution, applying the superstatistics to the Langevin model subjected to spatially-correlated inputs.

cond-mat.stat-mech↗

arXiv2009

Generalized information entropies in nonextensive quantum systems: The interpolation approach

We discuss the generalized von Neumann (Tsallis) entropy and the generalized Fisher information (GFI) in nonextensive quantum systems, by using the interpolation approximation (IA) which has been shown to yield good results for the quantal distributions within $O(q-1)$ and in high- and low-temperature limits, $q$ being the entropic index [H. Hasegawa, Phys. Rev. E 80 (2009) 011126]. Three types of GFIs which have been proposed so far in the nonextensive statistics, are discussed from the viewpoint of their metric properties and the Cramér-Rao theorem. Numerical calculations of the $q$- and temperature-dependent Tsallis entropy and GFIs are performed for the electron band model and the Debye phonon model.

cond-mat.stat-mech↗

arXiv2009

The interpolation approach to nonextensive quantum systems

Recently it has been shown by the present author [H. Hasegawa, Phys. Rev. E (in press): arXiv:0904.2399] that the interpolation approximation (IA) to the generalized Bose-Einstein and Femi-Dirac distributions yields results in agreement with the exact ones within the $O(q-1)$ and in high- and low-temperature limits, where $(q-1)$ expresses the non-extensivity: the case of $q=1$ corresponding to the conventional quantal distributions. In this study, we have applied the generalized distributions in the IA to typical nonextensive subjects: (1) the black-body radiation, (2) the Bose-Einstein condensation, (3) the BCS superconductivity and (4) itinerant-electron (metallic) ferromagnetism. Effects of the non-extensivity on physical quantities in these nonextenisive quantum systems have been investigated. A critical comparison is made between results calculated by the IA and the factorization approximation (FA) which has been so far applied to many nonextensive systems. It has been pointed out that the FA overestimates the non-extensivity and that it leads to an inappropriate results for fermion systems like the subjects (3) and (4).

cond-mat.stat-mech↗

arXiv2009

Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches

Generalized Bose-Einstein (BE) and Fermi-Dirac (FD) distributions in nonextensive quantum statistics have been discussed by the maximum-entropy method (MEM) with the optimum Lagrange multiplier based on the exact integral representation [Rajagopal, Mendes, and Lenzi, Phys. Rev. Lett. {\bf 80}, 3907 (1998)]. It has been shown that the $(q-1)$ expansion in the exact approach agrees with the result obtained by the asymptotic approach valid for $O(q-1)$. Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. Based on the result of the exact approach, we have proposed the {\it interpolation approximation} to the generalized distributions, which yields results in agreement with the exact approach within $O(q-1)$ and in high- and low-temperature limits. By using the four methods of the exact, interpolation, factorization and superstatistical approaches, we have calculated coefficients in the generalized Sommerfeld expansion, and electronic and phonon specific heats at low temperatures. A comparison among the four methods has shown that the interpolation approximation is potentially useful in the nonextensive quantum statistics. Supplementary discussions have been made on the $(q-1)$ expansion of the generalized distributions based on the exact approach with the use of the un-normalized MEM, whose results also agree with those of the asymptotic approach.

cond-mat.stat-mech↗

arXiv2009

Nonextensive quantum method for itinerant-electron ferromagnetism: Fractorization approach

Magnetic and thermodynamical properties of itinerant-electron (metallic) ferromagnets described by the Hubbard model have been discussed with the use of the generalized Fermi-Dirac (GFD) distribution for nonextensive quantum systems. We have derived the GFD distribution within the superstatistics, which is equivalent to that obtained by the maximum-entropy method to the Tsallis entropy with the factorization approimation. By using the Hartree-Fock approximation to the electron-electron interaction in the Hubbard model, we have calculated magnetic moment, energy, specific heat and Curie-Weiss-type spin susceptibility, as functions of the temperature and entropic index $q$ expressing the degree of the nonextensivity: $q=1.0$ corresponds to the Boltzmann-Gibbs statistics. It has been shown that with increasing the nonextensivity of $| q-1 |$, the temperature dependence of magnetic moment becomes more significant and the low-temperature electronic specific heat is much increased. This is attributed to enlarged Stoner excitations in the GFD distribution, which is elucidated by an analysis with the use of the generalized Sommerfeld expansion. We discuss the difference and similarity between the effects of the nonextensivity on metallic and insulating ferromagnets.

cond-mat.stat-mech↗

arXiv2008

A moment approach to analytic time-dependent solutions of the Fokker-Planck equation with additive and multiplicative noise

An efficient method is presented as a means of an approximate, analytic time-dependent solution of the Fokker-Planck equation (FPE) for the Langevin model subjected to additive and multiplicative noise. We have assumed that the dynamical probability distribution function has the same structure as the exact stationary one and that its parameters are expressed in terms of first and second moments, whose equations of motion are determined by the FPE. Model calculations have shown that dynamical distributions in response to applied signal and force calculated by our moment method are in good agreement with those obtained by the partial difference equation method. As an application of our method, we present the time-dependent Fisher information for the inverse-gamma distribution which is realized in the FPE including multiplicative noise only.

cond-mat.stat-mech↗

arXiv2008

Population rate codes carried by mean, fluctuation and synchrony of neuronal firings

A population of firing neurons is expected to carry information not only by mean firing rate but also by fluctuation and synchrony among neurons. In order to examine this possibility, we have studied responses of neuronal ensembles to three kinds of inputs: mean-, fluctuation- and synchrony-driven inputs. The generalized rate-code model including additive and multiplicative noise (H. Hasegawa, Phys. Rev. E {\bf 75} (2007) 051904) has been studied by direct simulations (DSs) and the augmented moment method (AMM) in which equations of motion for mean firing rate, fluctuation and synchrony are derived. Results calculated by the AMM are in good agreement with those by DSs. The independent component analysis (ICA) of our results has shown that mean firing rate, fluctuation (or variability) and synchrony may carry independent information in the population rate-code model. The input-output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise, as recently observed in prefrontal cortex. A comparison is made between results obtained by the integrate-and-fire (IF) model and our rate-code model.

q-bio.NC↗

arXiv2008

Dynamics of Information Entropies in Nonextensive Systems

We have discussed dynamical properties of the Tsallis entropy and the generalized Fisher information in nonextensive systems described by the Langevin model subjected to additive and multiplicative noise. Analytical expressions for the time-dependent Tsallis entropy and generalized Fisher information have been obtained with the use of the $q$-moment approach to the Fokker-Planck equation developed in a previous study [H. Hasegawa, Phys. Rev. E {\bf 77}, 031133 (2008)]. Model calculations of the information entropies in response to an applied pulse and sinusoidal inputs have been presented.

cond-mat.stat-mech↗

arXiv2008

Synchrony and variability induced by spatially correlated additive and multiplicative noise in the coupled Langevin model

The synchrony and variability have been discussed of the coupled Langevin model subjected to spatially correlated additive and multiplicative noise. We have employed numerical simulations and the analytical augmented-moment method which is the second-order moment method for local and global variables [H. Hasegawa, Phys. Rev. E {\bf 67}, 041903 (2003)]. It has been shown that the synchrony of an ensemble is increased (decreased) by a positive (negative) spatial correlation in both additive and multiplicative noise. Although the variability for local fluctuations is almost insensitive to spatial correlations, that for global fluctuations is increased (decreased) by positive (negative) correlations. When a pulse input is applied, the synchrony is increased for the correlated multiplicative noise, whereas it may be decreased for correlated additive noise coexisting with uncorrelated multiplicative noise. An application of our study to neuron ensembles has demonstrated the possibility that information is conveyed by the variance and synchrony in input signals, which accounts for some neuronal experiments.

cond-mat.dis-nn↗

arXiv2008

Effects of correlated variability on information entropies in nonextensive systems

We have calculated the Tsallis entropy and Fisher information matrix (entropy) of spatially-correlated nonextensive systems, by using an analytic non-Gaussian distribution obtained by the maximum entropy method. Effects of the correlated variability on the Fisher information matrix are shown to be different from those on the Tsallis entropy. The Fisher information is increased (decreased) by a positive (negative) correlation, whereas the Tsallis entropy is decreased with increasing an absolute magnitude of the correlation independently of its sign. This fact arises from the difference in their characteristics. It implies from the Cramér-Rao inequality that the accuracy of unbiased estimate of fluctuation is improved by the negative correlation. A critical comparison is made between the present study and previous ones employing the Gaussian approximation for the correlated variability due to multiplicative noise.

cond-mat.stat-mech↗

arXiv2008

Stationary and dynamical properties of information entropies in nonextensive systems

The Tsallis entropy and Fisher information entropy (matrix) are very important quantities expressing information measures in nonextensive systems. Stationary and dynamical properties of the information entropies have been investigated in the $N$-unit coupled Langevin model subjected to additive and multiplicative white noise, which is one of typical nonextensive systems. We have made detailed, analytical and numerical study on the dependence of the stationary-state entropies on additive and multiplicative noise, external inputs, couplings and number of constitutive elements ($N$). By solving the Fokker-Planck equation (FPE) by both the proposed analytical scheme and the partial difference-equation method, transient responses of the information entropies to an input signal and an external force have been investigated. We have calculated the information entropies also with the use of the probability distribution derived by the maximum-entropy method (MEM), whose result is compared to that obtained by the FPE. The Cramér-Rao inequality is shown to be expressed by the {\it extended} Fisher entropy, which is different from the {\it generalized} Fisher entropy obtained from the generalized Kullback-Leibler divergence in conformity with the Tsallis entropy. The effect of additive and multiplicative {\it colored} noise on information entropies is discussed also.

cond-mat.stat-mech↗

arXiv2008

Dynamics of the Langevin model subjected to colored noise: Functional-integral method

We have discussed the dynamics of Langevin model subjected to colored noise, by using the functional-integral method (FIM) combined with equations of motion for mean and variance of the state variable. Two sets of colored noise have been investigated: (a) one additive and one multiplicative colored noise, and (b) one additive and two multiplicative colored noise. The case (b) is examined with the relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate-and-fire model including stochastic excitatory and inhibitory synapses and a noisy input. We have studied the stationary probability distribution and dynamical responses to time-dependent (pulse and sinusoidal) inputs of the linear Langevin model. Model calculations have shown that results of the FIM are in good agreement with those of direct simulations (DSs). A comparison is made among various approximate analytic solutions such as the universal colored noise approximation (UCNA). It has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the UCNA are rather different from those of DS and the FIM, although they yield the same stationary distribution.

cond-mat.stat-mech↗

arXiv2007

Stochastic bifurcation of FitzHugh-Nagumo ensembles subjected to additive and/or multiplicative noises

We have studied the dynamical properties of finite $N$-unit FitzHugh-Nagumo (FN) ensembles subjected to additive and/or multiplicative noises, reformulating the augmented moment method (AMM) with the Fokker-Planck equation (FPE) method [H. Hasegawa, J. Phys. Soc. Jpn. {\bf 75}, 033001 (2006)]. In the AMM, original $2N$-dimensional stochastic equations are transformed to eight-dimensional deterministic ones, and the dynamics is described in terms of averages and fluctuations of local and global variables. The stochastic bifurcation is discussed by a linear stability analysis of the {\it deterministic} AMM equations. The bifurcation transition diagram of multiplicative noise is rather different from that of additive noise: the former has the wider oscillating region than the latter. The synchronization in globally coupled FN ensembles is also investigated. Results of the AMM are in good agreement with those of direct simulations (DSs).

cond-mat.stat-mech↗