arXiv subjects

Masamichi Ishihara

Publications and source records attributed to Masamichi Ishihara.

At least 19 recordsLinked to original sources

arXiv2026

Momentum distribution and correlation function of free particles in the Tsallis statistics using conventional expectation value and equilibrium temperature

We applied the Tsallis statistics with the conventional expectation value to a system of free particles, adopting the equilibrium temperature, which is often called the physical temperature. The entropic parameter $q$ in the Tsallis statistics is less than one for power-law-like distribution. The well-known relation between the energy and the temperature in the Boltzmann--Gibbs statistics holds in the Tsallis statistics, when the equilibrium temperature is adopted. We derived the momentum distribution and the correlation function in the Tsallis statistics. The momentum distribution and the correlation function in the Tsallis statistics are different from those in the Boltzmann--Gibbs statistics, even when the equilibrium temperature is adopted. These quantities depend on $q$ and $N$, where $N$ is the number of particles. It is also shown that the parameter $q$ is required to satisfy $1-1/(3N/2+1) < q < 1$.

cond-mat.stat-mech↗

arXiv2025

Description using equilibrium temperature in the canonical ensemble within the framework of the Tsallis statistics employing the conventional expectation value

We studied the thermodynamic quantities and the probability distribution, expressing the probability distribution as a function of the energy, in the canonical ensemble within the framework of the Tsallis statistics, which is characterized by the entropic parameter $q$, employing the conventional expectation value (the linear average). We treated the power-law-like distribution. The equilibrium temperature, which is often called the physical temperature, was employed, and the probability distribution described with the equilibrium temperature was derived. The Tsallis statistics represented by the equilibrium temperature was applied to $N$ harmonic oscillators, where $N$ is the number of the oscillators. The expressions of the energy, the Tsallis entropy, and the heat capacity were obtained. The expressions of these quantities and the expression of the probability distribution were obtained when the differences between adjacent energy levels are the same. These quantities and the distributions were numerically calculated. The $q$ dependences of the energy, the Rényi entropy, and the heat capacity are weak. In contrast, the Tsallis entropy depends on $q$. The probability distribution as a function of the energy depends on $N$ and $q$. The results provide a basis for describing power-law-like phenomena in the Tsallis statistics. The present formulation is expected to apply to various phenomena, because the harmonic oscillator plays a fundamental role in describing classical and quantum systems.

cond-mat.stat-mech↗

arXiv2024

Multiple quantum harmonic oscillators in the Tsallis statistics

We studied multiple quantum harmonic oscillators in the Tsallis statistics of entropic parameter $q$ in the cases that the distributions are power-like, separately applying the conventional expectation value, the unnormalized $q$-expectation value, and the normalized $q$-expectation value (escort average). We obtained the expressions of the energy and the Tsallis entropy, using the Barnes zeta function. For the same oscillators, we obtained the expressions of the energy, the Tsallis entropy, the average level of the oscillators, and the heat capacity. Numerically, we calculated the energy, the Tsallis entropy, and the heat capacity for various $N$ and $q$, using the expansion of the Barnes zeta function with the Hurwitz zeta function, where $N$ is the number of independent oscillators. The parameter $q$ is less than one in the Tsallis statistics with the conventional expectation value. The parameter $q$ is greater than one in both sets of the Tsallis statistics, each of which is defined with a different $q$-expectation value. These limitations of $q$ arise from the requirements that the distributions are power-like. It was shown from the requirements for the Barnes zeta function that $q$ is greater than $N/(N+1)$ for the conventional expectation value and that $q$ is less than $(N+1)/N$ for both of the $q$-expectation values. In the Tsallis statistics with the conventional expectation value, the energy, the Tsallis entropy, and the heat capacity decrease with $q$. These quantities per oscillator increase with $N$. In the Tsallis statistics with the unnormalized $q$-expectation value, the energy, the Tsallis entropy, and the heat capacity increase with $q$ at low temperature, while decrease with $q$ at high temperature. These quantities per oscillator increase with $N$ at low temperature, while decrease with $N$ at high temperature. The heat capacity is the Schottky-type. The quantities are affected by the zero-point energy. In the Tsallis statistics with the normalized $q$-expectation value, the $N$ dependence of the energy per oscillator and that of the heat capacity per oscillator are quite weak, and the $q$ dependence of the energy and that of the heat capacity are also weak, when the equilibrium temperature, which is often called the physical temperature, is adopted. The Tsallis entropy per oscillator decreases with $N$ and the Tsallis entropy decreases with $q$.

cond-mat.stat-mech↗

arXiv2023

Relation between the escort average in microcanonical ensemble and the escort average in canonical ensemble in the Tsallis statistics

We studied the escort averages in microcanonical and canonical ensembles in the Tsallis statistics of entropic parameter $q>1$. The quantity $(q-1)$ is the measure of the deviation from the Boltzmann-Gibbs statistics. We derived the relation between the escort average in the microcanonical ensemble and the escort average in the canonical ensemble. Conditions arise by requiring that the integrals appeared in the canonical ensemble do not diverge. A condition is the relation between the heat capacity $C_V^{\mathrm{CE}}$ at constant volume in the canonical ensemble and the entropic parameter $q$: $0 < (q-1) C_V^{\mathrm{CE}} < 1$. This condition gives the known condition when $C_V^{\mathrm{CE}}$ equals the number of ingredients $N$. With the derived relation, we calculated the energy, the energy fluctuation, and the difference between the canonical ensemble and the microcanonical ensemble in the expectation value of the square of Hamiltonian. The difference between the microcanonical ensemble and the canonical ensemble in energy is small because of the condition. The heat capacity $C_V^{\mathrm{CE}}$ and the quantity $(q-1)$ are related to the energy fluctuation and the difference. It was shown that the magnitude of the relative difference $|(S^{\mathrm{CE}}_{\mathrm{R}q}-S^{\mathrm{ME}}_{\mathrm{R}q})/S^{\mathrm{ME}}_{\mathrm{R}q}|$ is small when the number of free particles is large, where $S^{\mathrm{ME}}_{\mathrm{R}q}$ is the Rényi entropy in the microcanonical ensemble and $S^{\mathrm{CE}}_{\mathrm{R}q}$ is the Rényi entropy in the canonical ensemble. The similar result was also obtained for the Tsallis entropy.

cond-mat.stat-mech↗

arXiv2023

Thermodynamic relations and fluctuations in the Tsallis statistics

The thermodynamic relations in the Tsallis statistics were studied with physical quantities. An additive entropic variable related to the Tsallis entropy was introduced by assuming the form of the first law of the thermodynamics. The fluctuations in the Tsallis statistics were derived with physical quantities with the help of the introduced entropic variable. It was shown that the mean squares of the fluctuations of the physical quantities in the Tsallis statistics are the same as those in the conventional statistics. The mean square of the fluctuation of the Tsallis entropy and the mean square of the fluctuation of the Tsallis temperature were also derived. The mean square of the relative fluctuation of the Tsallis entropy and the mean square of the relative fluctuation of the Tsallis temperature are represented with heat capacities. It was shown that these fluctuations of the Tsallis quantities have the $q$-dependent terms in the Tsallis statistics of the entropic parameter $q$.

cond-mat.stat-mech↗

arXiv2022

Thermodynamic quantities of independent harmonic oscillators in microcanonical and canonical ensembles in the Tsallis statistics

We study the energy and entropies for $N$ independent harmonic oscillators in the microcanonical and the canonical ensembles in the Tsallis classical and the Tsallis quantum statistics of entropic parameter $q$, where $N$ is the number of the oscillators and the value of $q$ is larger than one. The energy and entropies are represented with the physical temperature, and the well-known expressions are obtained for the energy and Rényi entropy. The difference between the microcanonical and the canonical ensembles is the existence of the condition for $N$ and $q$ in the canonical ensemble: $N(q-1)<1$. The condition does not appear in the microcanonical ensemble. The entropies are $q$-dependent in the canonical ensemble, and are not $q$-dependent in the microcanonical ensemble. For $N(q-1)<1$, this difference in entropy is quite small, and the entropy in the canonical ensemble does not differ from the entropy in the microcanonical ensemble substantially.

cond-mat.stat-mech↗

arXiv2021

Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics

We study the thermodynamic quantities in the system of the $N$ independent harmonic oscillators with different frequencies in the Tsallis statistics of the entropic parameter $q$ ($1<q<2$) with escort average. The self-consistent equation is derived, and the physical quantities are calculated with the physical temperature. It is found that the number of oscillators is restricted below $1/(q-1)$. The energy, the Rényi entropy, and the Tsallis entropy are obtained by solving the self-consistent equation approximately at high physical temperature and/or for small deviation $q-1$. The energy is $q$-independent at high physical temperature when the physical temperature is adopted, and the energy is proportional to the number of oscillators and physical temperature at high physical temperature. The form of the Rényi entropy is similar to that of von-Neumann entropy, and the Tsallis entropy is given through the Rényi entropy. The physical temperature dependence of the Tsallis entropy is different from that of Rényi entropy. The Tsallis entropy is bounded from the above, while the Rényi entropy increases with the physical temperature. The ratio of the Tsallis entropy to the Rényi entropy is small at high physical temperature.

cond-mat.stat-mech↗

arXiv2020

Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

We derived the equation of the density operator for generalized entropy and generalized expectation value with quantum analysis when conserved quantities exist. The derived equation is simplified when the conventional expectation value is employed. The derived equation is also simplified when the commutation relations, $[\hatρ, \hat{H}]$ and $[\hatρ, \hat{Q}^{[a]}]$, are the functions of the density operator $\hatρ$, where $\hat{H}$ is the Hamiltonian, and $\hat{Q}^{[a]}$ is the conserved quantity. We derived the density operators for the von Neumann entropy, the Tsallis entropy, and the Rényi entropy in the case of the conventional expectation value. We also derived the density operators for the Tsallis entropy and the Rényi entropy in the case of the escort average (the normalized $q$-expectation value), when the density operator commutes with the Hamiltonian and the conserved quantities. We found that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in the case of the conventional expectation value, even when conserved quantities do not commute. The simple extension of the argument is also shown in the case of the escort average, when the density operator $\hatρ$ commutes with the Hamiltonian $\hat{H}$ and the conserved quantity $\hat{Q}^{[a]}$: $[\hatρ, \hat{H}] = [\hatρ, \hat{Q}^{[a]}]=0$. These findings imply that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in some systems.

cond-mat.stat-mech↗

arXiv2019

Chiral phase transition in the linear sigma model within Hartree factorization in the Tsallis nonextensive statistics

We studied chiral phase transition in the linear sigma model within the Tsallis nonextensive statistics in the case of small deviation from the Boltzmann-Gibbs (BG) statistics. The statistics has two parameters: the temperature $T$ and the entropic parameter $q$. The normalized $q$-expectation value and the physical temperature $\Tph$ were employed in this study. The normalized $q$-expectation value was expanded as a series of the value $(1-q)$, where the absolute value $|1-q|$ is the measure of the deviation from the BG statistics. We applied the Hartree factorization and the free particle approximation, and obtained the equations for the condensate, the sigma mass, and the pion mass. The physical temperature dependences of these quantities were obtained numerically. We found following facts. The condensate at $q$ is smaller than that at $q'$ for $q>q'$. The sigma mass at $q$ is lighter than that at $q'$ for $q>q'$ at low physical temperature, and the sigma mass at $q$ is heavier than that at $q'$ for $q>q'$ at high physical temperature. The pion mass at $q$ is heavier than that at $q'$ for $q>q'$. The difference between the pion masses at different values of $q$ is small for $\Tph \le 200$ MeV. That is to say, the condensate and the sigma mass are affected by the Tsallis nonextensive statistics of small $|1-q|$, and the pion mass is also affected by the statistics of small $|1-q|$ except for $\Tph \le 200$ MeV.

hep-ph↗

arXiv2019

Derivation of density operators for generalized entropies with quantum analysis

We gave a simple derivation of density operator with the quantum analysis. We dealt with the functional of a density operator, and applied maximum entropy principle. We obtained easily the density operators for the Tsallis entropy and Rényi entropy with the $q$-expectation value (escort average), and also obtained easily the density operators for the Boltzmann-Gibbs entropy and the Burg entropy with the conventional expectation value. The quantum analysis works effectively in the calculation of the variation of the functional which includes trace.

cond-mat.stat-mech↗

arXiv2019

Chiral phase transition within the linear sigma model in the Tsallis nonextensive statistics based on density operator

We studied the chiral phase transition for small $|1-q|$ within the Tsallis nonextensive statistics of the entropic parameter $q$, where the quantity $|1-q|$ is the measure of the deviation from the Boltzmann-Gibbs statistics. We adopted the normalized $q$-expectation value in this study. We applied the free particle approximation and the massless approximation in the calculations of the expectation values. We estimated the critical physical temperature, and obtained the chiral condensate, the sigma mass, and the pion mass, as functions of the physical temperature $T_{\mathrm{ph}}$ for various $q$. We found the following facts. The $q$-dependence of the critical physical temperature is $1/\sqrt{q}$. The chiral condensate at $q$ is smaller than that at $q'$ for $q>q'$. The $q$-dependence of the pion mass and that of the sigma mass reflect the $q$-dependence of the condensate. The pion mass at $q$ is heavier than that at $q'$ for $q>q'$. The sigma mass at $q$ is heavier than that at $q'$ for $q>q'$ at high physical temperature, while the sigma mass at $q$ is lighter than that at $q'$ for $q>q'$ at low physical temperature. The quantities which are functions of the physical temperature $T_{\mathrm{ph}}$ and the entropic parameter $q$ are described by only the effective physical temperature defined as $\sqrt{q} T_{\mathrm{ph}}$ under the approximations.

hep-ph↗

arXiv2018

Momentum distribution and correlation for a free scalar field in the Tsallis nonextensive statistics based on density operator

We derived the expression of the normalized $q$-expectation value based on the density operator to the order $1-q$ with the physical temperature in the Tsallis nonextensive statistics of entropic parameter $q$. With the derived expression of the normalized $q$-expectation value, we calculated the momentum distribution and the correlation to the order $1-q$ as functions of the inverse physical temperature for a free scalar field. To the order $1-q$, the momentum distribution derived by using the density operator coincides with the momentum distribution derived from the entropic measure described with the distribution, when the physical temperature equals the temperature in the distribution derived from the entropic measure. The correlation depends on the momentums for $q \neq 1$. The factor two appears in the correlation for the same momentums, and indicates that the effects of boson at $q \neq 1$ and those at $q=1$ are similar for the correlation.

cond-mat.stat-mech↗

arXiv2017

Phase transition for the system of small volume in the $ϕ^4$ theory in the Tsallis nonextensive statistics

We studied the effects of the nonextensivity on the phase transition for the system of small volume $V$ in the $ϕ^4$ theory in the Tsallis nonextensive statistics of entropic parameter $q$ and temperature $T$, when the deviation from the Boltzmann-Gibbs statistics, $|q-1|$, is small. We calculated the condensate and the mass to the order $q-1$ with the normalized $q$-expectation value under the massless free particle approximation. The following facts were found. The condensate $Φ$ divided by $v$, $Φ/v$, at $q$ is smaller than that at $q'$ for $q>q'$ as a function of $T_{\mathrm{ph}}/v$ which is the physical temperature $T_{\mathrm{ph}}$ divided by $v$, where $T_{\mathrm{ph}}$ at $q=1$ coincides with $T$ and $v$ is the value of the condensate at $T=0$. The mass decreases, reaches minimum, and increases after that, as $T_{\mathrm{ph}}$ increases. The mass at $q>1$ is lighter than the mass at $q=1$ at low physical temperature and heavier than the mass at $q=1$ at high physical temperature. The effects of the nonentensivity on the physical quantity as a function of $T_{\mathrm{ph}}$ become strong as $|q-1|$ increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.

cond-mat.stat-mech↗

arXiv2017

Transverse momentum fluctuation under the Tsallis distribution at high energies

We studied the effects of the Tsallis distribution on the transverse momentum fluctuation in high energy collisions. The parton-hadron duality and the Bose-Einstein type correlation between partons were assumed. The fluctuation was calculated in the boost-invariant picture for the expectation value used in the Boltzmann-Gibbs statistics and for the expectation value used in the Tsallis nonextensive statistics. It was shown that the fluctuation is a function of $η$ which is the ratio of the inverse temperature to the correlation length. We found the following points: (1) the fluctuation depends on the form of the distribution and depends weakly on the definition of the expectation value used in the statistics,(2) the fluctuation increases as the entropic parameter value of the Tsallis distribution increases, and (3) the variation of the fluctuation as a function of the entropic parameter for the expectation value used in the Boltzmann-Gibbs statistics is larger than that for the expectation value used in the Tsallis nonextensive statistics in the wide range of $η$.

hep-ph↗

arXiv2017

Event-by-event mean $p_{\rm T}$ fluctuations and transverse size of color flux tube generated in $p$-$p$ collisions at $\sqrt{s}$=0.90TeV

We propose a novel phenomenological model of mean transverse momentum fluctuations based on the Geometrical Scaling hypothesis. Bose-Einstein correlations between two gluons generated from an identical color flux tube are taken into account as a source of the fluctuation. We calculate an event-by-event fluctuation measure $\sqrt{C_m}/\langle p_{\rm T}\rangle$ and show that ALICE data observed at $\sqrt s=$0.90 TeV for $p$+$p$ collisions are reproduced. By fitting our model to the experimental data, we evaluate the transverse size of the color flux tube as a function of the multiplicity.

hep-ph↗

arXiv2017

Momentum Distribution and Correlation due to mass difference caused by power-like distribution

The momentum distribution and particle correlation due to the mass difference were studied both in the case of the conventional expectation value and in the case of $q$-expectation value, when the momentum distribution is described by a Tsallis distribution with the entropic parameter $q \ge 1$. The magnitude of the momentum distribution for hard modes increases as $q$ increases, and the $q$-dependence of the momentum distribution is quite weak for soft modes. The correlation at $q>1$ is larger than that at $q=1$ for soft modes, while the correlation at $q>1$ is smaller than that at $q=1$ for hard modes. The $q$-dependence of these quantities in the case of $q$-expectation value is weaker than that in the case of the conventional expectation value, respectively.

hep-ph↗

arXiv2016

Chiral phase transitions in the linear sigma model in the Tsallis nonextensive statistics

We studied chiral phase transitions in the Tsallis nonextensive statistics which has two parameters, the temperature $T$ and entropic parameter $q$. The linear sigma model was used in this study. The critical temperature, condensate, masses, and energy density were calculated under the massless free particle approximation. The critical temperature decreases as $q$ increases. The condensate at $q>1$ is smaller than that at $q=1$. The sigma mass at $q>1$ is heavier than the mass at $q=1$ at high temperature, while the sigma mass at $q>1$ is lighter than the mass at $q=1$ at low temperature. The pion mass at $q>1$ is heavier than the mass at $q=1$. The energy density increases remarkably as $q$ increases. The $q$ dependence in the case of the $q$-expectation value is weaker than that in the case of the conventional expectation value with a Tsallis distribution. The parameter $q$ should be smaller than $4/3$ from energetic point of view. The validity of the Tsallis statistics can be determined by the difference in $q$ of the restriction for $5/4 < q < 4/3$ when the interaction is weak, because the parameter $q$ is smaller than $5/4$ in the case of the conventional expectation value with a Tsallis distribution.

hep-ph↗

arXiv2016

Effects of Tsallis distribution on parametric resonance in chiral phase transitions

The parametric resonance was studied in chiral phase transitions when the momentum distribution is described by a Tsallis distribution. A Tsallis distribution has two parameters, the temperature $T$ and the entropic index $q$. The amplification was estimated in two cases: 1) expansionless case and 2) one dimensional expansion case. In an expansionless case, the temperature $T$ is constant, and the amplified modes as a function of $T$ were calculated for various $q$. In one dimensional expansion case, the temperature $T$ decreases as a function of the proper time, and the amplification as a function of the transverse momentum was calculated for various $q$. In the expansionless case, the following facts were found: 1) the larger the value $q$ is, the softer the amplified modes are for the first and second resonance bands, 2) the amplified mode of the first resonance band decreases and vanishes, as the temperature $T$ increases, and 3) the amplified mode of the second resonance band decreases and approaches to zero, as the temperature $T$ increases. In one dimensional expansion case, the following facts were found: 1) the soft mode is amplified, 2) the amplification is extremely strong around the amplified mode of the first resonance band at $T=0$, and 3) the magnitude of the amplification as a function of transverse momentum oscillates around the amplified mode of the first resonance band at $T=0$.

hep-ph↗