Critical Angular Momentum Research Articles

Dynamics and collision of particles in modified black-bounce geometry

In the present work, we first regularize a black hole spacetime in modified gravity (MOG) in the presence of the scalar-tensor-vector (STV) field, called the Schwarzschild MOG black hole, under the transformation r2→r2+a2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r^2 \\rightarrow r^2 + a^2$$\\end{document}, known as the Simpson–Visser (SV) spacetime (where a is regularization or black-bounce parameter). The spacetime can represent a black hole and a wormhole. We analyze horizon properties and calculate the effective mass of the spacetime. Also, we find black hole-wormhole regions in black-bounce and MOG parameter spacetime. We also analyze scalar invariants of spacetime, such as the Ricci scalar, the square of the Ricci tensor, and the Kretchmann scalar. We study test particle motion in the SV-MOG spacetime by considering the interaction between the particle and the STV field. We investigate how the STV fields change the innermost stable circular orbits (ISCOs), energy, and angular momentum of the test particle’s ISCO. It is shown that the ISCO decreases in the presence of the black bounce parameter and increases in the STV field. We also study the collisions of test particles and analyze how the MOG and black-bounce parameters influence the critical angular momentum of colliding particles and their center of mass energy.

Circular motion and collisions of spinning test particles around Kerr–Kiselev black holes

Exploring the effect of exotic fields around black holes on particle dynamics may help to understand the nature of dark matter and energy. The quintessential field can be treated as one of such fields. In this work, we investigate the motion of spinning particles in the vicinity of rotating black holes immersed in quintessential dark energy, characterized by the equation of state (EoS) parameter ω∈(−1;−1/3) governing the equation of state of the dark energy and dimensionless quintessential field parameter C. Using the Mathisson–Papapetrou–Dixon (MPD) equations, we derive the effective potential and study superluminal bound values for the particle spin. Also, we investigate the behaviors of the innermost and outermost stable circular orbit (ISCO & OSCO) of spinning test particles and their energy and angular momentum at the orbits. Note that the OSCO exists due to the third cosmological-like horizon caused by the quintessential field and shows that the ISCO and OSCO coincide at critical values of the quintessential field and EoS parameters, which also depend on the particle and black hole spin. Finally, we explore collisions of spinning particles and analyze the center-of-mass energies and critical angular momentum, which allows the collisions of the particles near the black hole.

Charged particles and quasiperiodic oscillations in Black–bounce–Reissner–Nordström geometry in braneworlds

We study the dynamics and oscillations of electrically charged particles along stable circular orbits around a nonrotating electrically charged spacetime. In this work, we consider a regularized Reissner–Nordström (RN) spacetime under the transformation r2→r2+l2, known as the black-bounce-RN (or Simpson–Visser RN) spacetime, which can represent both a black hole and a wormhole depending on the values of l, in the presence of braneworld effects. In our analyses, we introduce a new parameter that shows a similar gravitational effect, which includes electric (Q) and tidal charges (B) of the spacetime as b=Q2+B. We investigate the horizon properties of the spacetime and estimate the parametric distributions that describe a black hole and a wormhole. Furthermore, we analyze the effective potential of the particles and the critical angular momentum for the cases dominated by the tidal charge b<0 and the electric charge b>0, respectively. Moreover, we apply the epicyclic frequencies in the relativistic precession model to fit twin high-frequency (HF) quasiperiodic oscillations (QPOs) observed in microquasars and active galactic nuclei. Finally, we apply a Monte Carlo Markov Chain (MCMC) simulation to constrain the multidimensional parameter space for the microquasars GRO J1655-40 & GRS 1915-105 and the galactic center, for being a black hole and wormhole candidate using observational data from QPOs. Our findings suggest that the central object of the Milky Way and the microquasars GRS 1915-105 may be a black hole and a wormhole with certain parameters. However, there are no constraints for the microquasar GRO J1655-40 obtained regarding the wormhole case. It implies that the central object fails to be a candidate for being a wormhole in the black-bounce charged spacetimes in the braneworlds model.

Circular motion and collisions of particles with magnetic dipole moment and electric charge in dipolar magnetosphere around Schwarzschild black holes

No-hair theorem indicates that black holes cannot have their own magnetic dipole moment. They can be weakly magnetized in binary systems with a neutron star companion and an accretion disc of charged particles. A simple model suggested by Petterson states that a current loop accreting a Schwarzschild black hole generates dipole-like magnetic fields in the outer region of the loop that are uniform in the inner region. This study considers circular motion and collisions of charged test particles with magnetic dipole moments in the inner and outer regions. First, we derive the effective potential taking into account the magnetic interactions between external magnetic fields with electric charge and the magnetic dipole moment of the particle. We investigate the possible innermost stable circular orbits (ISCOs) of the charged and magnetized particles orbiting the magnetized Schwarzschild black hole inside and outside the current loop. Finally, we explore the collisional processes of these particles near the black hole horizons, examining the effects of magnetic interactions on the critical angular momentum of particles that may collide and the center of mass energy of the colliding particles. We discuss astrophysical relevant objects with magnetic dipole moment and electric charge: magnetized neutron stars, white dwarfs, rotating stellar-mass black holes, electrons, and protons, and also estimate the interaction parameters for them.

Particle acceleration near a rotating charged black hole in 4D Einstein-Gauss-Bonnet gravity

We investigate the horizon structure and ergosphere in a charged rotating black hole within 4D Einstein-Gauss-Bonnet gravity, which introduces additional parameters (Q) because of the charge and Gauss-Bonnet parameter (β), besides the mass (M) and rotation parameter (a). Interestingly, for each value of the parameter Q (β), there is a critical GB parameter β=βE (Q=QE) that corresponds to an extremal black hole with degenerate horizons. For β<βE (Q<QE), it describes a non-extremal black hole with two horizons, and for β>βE (Q>QE), no black hole exists. The extremal value βE (QE) is also affected by the GB parameter α and the ergosphere. We also study the collision of two equal-mass particles near the horizon of this black hole and explicitly show the effect of the parameter β (Q). The innermost stable circular orbits (ISCO) and the effective potential, which governs the motion of particles in spacetime, have been analyzed for different parameter values. The centre-of-mass energy (ECM) depends on the rotation parameter a and the parameters β and Q. We investigate the ECM of two colliding particles near the horizon for both extremal and non-extremal cases. It is shown that in extremal cases, when one of the colliding particles has a critical angular momentum, the ECM can be arbitrarily high, suggesting that the charged rotating in 4D Einstein-Gauss-Bonnet gravity can function as a particle accelerator. Despite the complexity of the BH solution, an exact expression for the thermodynamic quantities of black holes, such as the mass, Hawking temperature, and entropy, is derived in terms of the horizon radius. These quantities show significant deviations from the Kerr solution because of the influence of the Gauss-Bonnet parameter and electric charge.

Collisions and dynamics of particles with magnetic dipole moment and electric charge near magnetized rotating Kerr black holes

Analysis of test magnetized and charged particles around black holes immersed in external magnetic fields may help to explain the observed astrophysical phenomena related to black holes, such as the acceleration of particles up to high energies. In this sense, we studied the circular motion of test-charged particles with magnetic dipole orbiting around magnetized rotating Kerr black holes. First, we derive the effective potential for the circular motion of such particles, including interactions between the external magnetic field and the electric charge, and the magnetic interaction between the magnetic dipole. In addition, we analyze the angular momentum and energy of particles corresponding to circular orbits. The effects of magnetic interaction and coupling parameters on the position of innermost stable circular orbits (ISCOs), the energy and angular momentum of the particles at ISCO, and the energy efficiency from the Novikov-Thorne accretion disc have been investigated. We also find cases of degeneracy between magnetic dipole interaction and magnetic coupling parameters, giving the same ISCO radius. Finally, we studied various cases of collisions of neutral, magnetized, and electrically charged particles near rotating Kerr black holes in the presence of external magnetic fields. The critical angular momentum of spinning charged particles is found in which the particles can collide. We also analyze the effects of both magnetic interactions on the center-of-mass energy of the colliding particles.

Rotating Bardeen black hole surrounded by perfect fluid dark matter as a particle accelerator

We study the event horizon of a rotating Bardeen black hole surrounded by perfect fluid dark matter and the black hole as a particle accelerator. The black hole is represented by four parameters: mass M, rotation parameter a, dark matter parameter α and magnetic charge g. It is interesting that when we determine the values of magnetic charge g and dark matter parameters α we can get a critical rotation parameter a E and then we get a contour plane with Δ = 0 taking three parameters as coordinates. We also derive the effective potential of the particle and the center-of-mass (CM) energy of the two particles outside the black hole by using the motion equations of the particle in the equatorial plane of the black hole. We find that the CM energy depends not only on the rotation parameter a, but also on the parameters g and α. We discuss the CM energy for two particles colliding at the black hole horizon in the extreme and non-extreme cases, respectively. It is found that the CM energy can become arbitrarily high when the angular momentum of one of the two particles is the critical angular momentum under the background of extreme black holes and there is no such result for non-extreme black holes, because the particles do not reach the black hole horizon when the angular momentum of the particles is critical angular momentum. Therefore, we prove that the rotating Bardeen black hole surrounded by perfect fluid dark matter can be used as a particle accelerator.

Towards Understanding Incomplete Fusion Reactions at Low Beam Energies: Modified Sum Rule Model

We investigated the enhanced production of nuclei formed via incomplete fusion (ICF) reactions near and above the Coulomb barrier energies (5–8 MeV/A). The cross-sections of the evaporation residues formed in the reactions—11B+124Sn, 10B+124Sn and 11B+122Sn—were measured using off-line gamma-ray spectrometry. The sum rule model (SRM) by Wilczyński et al. predicted the cross-section values too low compared to our experimental results. In earlier studies, the same model has been very successful in explaining ICF reactions at high beam energies (&gt;10 MeV/A). We, therefore, modified the SRM, specifically incorporating the energy dependence in the definition of critical angular momentum ℓcr. The resulting modified SRM gave an improved theoretical estimate for the reactions we studied.

Charged particles and Penrose process near charged black holes in Einstein–Maxwell-scalar theory

We study the dynamics of charged test particles around an electrically charged black hole in Einstein–Maxwell-scalar (EMS) gravity. The event horizon properties of the spacetime around the black hole are explored and the upper limit for the EMS theory parameters corresponding to extreme charge and minimal value of the event horizon are found. The effective potential for the radial motion of the charged particles at the equatorial plane is investigated. Specific energy and angular momentum of the particles corresponding to circular stable orbits are also studied. We also investigate the effects of the EMS parameter and the black hole charge on innermost stable circular orbits (ISCOs). We also investigate synchrotron radiation of charged particles in the spacetime of the charged black hole in EMS gravity. We also explore electric Penrose and Bañados–Silk–West processes near the black hole horizon, where we analyse in detail the effects of EMS parameters on energy efficiency in the Penrose process and critical angular momentum that allows colliding particles near the horizon, together with the center of mass energy in charged particles collisions.

Nuclear structure effects on over-barrier fusion reactions investigated with a new phenomenological model

We investigate the occurrence of nuclear structure effects in the cross section of heavy-ion fusion reactions at energies above the Coulomb barrier. To this end, we initially develop a universal phenomenological model capable to reproduce, with an unprecedented accuracy, all previously published experimental fusion excitation functions with a few parameters. The new model, which foresees exclusively charge, mass, and energy of the colliding systems, shows a clear saturation of the critical angular momentum and avoids analytical non-regularities. The predictions of the newly developed model are then inspected to pin down residual discrepancies with the data, which could be ascribed to the structure of the colliding systems. In this framework, we obtain the following findings: (1) for the first time, we suggest an anomaly in the optimum value of the fusion cross section for systems having nearly-zero fusion Q-values; (2) we point out the occurrence of shell closure effects in the fusion of light systems; (3) we suggest that shell effects are washed-out at relative velocities vrel≳0.07c; (4) in the higher energy part of the fusion excitation function, the cross section for colliding systems involving fluorine or neon isotopes impinging on 2p3/2-1f5/2 nuclei is suppressed, possibly due to the occurrence of α-clustering effects enhancing α-transfer reactions.

Magnetized and Magnetically Charged Particles Motion around Regular Bardeen Black Hole in 4D Einstein Gauss–Bonnet Gravity

In this paper, we study the horizon properties and scalar invariants of the spacetime around a regular black hole (BH) in 4D Einstein Gauss-Bonnet (4D EGB) gravity. It is observed that the presence of both Gauss-Bonnet (GB) coupling and magnetic charge parameters causes the shrinking of the outer horizon. We find that the range of the GB parameter α/M2∈(−0.15869,1), and the extreme value of magnetic charge reaches up to gextr=0.886M, which allows for the existence of a BH horizon, while it is gextr=0.7698M for pure Bardeen BH. We also investigate the dynamics of magnetized particles around the magnetically charged Bardeen BH, assuming the particle’s motion occurs in the equatorial plane in the proper observation frame, and the direction of the magnetic dipole moment of the particles is always kept radially and its magnitude is constant. Moreover, the dynamics of magnetically charged particles are also studied, and it is shown that both the energy and angular momentum of the particles corresponding to circular orbits increases with the increase of their magnetic charge. Finally, we also study collisions of magnetized, electrically neutral, and magnetically charged particles around the Bardeen BHs, where we provide analyses of critical angular momentum that may allow collision of the particles near-horizon radius, producing enormous values of center of mass energy of the collisions.

Effects of entrance channels on breakup fusion induced by F19 projectiles

The study of breakup fusion of $^{19}\mathrm{F}$ with $^{154}\mathrm{Sm}$ target was studied through offline $\ensuremath{\gamma}$ ray spectrometry. Partial cross sections of evaporation residues produced in this reaction were measured in center-of-mass energies ranging $\ensuremath{\approx}3--30$ MeV above the fusion barrier. The excitation functions of the evaporation residues populated through $xn/pxn$ channels were found to be satisfactorily reproduced by statistical model calculations, whereas for the $\ensuremath{\alpha}$ emitting channels the cross sections show an enhancement over the theoretical predictions. The critical angular momentum deduced from the measured cross sections was found to be in good agreement with statistical model calculations. The degree of fusion incompleteness in the $^{19}\mathrm{F}+^{154}\mathrm{Sm}$ reaction is estimated by comparing the fusion excitation functions with coupled channels calculations and the extracted fusion function with the universal fusion function. The large cross sections observed for incomplete fusion products support the interpretation that this suppression of fusion is caused by $^{19}\mathrm{F}$ breaking up into charged fragments before reaching the fusion barrier. The incomplete fusion probability was also found to increase with the reduced mass and charge of the entrance channel, indicating the influence of entrance channel mass asymmetry and Coulomb repulsion on incomplete fusion. The present analysis shows the presence of strong clustering in the $^{19}\mathrm{F}$ projectile as $\ensuremath{\alpha}$ and $^{15}\mathrm{N}$.

Diffusion as a possible mechanism controlling the production of superheavy nuclei in cold fusion reactions

The fusion probability for the production of superheavy nuclei in cold fusion reactions was investigated and compared with recent experimental results for $^{48}$Ca, $^{50}$Ti, and $^{54}$Cr incident on a $^{208}$Pb target. Calculations were performed within the fusion-by-diffusion model (FbD) using new nuclear data tables by Jachimowicz et al. It is shown that the experimental data could be well explained within the framework of the FbD model. The saturation of the fusion probability at bombarding energies above the interaction barrier is reproduced. It emerges naturally from the physical effect of the suppression of contributions of higher partial waves in fusion reactions and is related to the critical angular momentum. The role of the difference in values of the rotational energies in the fusion saddle point and contact (sticking) configuration of the projectile-target system is discussed.

Binary orbital evolution driven by a circumbinary disc

The question whether the interaction of a circumbinary disc with the central binary system leads to a shrinking or to an expansion of the binary orbit has attracted considerable interest as it impacts the evolution of binary black holes and stellar binary stars in their formation phase. We performed two-dimensional hydrodynamical simulations of circumbinary discs for a large parameter set of disc viscosities and thicknesses and for two different binary mass ratios for binaries on circular orbits. We measured the net angular momentum and mass transfer between disc and binary system, and evaluated the normalised specific angular momentum accretion, js. This was compared to the theoretical, critical specific angular momentum change js,crit that separates contracting from expanding cases, which depends on the binary mass ratio and on the relative accretion onto the two stars. Using finite and infinite disc models, we show that the inferred binary evolution is very similar for both setups, and we confirm that js can be measured accurately with cylindrical simulations that do not include the central binary. However, to obtain the relative accretion onto the stars for non-equal mass binaries, simulations that cover the whole domain including the binary are required. We find that for thick discs with aspect ratio h = 0.1, the binaries expand for all viscosities, while discs with h = 0.05 lead to an expansion only for higher viscosities with α exceeding ∼0.005. Overall, the regime of binary expansion extends to a much wider parameter space than previously anticipated, but for thin, low-viscosity discs, the orbits shrink.

Five dimensional rotating regular black holes and shadow

Higher-dimensional theories admit that astrophysical objects like supermassive black holes are somewhat different from standard ones and that their optical properties deviate from general relativity. A black hole casts a shadow as an optical appearance because of its strong gravitational field, and a black hole shadow could be used to determine which theory of gravity is consistent with observations. Measurements of the shadow sizes around the black holes can help evaluate various parameters of the black hole metric and may provide an opportunity to find signatures of extra dimensions. We present an exact five-dimensional (5D) rotating regular black hole metric, with a deviation parameter \(k\ge 0\), that interpolates between the 5D Kerr black hole (\(k=0\)) and 5D Kerr–Newman (\(r \gg k\)). This 5D rotating regular black hole is an exact general relativity solution coupled to nonlinear electrodynamics. Interestingly, for a given value of parameter k, there exits a critical angular momentum \(a=a_E\), which corresponds to an extremal rotating regular black hole with degenerate horizons. At the same time, for \(a<a_E\), one has a non-extremal rotating regular black hole with outer and inner horizons. Owing to the correction factor (\(e^{-k/r^2}\)), which is motivated by the quantum arguments, the ergoregion and shadow of 5D black holes are modified.

Rotating 4D Gauss–Bonnet black hole as a particle accelerator

We demonstrate that the rotating four-dimensional Gauss– Bonnet black hole can act as a particle accelerator with arbitrarily high centre-of-mass (CM) energy, when collision of two general particles takes place near the event horizon. The particles are at rest initially at infinity, and by fine tuning their angular momenta within a finite range, they are released so that they follow the time-like geodesics in the black hole spacetime, and the collision taking place on the equatorial plane is observed. The Gauss–Bonnet coupling constant α, provides a deviation in the results, from that observed in the Kerr black hole. The horizon structure, the range of allowed angular momentum and the critical angular momentum depend on the value of α. Our results show that the CM energy depends on the coupling parameter α in addition to the black hole spin a. For extremal cases, the CM energy diverges at the horizon, suggesting that Gauss–Bonnet black hole can also act as a particle accelerator similar to a Kerr black hole. For the non-extremal case, there exists a finite upper bound on the CM energy, the maximal value of which depends on the parameter α.

Probing the influence of incomplete fusion in the Li6+Y89 reaction up to 7.2 MeV/nucleon energy

The nuclear reaction study with weakly bound heavy nuclei is an enduring open question due to the scarcity of experimental data. In view of this, an experiment has been performed to study the fusion dynamics of weakly bound $^{6}\mathrm{Li}$ projectile in $^{6}\mathrm{Li}+^{89}\mathrm{Y}$ reaction at energies slightly above the Coulomb barrier. The off-beam $\ensuremath{\gamma}$-spectroscopic technique has been used to measure the cross sections of populated residues in $^{6}\mathrm{Li}+^{89}\mathrm{Y}$ reaction, and the data are analyzed using equilibrium and pre-equilibrium models in the framework of the empire3.2.2 code. It has been found that complete fusion (CF) and incomplete fusion (ICF) of $^{6}\mathrm{Li}$ play a substantial role in producing the residues. Subsequently, the ICF cross section, which is a model-dependent quantity, has been extracted. The strength of incomplete fusion has been deduced and compared with the existing data from $^{7}\mathrm{Li}+^{89}\mathrm{Y}$ reaction. The ICF strength is found to be higher for $^{6}\mathrm{Li}+^{89}\mathrm{Y}$ as compared with the $^{7}\mathrm{Li}+^{89}\mathrm{Y}$ system, which possibly indicates the role of the projectile's structure on the ICF occurrence probability. The occurrence of ICF below a critical angular momentum is also anticipated for both $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$ interaction with $^{89}\mathrm{Y}$.

Effect of neutron excess in the entrance channel on the O18+Nb93 system: An experimental study relevant to incomplete-fusion dynamics

The incomplete fusion dynamics in the $^{18}\mathrm{O}+^{93}\mathrm{Nb}$ system at energies above the Coulomb barrier has been investigated. The experimentally measured cross sections have been compared with the theoretical predictions of the statistical model code pace4. To examine the effect of entrance channel parameters on the onset and strength of incomplete fusion, relative contributions of complete and incomplete fusion have been deduced from the analysis of measured excitation functions. The contribution of incomplete fusion deduced from the analysis of excitation functions has been studied in terms of various entrance channel parameters, namely, entrance channel mass asymmetry (${\ensuremath{\mu}}_{A}$) of interacting projectile and target combination, Coulomb factor (${Z}_{P}{Z}_{T}$), ground state $\ensuremath{\alpha}\text{\ensuremath{-}}Q$ value of the reaction, and neutron skin thickness of target nuclei. It has been found that the probability of incomplete fusion depends strongly on entrance channel parameters. Further, the incomplete fusion contribution for the $^{18}\mathrm{O}$ projectile with two excess neutrons is noticed to be relatively larger as compared to $^{16}\mathrm{O}$. This may be due to the larger probability of breakup for the $^{18}\mathrm{O}$ projectile resulting in rather weak binding forces as compared to $^{16}\mathrm{O}$. The existence of incomplete fusion below critical angular momentum (${\ensuremath{\ell}}_{\mathrm{crit}}$), i.e., $\ensuremath{\ell}\ensuremath{\leqslant}\phantom{\rule{4pt}{0ex}}{\ensuremath{\ell}}_{\mathrm{crit}}$, has also been observed for the studied system.

Collision of particles near a three-dimensional rotating Ho\u0159ava AdS black hole

We consider a three-dimensional rotating AdS black hole, which is a solution of Hořava gravity in the low-energy limit that corresponds to a Lorentz-violating version of the BTZ black hole, and we analyze the effect of the breaking of Lorentz invariance on the possibility that the black hole can act as a particle accelerator by analyzing the energy in the center-of-mass (CM) frame of two colliding particles in the vicinity of its horizons. We find that the critical angular momentum of particles increases when the Hořava parameter xi increases and when the aether parameter b increases. Also, the particles can collide on the inner horizon with arbitrarily high CM energy if one of the particles has a critical angular momentum, possible for the BSW process. Here it is essential that, while for the extremal BTZ black hole the particles with critical angular momentum only can exist on the degenerate horizon, for the Lorentz-violating version of the BTZ black hole the particle with critical angular momentum can exist in a region away from the degenerate horizon. It is worth mentioning that the results exposed in this manuscript can be applied for the covariant version of Hořava gravity, where the covariant definition of the center-of-mass energy is well defined.

Effect of high angular momentum on η/s of nuclear matter

The shear viscosity ($\ensuremath{\eta}$) of nuclear matter is investigated in different nuclei (nuclear mass $A\ensuremath{\approx}59--194$) using experimental giant dipole resonance (GDR) width ($\mathrm{\ensuremath{\Gamma}}$) at high angular momenta ($J=12--54$ $\ensuremath{\hbar}$) and temperatures ($T=1.2--2.1$ MeV) collected from the existing literature. $\ensuremath{\eta}$, calculated from $\mathrm{\ensuremath{\Gamma}}$, is found to increase with $T$ and $J$. We show that critical temperature included fluctuation model (CTFM) successfully describes $J$-induced $\ensuremath{\eta}$ even beyond critical angular momentum ${J}_{c}$ at different values of $T$. However, the Fermi liquid drop model (FLDM) could not explain the data at higher angular momenta. We propose the addition of a $J$-dependent term with the FLDM $\ensuremath{\eta}$ to improve the prediction at such high-$J$ region. The $\ensuremath{\eta}/s$ ratio, highly important for measuring fluidity, is calculated using $\ensuremath{\eta}$ and the entropy density $s$. The latter is estimated using the Fermi gas formula. Interestingly, the experimental value of the ratio is independent of $J$ and $A$ and comes within 2.6--6.0 $\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{\mathrm{B}}$, which is very close to those of a partonic system like quark gluon plasma at high temperature.