Entropy quantization of Reissner-Nordström de Sitter black hole via adiabatic covariant action

Based on the ideas of adiabatic invariant quantity, we attempt to quantize the entropy of a charged black hole in de Sitter spacetime in two different coordinates. The entropy spectrum is obtained by imposing Bohr-Sommerfeld quantization rule and the laws of black hole thermodynamics to the modified adiabatic covariant action of the charged black hole. The result shows that the spacing of entropy spectrum is equidistant, and the corresponding horizon area quantum is identical to Bekenstein’s result. Interestingly, in contrast to the quasinormal mode analysis, we note that there is no need to impose the small charge limit for the obtained entropy spectrum of the charged black hole. We also note that the modified adiabatic covariant action gives the same value for the black hole entropy spectrum in different coordinate frames. This is a physically desired result since the entropy spectrum should be invariant under the coordinate transformations.     

Area and entropy spectra of black holes via an adiabatic invariant

By considering and using an adiabatic invariant for black holes, the area and entropy spectra of static spherically-symmetric black holes are investigated. Without using quasi-normal modes of black holes, equally-spaced area and entropy spectra are derived by only utilizing the adiabatic invariant. The spectra for non-charged and charged black holes are calculated, respectively. All these results are consistent with the original Bekenstein spectra.     

Quantization of the Kerr-Sen Black Hole Entropy via Adiabatic Invariance

Imposing the Bohr-Sommerfeld quantization rule to the adiabatic invariant quantity, we study the quantum spectrum of entropy for the Kerr-Sen black hole. The equally spaced entropy spectrum is derived. However, whether the spacing of the area spectrum is equidistant spaced or not, is still somewhat controversial.     

Black hole spectroscopy via adiabatic invariance

During the last years, one had to combine the proposal about how quasinormal frequencies are related with black holes and the proposal about the adiabatic invariance of black holes in order to derive the quantized entropy spectrum and its minimum change for several black holes. In this Letter we exclusively utilize the statement that the black hole horizon area is an adiabatic invariant and derive an equally spaced entropy spectrum of a black hole with its quantum to be equal to the one given by Bekenstein. Interestingly, in our approach no concept of quasi-normal mode is needed.     

Spectroscopy of the Rotating Kaluza-Klein Spacetime via Revisited Adiabatic Invariant Quantity

In this paper, we have investigated the spectroscopy of the rotating Kaluza-Klein spacetime by applying Bohr–Sommerfeld quantization rule and the first law of thermodynamics. we derived the expression of the adiabatic invariant quantity in the dragged–Painlevé coordinate system. Then, via revisited adiabatic invariant quantity, we derive the area and entropy spectra of the spacetime. We obtained the area spectrum of the Kaluza-Klein spacetime is \({\Delta } A=8\pi {l_{P}^{2}}\), and the entropy spectrum is ΔS = 2π. This result is consistent with the Bekenstein’s original result, which imply the entropy and horizon area are discrete and equidistant for the spacetime.     

Self-adjoint formulation of spherically symmetric hydrodynamics

We present here a new formalism applicable to spherically symmetric gas flow, in which all the known symmetries of the Euler equations of inviscid adiabatic gas flow are made manifest (including the symmetry t ↔ 1/t which holds for monatomic gases only).The formalism is manifestly invariant under a duality transformation called here (T̃), at least for the flow of monatomic gases with a particular entropy distribution.The associated self-similar problem is found to be entirely integrable in closed form; the axisymmetric case with combined expansion and rotation is found to be integrable too.This raises the question of whether the “Painlevé conjecture” of Ablowitz, Ramani and Segur applies, i.e. whether the corresponding partial differential systems are completely integrable, in particular through the inverse scattering transform or Bäcklund transformations. In this connection it must be noted that, in the case of plane symmetry, we have already shown the existence of an infinite number of conversition laws (ref. 2).     

Entropy Quantization from a Neutral Black String

Taking into account the Bekenstein-Hawking area law,based on the analysis of Zeng and Liu et al.that area spectrum is determined by the periodicity of an outgoing wave,we discuss on the quantization of entropy from a neutral black string.In addition,applying the adiabatic invariant quantity method proposed by Majhi and Vagenas,we further verify the entropy quantum of the neutral black string.As a result,two different methods show that the quantum of entropy is Δ S=2π,which is in agreement with Bekenstein's proposal.     

Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

Quantization of Horizon Area from the NUT-Kerr-Newman Black Hole

The aim of this Letter is to investigate the spectroscopy of the NUT-Kerr-Newman black hole by improving the method of revisited adiabatic invariant quantity. We present the modified expression of the adiabatic invariant quantity in the dragged–Painlevé coordinate system, and derive the spectroscopy of the black hole via revisited adiabatic invariant quantity, using Bohr–Sommerfeld quantization rule and the first law of the black hole thermodynamics. The result shows that the area and entropy spectra are respectively equally spaced and independent of black hole parameters and the area spectrum of the black hole is $\Delta A=8\pi l_{P}^{2}$ , which confirms the initial proposal of Bekenstein. It is noteworthy that there is no need to impose the small angular momentum limit and small charge limit in contrast to the quasinormal mode method.     

El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.     

Temperature, entropy, and kinetic theory

The relation between the concepts of temperature and entropy and the kinetic theory of gases is discussed, with particular attention to the aspects which are frequently treated as obvious or not even mentioned. In order to show that the usual thermodynamic relations are by no means obvious and may be contradictory, the model of a discrete velocity gas is used. It is also shown that the usual relation between the entropy rate and the heat supplied to a gas is not valid (even close to equilibrium) unless the theory is Galilei invariant (which is obviously not the case for a discrete velocity gas) and must be replaced by another one that eliminates all the paradoxical aspects of the matter.     

Accuracy of the conservation of the adiabatic invariant of a charged particle in plane waves

The conservation of the adiabatic invariant of a charged particle, entrapped by a transverse plane electromagnetic wave, is studied using a well-known method, both for the case when the phase velocity of the wave does not exceed the velocity of light, and for the opposite case. An expression is obtained for the variation of the adiabatic invariant, from which it is seen that the latter is conserved with exponential accuracy in the region of its existence.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 61–66, April, 1979.     

The effects of boundary layer phenomena on the performance of diskCCMHD generator

Two dimensional calculations were carried out to clarify the behavior of boundary layer and its effects on performance of closed cycle MHD (CCMHD) generator and to investigate the relation between enthalpy extraction ratio and adiabatic efficiency. Calculation results suggest that the large Lorentz force causes propagation and separation of boundary layer where reverse current flows, because of small electromotive force. For large load resistance boundary layer becomes very thick and the eddy current arises in broad region. The push work of working gas against Lorentz force is effectively converted into electric energy under the condition at which the Lorentz force decelerates the working gas to Mach number in the range between 1.0 and 1.5 in this case of the generator. Stagnation pressure loss increases with load resistance until enthalpy extraction ratio takes maximum value. The entropy production due to Joule heating and viscosity increases with load resistance. The difference between the load resistances for which the enthalpy extraction ratio and the adiabatic efficiency take maximum value can be explained with the entropy production of Joule heating and viscosity     

THE APPLICATION OF ADIABATIC THEOREM TO THE BEAA DYNAMICS IN ACCELERATORS

In this paper, the adiabatic criterion and the formulae for calculating the increment of adiabatic invariant due to the nonadiabatic change of paramters in Hamiltonian are derived and we have found that this increment is quite small, even in the case where the Landau's adiabatic criterion^[1] is violated.     

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

Exponential decrease of thermodynamic functions of photon gas in cubic and spherical cavities with low adiabatic invariants

The Planck radiation spectrum of ideal cubic and spherical cavities with low adiabatic invariants, γ = TV ^1/3, is discrete and strongly dependent on the cavity geometry and temperature. This behavior is a consequence of the random distribution of the state weights in the cubic cavity and of the random overlapping of successive multiplet components, in the case of a spherical cavity. The total energy density of cavities with low adiabatic invariant, γ (obtained by summing up the exact contributions of the eigenvalues and their weights) no longer obeys the Stefan-Boltzmann law. The new law includes a corrective factor depending on γ, which imposes an exponential decrease of the total energy density to zero, when γ → 0. A similar behavior is demonstrated for specific heat and for all other thermodynamic functions of photon gas in cubic and spherical cavities with low adiabatic invariants. This special quantum regime, defined by the limits of principal quantum numbers or by adiabatic invariants, is shown to be similar for cubic and spherical cavities.     

Contribution to the symplectic structure in the quantization rule due to noncommutativity of adiabatic parameters

A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fast-slow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2-form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.     

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow-fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.     

El-Nabulsi动力学模型下非Chetaev型非完整系统的精确不变量与绝热不变量

研究El-Nabulsi动力学模型下非Chetaev型非完整系统精确不变量与绝热不变量问题. 首先, 导出El-Nabulsi-d'Alembert-Lagrange原理并建立系统的运动微分方程. 其次, 建立El-Nabulsi模型下未受扰动的非Chetaev 型非完整系统的Noether对称性与Noether对称性导致的精确不变量之间的关系; 再次, 引入力学系统的绝热不变量概念, 研究受小扰动作用下非Chetaev型非完整系统Noether对称性的摄动导致绝热不变量问题, 给出了绝热不变量存在的条件及其形式. 作为特例, 本文讨论了El-Nabulsi模型下Chetaev型非完整系统的精确不变量与绝热不变量问题. 最后分别给出非Chetaev型和Chetaev型两种约束下的算例以说明结果的应用.