Fermion entropy of non-uniformly rectilinearly accelerating black hole with electric and magnetic charges

Using the thin film brick-wall model, we calculate the fermion entropy on event horizon and the surface density of the entropy on the Rindler Horizon to a rectilinearly accelerating non-stationary black hole with electric and magnetic charges. The conclusion that black hole entropy is proportional to its area can still be applied by regulating the cut-off factor ?$?$ and the film's thickness δ$\delta$, which are time dependent.     

Generalized uncertainty principle and entropy of three-dimensional rotating acoustic black hole

Using the new equation of state density from the generalized uncertainty principle, we investigate statistics entropy of a 3-dimensional rotating acoustic black hole. When λ introduced in the generalized uncertainty principle takes a specific value, we obtain an area entropy and a correction term associated with the acoustic black hole. In this method, there does not exist any divergence and one needs not the small mass approximation in the original brick-wall model.     

Universal canonical black hole entropy

Nonrotating black holes in three and four dimensions are shown to possess a canonical entropy obeying the Bekenstein-Hawking area law together with a leading correction (for large horizon areas) given by the logarithm of the area with a universal finite negative coefficient, provided one assumes that the quantum black hole mass spectrum has a power-law relation with the quantum area spectrum found in nonperturbative canonical quantum general relativity. The thermal instability associated with asymptotically flat black holes appears in the appropriate domain for the index characterizing this power-law relation, where the canonical entropy (free energy) is seen to turn complex.     

正则黑洞熵与相变

应用隧道效应对黑洞Hawking辐射研究得到的辐射谱,对非旋转黑洞的正则熵进行讨论.所得熵遵守Bekenstein-Hawking 面积定律,并带有修正项,主要修正项与面积的对数成正比,另有与黑洞热容量有关的修正项.利用所给出的正则熵,对黑洞的相变进行讨论,得到当黑洞的热容量出现发散时,正则熵在该处并不出现复数,由此认为此相变为二级相变. 关键词： 正则系综 量子修正 黑洞相变     

The entropy function for the extremal Kerr-(anti-)de Sitter black holes

Based on Sen's entropy function formalism, we consider the Bekenstein-Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein-Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J.We also study the Gauss-Bonnet correction to the entropy exploiting the entropy function formalism. We show that the term, though being topological thus does not affect the solution, contributes a constant addition to the entropy because the term shifts the Hamiltonian by that amount.     

Canonical versus grand canonical occupation numbers for simple systems

An ideal gas ofN indistinguishable particles is described by a canonical ensemble (c.e.) and also by a grand canonical ensemble (g.c.e.) which hasN as themean total number of particles, the temperature and volume being the same in both cases. Exact mean occupation numbersn _j(N) are found if the system has only two states 1 and 2 of energiesE ₂E ₁. This should apply to quantum wells and similar simple systems. For systems which have captured one particle, the theory gives the simplest answers, and one find a maximum discrepancy of 17% between the two ensembles for the fermion case. It occurs whenE ₂–E ₁53 meV at room temperature. ForN=1 the mean occupation number for the c.e. is identical for fermions and for bosons, being in both cases given byn ₂(1)={exp[(E ₂-E ₁)/kT]+1}^-1,n ₁(1)=1-n ₂(1) For largeN one reverts to the usual situation and the discrepancy between the ensembles becomes small.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

Letter: Gravitational Thermodynamics of a Vaidya Black Hole

The path integral approach is applied to the statistical thermodynamics of a radiating Vaidya black hole. The entropy still satisfies the Bekenstein-Hawking formula, except for a negligible term. The entropy production, as a measurement of the irreversibility, is also obtained.     

Black hole entropy function,attractors and precision counting of microstates

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.     

Information entropy for static spherically symmetric black holes

By using the new equation of state density derived from the generalized uncertainty relation, the number of the quantum states near event horizon is obtained, with which then the information entropy of static spherically symmetric black holes has been discussed. It is found that the divergent integral of quantum states near the event horizon can be naturally avoided if using the new equation of state density without introducing the ultraviolet cut-off. The information entropy of black holes can be obtained precisely by the residue theorem, which is shown to be proportional to the horizon area. The information entropy of black holes obtained agrees with the Bechenstein--Hawking entropy when the suitable cutoff factor is adopted.     

Microcanonical and Canonical Ensembles for fMRI Brain Networks in Alzheimer’s Disease

This paper seeks to advance the state-of-the-art in analysing fMRI data to detect onset of Alzheimer’s disease and identify stages in the disease progression. We employ methods of network neuroscience to represent correlation across fMRI data arrays, and introduce novel techniques for network construction and analysis. In network construction, we vary thresholds in establishing BOLD time series correlation between nodes, yielding variations in topological and other network characteristics. For network analysis, we employ methods developed for modelling statistical ensembles of virtual particles in thermal systems. The microcanonical ensemble and the canonical ensemble are analogous to two different fMRI network representations. In the former case, there is zero variance in the number of edges in each network, while in the latter case the set of networks have a variance in the number of edges. Ensemble methods describe the macroscopic properties of a network by considering the underlying microscopic characterisations which are in turn closely related to the degree configuration and network entropy. When applied to fMRI data in populations of Alzheimer’s patients and controls, our methods demonstrated levels of sensitivity adequate for clinical purposes in both identifying brain regions undergoing pathological changes and in revealing the dynamics of such changes.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

Canonical Entropy of Reissner-Nordstrom Black Hole

Recently, Hawking radiation of the black hole has been studied using the tunnel effect method. It is found the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How the departure from pure thermal spectrum affects the entropy? This is a very interesting problem. In this paper, we calculate the partition function by energy spectrum obtained by tunnel effect. Using the relation between the partition function and entropy, we derive the expression of entropy the general charged black hole. In our calculation, we not only consider the correction to the black hole entropy due to fluctuation of energy but also consider the effect of the change of the black hole charges on entropy. We discuss Reissner-Nordstrom black hole and obtain that Reissner-Nordstrom black hole cannot approach the extreme black hole by changing its charges.     

Black hole entropy and the modified uncertainty principle: A heuristic analysis

Recently Ali et al. (2009) proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Plank length). Inspired by this idea here we calculate the quantum corrected value of a Schwarzschild black hole entropy and a Reissner-Nordström black hole with double horizon by utilizing the proposed generalized uncertainty principle. We find that the leading order correction goes with the square root of the horizon area contributing positively. We also find that the prefactor of the logarithmic contribution is negative and the value exactly matches with some earlier existing calculations. With the Reissner-Nordström black hole we see that this model-independent procedure is not only valid for single horizon spacetime but also valid for spacetimes with inner and outer horizons.     

高自旋场对Vaidya-Bonner黑洞熵的贡献

利用改进的brick-wall模型推导出自旋为2的引力场对动态Vaidya-Bonner 黑洞熵的贡献， 在自然单位制中其值为A/2；这表明，当选择与标量场相同的截断因子时，其熵为标量场的2 倍，为Dirac 场的4/7. 关键词： Vaidya-Bonner黑洞 黑洞熵 高自旋场 薄膜brick-wall模型     

Difference between canonical and grand canonical ensembles in discrete lattice gas models

We calculate the site occupation probabilities of one-dimensional lattice gas models within the canonical and grand canonical ensembles. The appearing differences do not vanish if we increase the system size keeping the site energies discrete. In this way one can explain the surprising numerical results of Barszczak and Kutner. This effect in the single-site occupation number disappears in higher dimensions.     

一类广义Birkhoff系统的广义正则变换

研究一类广义Birkhoff系统的广义正则变换.建立这类广义Birkhoff系统的运动微分方程,得到了该系统的广义正则变换以及保持广义正则变换的条件.最后,举例说明结果的应用.     

黑洞的统计熵

运用量子统计的方法,直接求解Schwarzschild时空背景下玻色场和费米场的配分函数,得到熵的积分表达式.按照最近的研究结果,认为黑洞的Hawking辐射过程是隧道效应过程,在考虑黑洞隧道效应产生过程中黑洞能量发生变化的基础上,给出积分的下限为黑洞的视界位置.由此得到黑洞熵的主要项为视界面积的1/4.不存在使人疑惑的紫外截断因子,并且由此可得黑洞辐射粒子的能量与辐射温度成正比的结论. 关键词： 黑洞熵 量子统计 隧道效应 反作用     

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈n₀〉/N and specific heat C_N around the transition temperature T_c (i.e., t = T/T_c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ? 0.068.