The thermodynamic model for nuclear multifragmentation

A great many observables seen in intermediate energy heavy ion collisions can be explained on the basis of statistical equilibrium. Calculations based on statistical equilibrium can be implemented in microcanonical ensemble (energy and number of particles in the system are kept fixed), canonical ensemble (temperature and number of particles are kept fixed) or grand canonical ensemble (fixed temperature and a variable number of particles but with an assigned average). This paper deals with calculations with canonical ensembles. A recursive relation developed recently allows calculations with arbitrary precision for many nuclear problems. Calculations are done to study the nature of phase transition in intermediate energy heavy ion collision, to study the caloric curves for nuclei and to explore the possibility of negative specific heat because of the finiteness of nuclear systems. The model can also be used for detailed calculations of other observables not connected with phase transitions, such as populations of selected isotopes in a heavy ion collision.The model also serves a pedagogical purpose. For the problems at hand, both the canonical and grand canonical solutions are obtainable with arbitrary accuracy hence we can compare the values of observables obtained from the canonical calculations with those from the grand canonical. Sometimes, very interesting discrepancies are found.To illustrate the predictive power of the model, calculated observables are compared with data from the central collisions of Sn isotopes.     

Density-functional theory of inhomogeneous fluids in the canonical ensemble

We present a density-functional approach for dealing with inhomogeneous fluids in the canonical ensemble. A general relation is proposed between the free-energy functionals in the canonical and the grand canonical ensembles. The minimization of the canonical-ensemble free-energy functional gives rise to Euler-Lagrange equations which involve averaged Ornstein-Zernike equations of second and third order. The theory is especially appropriate for systems with a small, fixed number of particles. As an example of application we obtain accurate results for the density profile of a hard-sphere fluid in a closed spherical cavity that contains only a few particles.     

Linear response,persistent currents and related problems

Based on a general linear response approach we provide a systematic and unified survey of existing theories on persistent currents. The central notions in this context are equilibrium and dynamic persistent currents which are analyzed with respect to their similarities and differences in the canonical and grand canonical ensemble. We present criteria which relate the existence of persistent currents to the equipartition law and ergodicity for current correlators. We find that in additive Fermion systems at low temperatures both kinds of persistent currents coincide in the canonical ensemble whereas they differ in the grand canonical ensemble. Comparing different works on averaged persistent currents in diffusive mesoscopic rings within our framework and discussing several methods of calculating canonical currents with the help of grand canonical ensembles, we clarify some misunderstandings which have arisen in methodologically different approaches to the phenomenon of persistent currents. Finally, we relate the presence of dynamic persistent currents to the Hall conductivity on a finite cylinder and the center coordinate Kubo formula for the Hall conductivity.     

近独立子系系统的统计规律(四)——巨正则分布函数

根据吉布斯系综理论导出了由任意数目刚性粒子构成的力学系统的巨正则分布函数、化学势及粒子数和能量涨落公式.并介绍了刚球模型下,对二维和三维近独立子系系统巨正则分布的计算机模拟,结果与理论完全一致.     

A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and Its Applications

We introduce the boson and the fermion point processes from the elementary quantum mechanical point of view. That is, we consider quantum statistical mechanics of the canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume. Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain. By taking the thermodynamic limit such that the particle density converges to a finite value, the boson/fermion processes are obtained. This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points. Random point processes corresponding to para-particles of order two are discussed as an application of the formulation. Statistics of a system of composite particles at zero temperature are also considered as a model of determinantal random point processes.     

Equivalence of Gibbs ensembles for quantum systems obeying Maxwell-Boltzmann statistics

A generating functional of the canonical ensemble is introduced for quantum systems of particles obeying Maxwell-Boltzmann statistics. Power estimates for the quantum analogs of the particle distribution functions of the canonical and grand canonical ensembles are obtained for systems of particles with hard cores. It is shown on the basis of the power estimates that limiting generating functionals exist for the systems under consideration and satisfy the same Bogolyubov equation. The ensembles are equivalent in this sense.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1984.The authors thank É. A. Arinshtein for useful discussion.     

Phase transitions in four-dimensional AdS black holes with a nonlinear electrodynamics source

In this work we consider black hole solutions to Einstein's theory coupled to a nonlinear power-law electromagnetic field with a fixed exponent value. We study the extended phase space thermodynamics in canonical and grand canonical ensembles, where the varying cosmological constant plays the role of an effective thermodynamic pressure. We examine thermodynamical phase transitions in such black holes and find that both first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, Hawking–Page and second-order phase transitions are allowed.     

Uniform upper bounds (and thermodynamic limit) for the correlation functions of symmetric coulomb-type systems

Uniform upper bounds are proven for the correlation functions in the strictly charge-neutral canonical and grand canonical ensembles for charge-symmetric two-component systems. For the grand canonical ensemble the increase of the correlation functions along the thermodynamic-limit sequence is shown as well, implying the existence of the states. The particles have bounded pair interactions of positive type. Both classical and quantum systems with Boltzmann statistics are considered. Coulomb systems with regularized interactions are included as a special case.     

Baryon number projection for an interacting hadron gas

Calculations of hadronic matter usually enforce conservation of the average baryon number density using the grand canonical ensemble. We have performed calculations for an interacting system in the canonical ensemble with fixed baryon numberN _b , as appropriate for a finite fireball of the type produced in ultra relativistic heavy ion collisions. These results are compared with those obtained from calculations in the grand canonical ensemble. For an interacting nucleon gas the two ensembles yield free energies which differ by approximately 5%.     

Representing Euclidean Quantum Fields as Scaling Limit of Particle Systems

We give a new representation of Euclidean quantum fields as scaling limits of systems of interacting, continuous, classical particles in the grand canonical ensemble.     

Magnetic moment of a two-dimensional degenerate electron gas in mesoscopic samples

The orbital magnetism of two-dimensional electrons in mesoscopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquette, bracelet with hard wall confinement and also a confinement with a parabolic potential. We calculate the average magnetic moment which means an average with respect to size fluctuations and de Haas-van Alphen oscillations which arise in the case of a sharp Fermi cutoff. We see three distinct ranges in the magnetic field: (i) small field region where perturbation theory applies; (ii) moderate fields where edge currents play a prominent role; and (iii) the high field range with a Landau type susceptibility. In a quasiclassical picture, the electronic orbits are not qualitatively changed by a magnetic field in (i); skipping orbits are important in (ii); and in (iii), the cyclotron radius is smaller than the sample size. As a rule, we find an enhancement of the magnetic response which increases with k_FL, that is, with sample size divided by the Fermi wave length. Also, we have found out that the quasiclassical approximation fails in the calculation of the magnetic properties; on the other hand, we have seen no essential differences between the canonical ensemble (fixed particle number) and the grand canonical ensemble (chemical potential given). In the case of plaquettes, in particular for samples in the form of squares, we have found agreement with experimental results by Lévy, Reich, Pfeiffer and West.     

Condensation for a Fixed Number of Independent Random Variables

A family of m independent identically distributed random variables indexed by a chemical potential φ∈[0,γ] represents piles of particles. As φ increases to γ, the mean number of particles per site converges to a maximal density ρ _c <∞. The distribution of particles conditioned on the total number of particles equal to n does not depend on φ (canonical ensemble). For fixed m, as n goes to infinity the canonical ensemble measure behave as follows: removing the site with the maximal number of particles, the distribution of particles in the remaining sites converges to the grand canonical measure with density ρ _c ; the remaining particles concentrate (condensate) on a single site.     

Thermodynamic properties of a finite Bose gas in a harmonic trap

We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of total number $N$ of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, specially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.     

(2+1)-Dimensional AdS Black Hole in Grand Canonical Ensemble

We investigate thermodynamics of the (2+1)-dimensional AdS black hole in grand canonical ensemble. In the York's formalism, the black hole is enclosed in a “box” with a finite radius and the boundary temperature, radius and potential are fixed in the grand canonical ensemble. We investigate the thermodynamical properties such as action, entropy, temperature, etc. We only find the stable solution for (2+1)-dimensional AdS black hole and do not find the instanton with the negative heat capacity.     

Density matrix for particle production processes in an external field

The general problem of finding the density matrices of particles and antiparticles produced by an external electromagnetic field fron an arbitrary initial state is solved. Expressions are obtained for the density matrices of particles and antiparticles produced from a state with given number of initial particles and from a state described by an initial grand canonical ensemble.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fiz., No. 6, pp. 77–81, June, 1980.     

Grand Canonical Ensemble Interpretation for Charged Particle Quantum Tunneling Radiation

In this paper, we present a grand canonical ensemble interpretation for the massive charged particles tunneling from a charged black hole. The probability distribution function corresponding the emission shell system is derived in details, and the expression is same as the tunneling rate in Parikh-Wilzeck framework. With this result, the statistical significance of the quantum tunneling radiation is discussed.     

Thermodynamic properties of electrons in finite systems

The aim of this work is to study quantitatively how the Coulomb interaction of the electrons leads to the canonical ensemble average instead of the grand canonical one. Within the “equal level spacing” model we are able to give explicit expressions for a finite model interaction, and it turns out that for real systems at low temperatures the canonical average is an appropriate approximation. Further, we compare our theoretical results to experimental datas on TCNQ salts, which we regard as a collection of interrupted strands.     

Derivation of Mayer Series from Canonical Ensemble

Mayer derived the Mayer series from both the canonical ensemble and the grand canonical ensemble by use of the cluster expansion method. In 2002, we conjectured a recursion formula of the canonical partition function of a fluid(X.Z. Wang, Phys. Rev. E66(2002) 056102). In this paper we give a proof for this formula by developing an appropriate expansion of the integrand of the canonical partition function. We further derive the Mayer series solely from the canonical ensemble by use of this recursion formula.     

Improvement of the grand canonical quantum Monte Carlo method at low temperatures

We introduce an improvement of the algorithm for the simulation of correlated fermi systems in the grand canonical ensemble. Using this new method the computer time grows no more with the square but essentially linearly with the inverse temperature. At low temperatures the number of operations is diminished by a factor typically between 5 and 20. We present results for correlation functions of the one-dimensional Hubbard model at various band fillings.