Novel approach to deriving the canonical generating functional in lattice QCD at a finite chemical potential

A novel approach to the problem of deriving the generating functional for the canonical ensemble in lattice QCD at a nonzero chemical potential is proposed. The derivation proceeds in several steps. First, the baryon density for imaginary values of the chemical potential is obtained. Then, again for imaginary values of the chemical potential, the generating functional of the grand canonical ensemble is derived. In this analysis, a fit of baryon density is employed toward simplifying the procedure of numerical integration. Finally, the generating potential for the canonical ensemble is derived using a high-precision numerical Fourier transform. The generating functional for the canonical ensemble is also derived using the known hopping-parameter expansion, and the results obtained with the two methods are compared for the deconfinement phase in the lattice QCD with two flavors.     

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.     

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%.     

Mesoscopic systems with fixed number of electrons

In this paper, we study the physics of mesoscopic systems with noninteracting electrons of fixed number. From a technical point of view, this means a discussion of the differences between the canonical and the grand canonical ensemble (fixed versus fluctuating number of particles). Such a discussion is not trivial since the grand canonical ensemble is the most convenient basis for the statistics of identical particles and one has to spend labour in order to retrieve the canonical ensemble. Specifically, we are considering ensembles of mesoscopic systems with disorder, either by atomic defects or by fluctuations in their geometric definitions and we discuss various forms of disorder averages.     

Finite size scaling on the phase diagram of QCD

In the last few years there has been remarkable progress in the comparison of experimental data on the shape of event-by-event distributions of conserved quantities and lattice thermodynamic predictions based on the grand canonical ensemble. In this talk we discuss how the QCD crossover temperature and the freezeout curve are extracted from the analysis of fluctuations. We report that one can also go further and locate the QCD critical point at ?? }~ 2T _c . We also list the systematics which must be brought under control in future.     

Brick Wall Model and the Spectrum of a Schwarzschild Black Hole

The quantum entropy of a scalar field near a Schwarzschild black hole is investigated by employing the brick-wall model in the grand canonical ensemble. A positive chemical potential is introduced if the cutoff is set to be of order of the Planck length. We also discuss the relation between the chemical potential and the mass quantum of the black hole.     

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.     

Clan properties in parton showers

We discuss the relationship of the forward matrix element of the operator $\bar \psi \psi $ , related to the so-called sigma term, to the quark number. We show that in the naive quark model in the canonical formalism these quantities coincide in the limit of small average quarkmomenta. In the QCD parton model defined through light-front quantization this result is preserved at leading perturbative order but it receives radiative corrections. We analyze the theoretical and phenomenological consequences of this result, which provides a bridge between a current algebra quantity, the sigma term, and a deepinelastic quantity, the parton number.     

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.     

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.     

Zero mode solutions of quark Dirac equations in QCD as the sources of chirality violating condensates

It is demonstrated, that chirality violating condensates in massless QCD arise from zero mode solutions of Dirac equations in arbitrary gluon fields. Basing of this idea, the model is suggested, which allows one to calculate quark condensate magnetic susceptibilities in the external constant electromagnetic field.     

Remarks on continuous phase transitions in molecules

We further develop the idea of continuous phase transitions in molecules applying it to the system SiH₃ and discuss the connection of this approach with the statistical mechanics of a grand canonical ensemble of molecules.     

Grand Canonical Ensemble Monte Carlo Simulation of Depletion Interactions in Colloidal Suspensions

Depletion interactions in colloidal suspensions confined between two parallel plates are investigated by using acceptance ratio method with grand canonical ensemble Monte Carlo simulation. The numerical results show that both the depletion potential and depletion force are affected by the confinement from the two parallel plates. Furthermore, it is found that in the grand canonical ensemble Monte Carlo simulation, the depletion interactions are strongly affected by the generalized chemical potential.     

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.     

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.