Application of the Monte Carlo method to the problem of surface segregation simulation

A generalization of the Monte Carlo method to the case of grand canonical ensemble allowing the elimination of the problem of determination of the chemical potential of alloy components was proposed. The method is particularly convenient for the calculations of surface segregations because it excludes time-consuming calculation of the temperature-dependent bulk chemical potential μ(T). The new method was used for calculating segregations at the (100), (110), and (111) surfaces of the Ni₅₀Pd₅₀ alloy using the Ising model with ab initio effective interatomic interaction potentials.     

Canonical hadron gas interpretation ofp-A E802 data

Hadron gas models have proved successful in predicting particle production in relativistic nucleus-nucleus collisions. The extension of these models to the smaller systems formed in proton-nucleus collisions requires that the finite size of the system be considered. We study two features introduced by the finite size: the need to conserve strangeness and baryon number exactly by performing calculations in the canonical ensemble, and the inclusion of a finite size geometrical correction term in the single particle density of states. We find significant differences between the grand canonical and canonical ensembles and a strong dependence on the baryon number of the system.     

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.     

The canonical and grand canonical models for nuclear multifragmentation

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, canonical ensemble or grand canonical ensemble. This paper deals with calculations with canonical and grand 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 nuclear matter.     

The Bose-Einstein condensation in a finite one-dimensional homogeneous system of noninteracting bosons

Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l (L ? l) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N(l/L)⁴ ? l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t=T/T ₁, where T ₁ is a certain characteristic temperature depending only on the longitudinal size L. Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ～ 1 and acquires the shape of the Bose-Einstein distribution for t ? 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N(l/L)⁴ ? 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.     

Study of weakly interacting trapped Bose gas

We have obtained expressions for single particle density and two particle density ofweakly interacting trapped quantum gases. These are valid for all temperature and in anydimension. These expressions have been simplified and expressed in terms ofnon-interacting single particle density. The ground fluctuations for T<T_cin grand canonical ensemble has been treated with care using the method of Kocharovskyet al. [Phys. Rev. A 61, 053606 (2000)]. Some numerical results are presentedin one and three dimension for isotropic harmonically trapped Bose gas with contactinteractions. It is seen that boson density decreases with increasing repulsiveinteractions. The expression for critical temperature is also shown to agree with earlierresult and is in accordance with experiments.     

Combinatorial and Topological Analysis of the Ising Chain in a Field

We present an alternative solution of the Ising chain in a field under free and periodic boundary conditions, in the microcanonical, canonical, and grand canonical ensembles, from a unified combinatorial and topological perspective. In particular, the computation of the per-site entropy as a function of the energy unveils a residual value for critical values of the magnetic field, a phenomenon for which we provide a topological interpretation and a connection with the Fibonacci sequence. We also show that, in the thermodynamic limit, the per-site microcanonical entropy is equal to the logarithm of the per-site Euler characteristic. The canonical and grand canonical partition functions are identified as combinatorial generating functions of the microcanonical problem, which allows us to evaluate them. A detailed analysis of the magnetic field-dependent thermodynamics, including positive and negative temperatures, reveals interesting features. Finally, we emphasize that our combinatorial approach to the canonical ensemble allows exact computation of the thermally averaged value <????> of the Euler characteristic associated with the spin configurations of the chain, which is discontinuous at the critical fields, and whose thermal behavior is expected to determine the phase transition of the model. Indeed, our results show that the conjecture <????>?(T _C)?=?0, where T _C is the critical temperature, is valid for the Ising chain.     

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.     

Abelian and non-Abelian anomalies in monopole-fermion interactions

Taking an example of the standard SU(5) theory, the monopole-fermion system is reduced to an effective 2-dimensional model. This is a generalized Schwinger model containing four Abelian gauge fields interacting with N generations of massless fermions through vector and axialvector couplings. We quantize such a system exactly in the canonical operator formalism. Then, analyzing the cluster property of operators carrying various chiral charges, the roles of the Abelian and non-Abelian anomalies are studied in monopole-induced baryon decay. We demonstrate that the Abelian anomaly and the charge-mixing boundary condition are the driving forces for monopole-induced baryon decay, though the conservation law suggests the importance of the non-Abelian anomaly.     

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.     

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.     

Ideal bose gas in an oscillator potential (exact solution)

Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.     

Effects of electron interaction on the change in sound velocity under de Haas-van Alphen conditions

Based on a temperature propagator technique in the grand ensemble of an interacting electron gas, the oscillatory sound velocity is examined under the de Haas-van Alphen conditions. In consideration of the oscillation of the Fermi energy (chemical potential) and the first order exchange effects, the dHvA oscillations of the sound velocity are shown to have the same one phase as in the case of an ideal electron gas, in agreement with experimental results. For large electron density, that is, for very small r_s, and by a proper renormalization of the Fermi energy, we have succeeded in eliminating one of the two oscillatory functions which have a phase difference of $\text{π}\text{2}$.     

The forces on a single interacting Bose–Einstein condensate

Using double parabola approximation for a single Bose–Einstein condensate confined between double slabs we proved that in grand canonical ensemble (GCE) the ground state with Robin boundary condition (BC) is favored, whereas in canonical ensemble (CE) our system undergoes from ground state with Robin BC to the one with Dirichlet BC in small-L region and vice versa for large-L region and phase transition in space of the ground state is the first order. The surface tension force and Casimir force are also considered in both CE and GCE in detail.     

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.     

Configurational entropy of adsorbed rigid rods: Theory and Monte Carlo simulations

The configurational entropy of straight rigid rods of length k (k-mers) adsorbed on square, honeycomb, and triangular lattices is studied by combining theory and Monte Carlo (MC) simulations in grand canonical and canonical ensembles. Three theoretical models to treat k-mer adsorption on two-dimensional lattices have been discussed: (i) the Flory-Huggins approximation and its modification to address linear adsorbates; (ii) the well-known Guggenheim-DiMarzio approximation; and (iii) a simple semi-empirical model obtained by combining exact one-dimensional calculations, its extension to higher dimensions and Guggenheim-DiMarzio approach. On the other hand, grand canonical and canonical MC calculations of the configurational entropy were obtained by using a thermodynamic integration technique. In the second case, the method relies upon the definition of an artificial Hamiltonian associated with the system of interest for which the entropy of a reference state can be exactly known. Thermodynamic integration is then applied to calculate the entropy in a given state of the system of interest. Comparisons between MC simulations and theoretical results were used to test the accuracy and reliability of the models studied.     

Lack of screening in the continuous dipole systems

We study continuous statistical systems interacting via a regularized dipole potential in the grand canonical ensemble. In the explicitly given region of high temperature (or low density) we show that the effective potential between two parallel dipoles isnot absolutely integrable (it is, however, square integrable), which implies that the effective potential does not fall off faster than |x|^–3 in some directions.Research supported in part by NSF PHY 76-80958     

A new analytic solution for the Zwanzig-Lauritzen model of polymer chain folding

A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L₂ functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.     

Screening of instantons in the quark gas

The grand partition function of quark matter is developed about an arbitrary classical gauge field configuration in a systematic weak coupling expansion. In the presence of a finite density massless quark gas the instanton induced effective quark interaction is modified by a factor exp[?2N_F(ω?)²], i.e. the baryon number chemical potential μ acts as an intrinsic infrared cutoff on the instanton scale size ?. The equation of state of the quark matter is also briefly discussed.