A comparison of the efficiency of Monte Carlo (MC) and molecular dynamics (MD) calculations at first order phase transitions

As a rule the canonical ensemble is used in the statistical analysis of phase transitions. There the system of interest is coupled to an infinite bath. It will be demonstrated that it is of considerable advantage to use a different ensemble where the size of the bath is comparable to the size of the system of interest. The advantages will be demonstrated for the example of a first order phase transition by making use of the density of states of the modifiedxy-model which has recently been determined in an MD-calculation for several lattice sizes. In contrast to the canonical ensemble the probability distribution does not suffer from the double hump structure and therefore allows an accurate determination of the properties at the transition. In the end we shall argue that the corresponding ensemble can also be realized in an MC-approach.     

Simple cases of spontaneous emission from an atom-photon cluster localized in a microcavity

The spontaneous emission from an atomic ensemble localized in a microcavity with the participation of microcavity photons and an external broadband quantized electromagnetic field at the Raman resonance of photons with an optically forbidden (two-photon) atomic transition has been studied. The average spontaneous decay intensity has been calculated for simple cases. It is shown that the dynamics of spontaneous emission from this atomic ensemble differs generally from the conventional superradiance (spontaneous emission of an atomic ensemble at a one-photon optically allowed transition from excited to the ground state. When the atomic ensemble is strongly excited, the delay times and the emission pulse shape differ significantly. The parameter ranges where the spontaneous emission from the atomic ensemble under consideration at a two-photon Raman transition can be described as conventional superradiance with renormalized parameters are found. In the case of single excitation the photon emission probability depends on the number of photons and atoms in the microcavity.     

Rules for transition rates in nonequilibrium steady states

Just as transition rates in a canonical ensemble must respect the principle of detailed balance, constraints exist on transition rates in driven steady states. I derive those constraints, by maximum information-entropy inference, and apply them to the steady states of driven diffusion and a sheared lattice fluid. The resulting ensemble can potentially explain nonequilibrium phase behavior and, for steady shear, gives rise to stress-mediated long-range interactions.     

System size resonance in coupled noisy systems and in the Ising model

We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles---lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance.     

Elementary-oscillator model for color centers with degenerate levels

Quantum transitions between nondegenerate and degenerate (with respect to angular momentum projection) levels of color centers in crystals are described in terms of classical electric-dipole oscillators. It is shown that, upon absorption of elliptically polarized exciting radiation, the system under study transfers to a definite state and the transition is described in general by an elliptical oscillator. An electric-dipole transition accompanied by the creation of a photon is described by a random vector, i.e., by an arbitrary elliptical oscillator lying in the corresponding plane. For an ensemble of n identically oriented centers undergoing a given transition, the luminescence intensity is described by a set of n/2 right-handed rotators and n/2 left-handed rotators. The results obtained are important for solving problems related to quantum information.     

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.     

Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviationtail, are studied numerically for two graph ensembles, for Erdös-Rényi random graphs withfinite connectivity and for two-dimensional bond percolation. Probabilities as small as10^-180 are accessed using an artificial finite-temperature (Boltzmann)ensemble. The distributions for the Erdös-Rényi ensemble agree well with previouslyobtained analytical results. The results for the percolation problem, where no analyticalresults are available, are qualitatively similar, but the shapes of the distributions aresomehow different and the finite-size corrections are sometimes much larger. Furthermore,for both problems, a first-order phase transition at low temperatures Twithin the artificial ensemble is found in the percolating regime, respectively.     

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.     

Dynamic criteria for melting in two dimensions

The two-dimensional (2D) melting transition is analyzed on the basis of the long-time behavior of a modified Lindemann parameter in 2D gamma(L)(t) and the bond-angular correlation function g(6)(t). Using video microscopy complete positional data are obtained over five decades in time for an ensemble of superparamagnetic colloidal particles confined to an air-water interface. We find that each of the three phases (solid/hexatic/isotropic liquid) is uniquely characterized by the long-time behavior of gamma(L)(t), g(6)(t), and the non-Gaussian parameter of the relative neighbor-neighbor displacement.     

Investigation into the dynamics and variations in the magnetic structure of an ensemble of ferromagnetic particles

The nonlinear dynamics of an ensemble of single-domain interacting particles upon excitation by radio-frequency pulses of a magnetic field is investigated. Reorientation of magnetic moments of ensemble particles under the effect of the pulses of the radio-frequency field is observed. The effect of varying the magnetic structure on the frequency dependences of susceptibility is studied. Temporal characteristics of precession in separate particles under the conditions of an orientation transition are considered.     

First order phase transitions in the canonical and the microcanonical ensemble

In the canonical ensemble any singularity of a thermodynamic function at a temperatureT _c is smeared over a temperature range of orderT _T /N. Therefore it is rather difficult to distinguish between a discontinuous and a continuous phase transition on the basis of numerical data obtained for finite systems in the canonical ensemble. It is demonstrated for four model systems that this problem cannot be circumvented by considering higher cumulants of the energy distribution or cumulant ratios. On the other hand, the distinction between first and a second order phase transition is rather direct if based on the microcanonical density of states which is readily obtainable in the dynamical ensemble.     

Transition from antibunching to bunching in cavity QED

The photon statistics of the light emitted from an atomic ensemble into a single field mode of an optical cavity is investigated as a function of the number of atoms. The light is produced in a Raman transition driven by a pump laser and the cavity vacuum, and a recycling laser is employed to repeat this process continuously. For weak driving, a smooth transition from antibunching to bunching is found for about one intracavity atom. Remarkably, the bunching peak develops within the antibunching dip. The observed behavior is well explained by a model describing an ensemble of independent emitters.     

Remarks on the use of a microcanonical ensemble to study phase transitions in lattice gauge theory

The phase transitions of the Z₂ and U₁ lattice gauge theories are described by sampling a microcanonical ensemble, in which the gauge field is in thermal equilibrium with a system of auxiliary variables called demons. It is shown that the microcanonical ensemble yields a more complete picture of a first-order transition than that obtained by sampling the Boltzmann distribution.     

Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence

The near field solution for the scattering of a plane monochromatic electromagnetic wave by an ensemble of parallel infinite dielectric cylinders at perpendicular incidence is presented in this paper. The solution is given for the calculation of the electric and magnetic near fields and the Poynting vector. A MATLAB program has been developed to solve the near field formulas which is introduced and validated. The near to far field transition as well as formation and transport of photonic nanojets have been calculated for multiple cylinder scattering.     

Atomic beam preparation for hydrogen maser operation with unpolarized atoms

The conventional atomic beam preparation technique for hydrogen maser operation consists of a six pole magnetic field focussing in combination with adiabatically changing magnetic fields in the transition region between focusser and maser cavity, which results in a state selected, polarized ensemble of hydrogen atoms. Hydrogen maser operation with state selected, but unpolarized atoms has the advantage to reduce considerably the sensitivity of hydrogen maser frequency on coherently excited, low frequency,Δm _F =±1 Zeeman transitions. The influence of the magnetic field pattern in the transition region on the energy level population probabilities for the focussed hydrogen beam has been analysed. Atomic beam preparation techniques using non-adiabatically changing magnetic field configurations in the transition region are described. The dependence of hydrogen maser oscillation on the energy level population within the atomic beam is theoretically investigated and the results are used for the analysis of the experimentally achieved atomic beam populations. It has been shown that a pulsed magnetic spin guidance field in the transition region, which switches between adiabatically changing magnetic field configurations and a sudden magnetic field reversal, results in an ensemble of hydrogen atoms with vanishing time averaged polarization, which does not deteriorate the conventionally achieved maser oscillation amplitude.     

Phase chaos in the dynamics of an ensemble of oscillators with time-modulated global coupling

The object of consideration is an ensemble of globally coupled self-sustained oscillating elements with a finite-width frequency distribution. The ensemble interacts with the field of a resonator, which is a linear oscillator with a frequency doubly exceeding the mean frequency of the oscillators in the ensemble. The global coupling is switched on and off alternately, so that the ensemble alternatively passes from synchrony to asynchrony (Kuramoto transition). At each stage of activity (synchronization), the field of the resonator causes the mean field of the ensemble to oscillate so that the phase doubles compared with the previous stage of excitation. Therefore, the mean field dynamics is chaotic and, as follows from numerical simulation data, can be associated with the Smale-Williams attractor. Systems of this type can be applied in electronics, specifically, in secure communication systems, noise location, etc.     

Transition from Poisson to circular unitary ensemble

Transitions to universality classes of random matrix ensembles have been useful in the study of weakly-broken symmetries in quantum chaotic systems. Transitions involving Poisson as the initial ensemble have been particularly interesting. The exact two-point correlation function was derived by one of the present authors for the Poisson to circular unitary ensemble (CUE) transition with uniform initial density. This is given in terms of a rescaled symmetry breaking parameter Λ. The same result was obtained for Poisson to Gaussian unitary ensemble (GUE) transition by Kunz and Shapiro, using the contour-integral method of Brezin and Hikami. We show that their method is applicable to Poisson to CUE transition with arbitrary initial density. Their method is also applicable to the more general ℓCUE to CUE transition where ℓCUE refers to the superposition of ℓ independent CUE spectra in arbitrary ratio.     

Nonuniform van der Waals theory

The liquid-vapor interface of a confined fluid at the condensation phase transition is studied in a combined hydrostatic/mean-field limit of classical statistical mechanics. Rigorous and numerical results are presented. The limit accounts for strongly repulsive short-range forces in terms of local thermodynamics. Weak attractive longer-range ones, like gravitational or van der Waals forces, contribute a self-consistent mean potential. Although the limit is fluctuationfree, the interface is not a sharp Gibbs interface, but its structure is resolved over the range of the attractive potential. For a fluid of hard balls with –r ^–6 interactions the traditional condensation phase transition with critical point is exhibited in the grand ensemble: A vapor state coexists with a liquid state. Both states are quasiuniform well inside the container, but wall-induced inhomogeneities show up close to the boundary of the container. The condensation phase transition of the grand ensemble bridges a region of negative total compressibility in the canonical ensemble which contains canonically stable proper liquid-vapor interface solutions. Embedded in this region is a new, strictly canonical phase transition between a quasiuniform vapor state and a small droplet with extended vapor atmosphere. This canonical transition, in turn, bridges a region of negative total specific heat in the microanonical ensemble. That region contains subcooled vapor states as well as superheated very small droplets which are microcanonically stable.