Mean Field Approximation

The mean field solution of the Ising model on a Barabási-Albert scale-free network with ferromagnetic coupling between linked spins is presented. The critical temperature Tc for the ferromagnetic to paramagnetic phase transition ( Curie temperature) is infinite and the effective critical temperature for a finite size system increases as the logarithm of the system size in agreement with recent numerical results of Aleksiejuk, Holyst and Stauffer.

We present a model for earthquake failure at intermediate scales (space: 100 m-100 km, time: 100 m/v shear -1000's of years). The model consists of a segmented strike-slip fault embedded in a 3-D elastic solid as in the framework of . The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of the regions around the fault, static/ kinetic friction laws with possible gradual healing, and stress transfer based on the solution of CHINNERY (1963) for static dislocations in an elastic half-space. As a new ingredient, we approximate the dynamic rupture on a continuous time scale using a finite stress propagation velocity (quasi-dynamic model) instead of instantaneous stress transfer (quasi-static model). We compare the quasi-dynamic model with the quasi-static version and its mean field approximation, and discuss the conditions for the occurrence of frequency-size statistics of the Gutenberg-Richter type, the characteristic earthquake type, and the possibility of a spontaneous mode switching from one distribution to the other. We find that the ability of the system to undergo a spontaneous mode switching depends on the range of stress transfer interaction, the cell size, and the level of strength heterogeneities. We also introduce time-dependent log ðtÞ healing and show that the results can be interpreted in the phase diagram framework. To have a flexible computational environment, we have implemented the model in a modular C++ class library. some parameters empirical values are available, while others can be used to tune the model dynamics towards an expected behavior. The model of of a fault in a 3-D elastic half-space appears to meet these criteria. Using this model, several observed frequency-size and temporal statistics could be explained in terms of structural properties of a given fault.

We analyze the dynamics of a Bose-Einstein condensate undergoing a continuous dispersive imaging by using a Lindblad operator formalism. Continuous strong measurements drive the condensate out of the coherent state description assumed within the Gross-Pitaevskii mean-field approach. Continuous weak measurements allow instead to replace, for timescales short enough, the exact problem with its mean-field approximation through a stochastic analogue of the Gross-Pitaevskii equation. The latter is used to show the unwinding of a dark soliton undergoing a continuous imaging. 03.75.Fi,05.30.Jp, The interplay between quantum and classical descriptions of the physical world and the role of the measurement process are still at the heart of the understanding of quantum mechanics [1]. The related theoretical debate has been greatly enriched in the last decades by the realization of new experimental techniques aimed at producing quantum states without classical analogue, such as entangled or squeezed states, or to explore phenomena which are intrinsically quantum mechanical, like quantum jumps. All this occurred also having in mind practical implications, such as the improvement of sensitivity of various devices operating at or near the quantum limit .

From quantitative measurement of the equilibrium terrace-width (`) distribution (TWD) of vicinal surfaces, one can assess the strength A of elastic step±step repulsions A/`2. Generally the TWD depends only onà A  step stiffness=k B T 2. From ideas of¯uctuation phenomena, TWDs should be describable by the``generalized Wigner distribution'' (GWD), essentially a power-law in`/h`i times a``Gaussian decay'' in`/h`i. The power-law exponent is related simply toÃ. Alternatively, the GWD gives the exact solution for a mean-®eld approximation. The GWD provides at least as good a description of TWDs as the standard ®t to a Gaussian (centered at h`i). It works well for weak elastic repulsion strengthsà (where Gaussians fail), as illustrated explicitly for vicinal Pt(1 1 0). Application to vicinal copper surfaces con®rms the viability of the GWD analysis. The GWD can be treated as a two-parameter ®t by scaling`using an adjustable characteristic width. With Monte Carlo and transfer-matrix calculations, we show that for physical values ofÃ, the GWD provides a better overall estimate than the Gaussian models. We quantify how a GWD approaches a Gaussian for largeà and present a convenient, accurate expression relating the variance of the TWD toÃ. We describe how discreteness of terrace widths impacts the standard continuum analysis.

Tables I-III. 1U Fart III: Real Space Methods for the XY Model Part IV: Transfer Matrix Methods for the Ising Chain in a Transverse Magnetic Field and members of the SLAC theory &roup for useful discussions •-especially Marvin Welnstein, who sugger.te-d the idea for the second part of the thesis. Many thanks to Jacqueline A. llahn, who typed the first manuscript. Finally, special thanks art-due to my family, and In particular to my father, mother and sister for their loving support Erum the very earliest stages of my unusual education to the present day. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United Stalin Govemncnt. Neither trie United States Government nor any agency thcrwf, nor iiny of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi bility for the accuracy, complctene&s, or usefuln«*«v of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer ence hcreir. to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom mendation, or favoring by the United Sta.-S Government or any agency thereof The view* and opinions of authors expressed herein do not neccssjr'k Male 'jr reflect those of the United States Government or any agency thereof.

The generalized Heusler compounds Mn2CoZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) with the Hg2CuTi structure are of large interest due to their half-metallic ferrimagnetism. The complex magnetic interactions between the constituents are studied by first principles calculations of the Heisenberg exchange coupling parameters, and Curie temperatures are calculated from those. Due to the direct Mn-Mn exchange interaction in Mn2CoZ, the Curie temperature decreases, while the total moment increases when changing Z from one group to another. The exchange interactions are dominated by a strong direct exchange between Co and its nearest neighbor Mn on the B site, which is nearly constant. The coupling between the nearest-neighbor Mn atoms scales with the magnetic moment of the Mn atom on the C site. Calculations with different lattice parameters suggest a negative pressure dependence of the Curie temperature, which follows from decreasing magnetic moments. Curie temperatures of more than 800 K are predicted for Mn2CoAl (890 K), Mn2CoGa (886 K), and Mn2CoIn (845 K).

The differential equation of Abrams and Strogatz for the competition between two languages is compared with agentbased Monte Carlo simulations for fully connected networks as well as for lattices in one, two and three dimensions, with up to 10 9 agents. In the case of socially equivalent languages, agent-based models and a mean-field approximation give grossly different results. r

A stochastic cellular automata model for the population dynamics of the army ant Eciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusionlike, although it is random. This has implications for the kind of model that is chosen.

The clear dominance of two-gender sex in recent species is a notorious puzzle of evolutionary theory. It has at least two layers: besides the most fundamental and challenging question why sex exists at all, the other part of the problem is equally perplexing but much less studied. Why do most sexual organisms use a binary mating system? Even if sex confers an evolutionary advantage (through whatever genetic mechanism), why does it manifest that advantage in two, and exactly two, genders (or mating types)? Why not just one, and why not more than two?

The global phase diagram of the Blume-Capel model in a random field obeying the bimodal symmetric distribution is determined by using the mean-field method. The phase diagram includes an isolated ordered critical end point and two lines of tricritical points. A new phase emerges for strong enough random fields: the ferromagnetic-nonmagnetic phase. It is argued that such a phase occurs in three dimensions.

With this paper we will try to introduce the foundations and the formalism of relativistic mean field theory and its applications. We begin by discussing the formulation of the theory of special relativity. Then we derive the Lagrangian formulation of a field from the continuous limit of a discrete system. Afterwards, we formulate a relativistically invariant Lagrangian for a field and use the previous formalism to investigate several problems of continuous systems. Finally, reference is made to the application of the mean field approximation to the nuclear model of Quantum Hadrodynamics (QHD).

This study aims to model the methane partial oxidation process in the burner and combustion chamber of autothermal reactor. The numerical simulation based on this model offers a powerful tool that can assist in reactor design and optimization and scale up of the process saving expensive pilot work. The steady-state governing equations were solved using the SIMPLE algorithm and the effect of turbulence on the mean flow field was accounted for using the RNG k–ε model. A two-step reaction mechanism was used for the gas combustion with CO as the intermediate species. The reaction rates were modeled using an Eddy-Dissipation Model. In terms of the geometrical model, a 3D model for burner was developed while an axis-symmetric model for the combustion chamber was implemented to reduce the computational costs. The model formulated was validated against a currently operating autothermal reactor and then has been used to investigate different aspects of these reactors. Results show that effect of oxygen to methane ratio is more than that of feed temperature. It is demonstrated that a 60% increase in O2/CH4 ratio causes a 15.4% decrease and 42.7% increase in H2/CO ratio and methane conversion, respectively. In contrast, a 60% increase in feed temperature does not have a significant effect on the process.

The complete tensor structure of the quark-gluon vertex in Landau gauge is determined at two kinematical points ('asymmetric' and 'symmetric') from lattice QCD in the quenched approximation. The simulations are carried out at β = 6.0, using a meanfield improved Sheikholeslami-Wohlert fermion action, with two quark masses ∼ 60 and 115 MeV. We find substantial deviations from the abelian form at the asymmetric point. The mass dependence is found to be negligible. At the symmetric point, the form factor related to the chromomagnetic moment is determined and found to contribute significantly to the infrared interaction strength.

We compared the performances of two methods that combine quantum mechanics and molecular mechanics for the study of liquid systems. They dier in the description of the solute±solvent interaction. One makes use of the mean ®eld approximation and the other does not. We show that the introduction of this approximation does not introduce sig-ni®cant errors into the induced dipole moment of the solute, the interaction energy or the solvent structure, while it permits a considerable reduction (from several thousand to four or ®ve) of the number of quantum calculations and hence of the computational demands. Ó

A simple three-state lattice model that incorporates two states for locally ordered and disordered forms of liquid water in addition to empty cells is introduced. The model is isomorphic to the Blume-Emery-Griffith model. The locally ordered (O) and disordered (D) forms of water are treated as two components, and we assume that the density of the D component is larger. The density of the sample is determined by the fraction of cells occupied by the O and D forms of water. Due to the larger density of the D state, the strength of the van der Waals (vdW) interactions increases in the direction O-O<O-D<D-D . On the other hand, the H-bond interactions are assumed only for the O-O pairs. For the vdW and H-bond interaction parameters and the density ratio of the close-packed and ice forms of water compatible with experimentally known values, we find liquid-vapor and liquid-liquid transitions and the corresponding critical points in good agreement with other approaches. Water anomalies are correctly predicted within the mean-field approximation on a qualitative level.

Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications.

A theory of inhomogeneous multicomponent systems containing weakly charged polyelectrolytes is developed. The theory treats the polymer conformation and the electrostatics simutaneously using a functional integral representation of the partition function. A mean-field approximation to the theory leads to two sets of coupled mean-field equations: a generalized Poisson-Boltzmann equation describing the electrostatic potential, and a set of self-consistent field equations describing the equilibrium densities. The theory can be used to study the interfacial properties, microphase structures, and adsorptions of a variety of weakly charged polyelectrolyte systems. As a simple application, the interfaces between the polymer-rich and polymer-poor phases of a polyelectrolyte solution is studied.

We study a model of freely cooling inelastic granular gas in one dimension, with a restitution coefficient which approaches the elastic limit below a relative velocity scale v. While at early times (t << 1/v) the gas behaves as a completely inelastic sticky gas conforming to predictions of earlier studies, at late times (t >> 1/v) it exhibits a new fluctuation dominated phase ordering state. We find distinct scaling behavior for the (i) density distribution function, (ii) occupied and empty gap distribution functions, (iii) the density structure function and (iv) the velocity structure function, as compared to the completely inelastic sticky gas. The spatial structure functions (iii) and (iv) violate the Porod law. Within a mean-field approximation, the exponents describing the structure functions are related to those describing the spatial gap distribution functions.

A strategy for nding approximate solutions to discrete optimization problems with inequality constraints using mean eld neural networks is presented. The constraints x 0 are encoded by x (x) terms in the energy function. A careful treatment of the mean eld approximation for the self-coupling parts of the energy is crucial, and results in an essentially parameter-free algorithm.

Dilute ferromagnetic oxides having Curie temperatures far in excess of 300 K and exceptionally large ordered moments per transition-metal cation challenge our understanding of magnetism in solids. These materials are high-k dielectrics with degenerate or thermally activated n-...

A statistical thermodynamic model of ordering in nonstoichiometric B2-phases is developed, based on the oldest and simplest theory of ordering, the mean-®eld approximation with constant pair-wise interaction energies between nearest-neighbor atoms. This approximation is limited to the description of the distribution of atoms and vacancies on the dierent sublattices, a possible shortrange ordering tendency of point defects is neglected. The expressions for the point defect concentrations as a function of composition and temperature are derived from the grand canonical ensemble. In B2-phases the vacancy concentration depends on composition and temperature and it is therefore necessary to consider an open system, where the transfer of atoms to or from the outside is allowed. In B2-phases not only one defect equilibrium is involved, but there are four defect equilibria to be considered. It is shown that, if a series of defect equilibria is involved, it is necessary to apply the principle of microscopic reversibility to formulate the required number of independent equilibrium conditions. Expressions for the activities and partial molar enthalpies of the alloy components as a function of composition and temperature are derived, and a comparison with experimental data for b H -FeAl is given, which shows the practicability of this approach. # Intermetallics 7 (1999) 141±151 0966-9795/99/$Ðsee front matter #

In order to study analytically the nature of the jamming transition in granular material, we have considered a cavity method mean field theory, in the framework of a statistical mechanics approach, based on Edwards' original idea. For simplicity we have applied the theory to a lattice model and a transition with exactly the same nature of the glass transition in mean field models for usual glass formers is found. The model is also simulated in three dimensions under tap dynamics and a jamming transition with glassy features is observed. In particular two step decays appear in the relaxation functions and dynamic heterogeneities resembling ones usually observed in glassy systems. These results confirm early speculations about the connection between the jamming transition in granular media and the glass transition in usual glass formers, giving moreover a precise interpretation of its nature.

Heisenberg exchange parameters for various Heusler compounds with L2 1 structure were calculated using the Korringa-Kohn-Rostoker method and by employing the magnetic-force theorem to calculate the total energy changes associated with a local rotation of magnetization directions. Random occupation was treated within the coherent potential approximation. Further, the Curie temperatures were calculated in the mean-field approximation and have been found to be in good agreement with the experiment. A procedure for evaluating the spin-stiffness constants for the case of multiple magnetic sublattices is given and the results were compared with measured values. Magnon dispersion curves were obtained by Fourier transforming the calculated exchange parameters.

We apply recently developed effective field theory nuclear models in mean field approximation (parameter sets G1 and G2) to describe ground-state properties of nuclei from the valley of β-stability up to the drip lines. For faster calculations of openshell nuclei we employ a modified BCS approach which takes into account quasi-bound levels owing to their centrifugal barrier, with a constant pairing strength. We test this simple prescription by comparing with available Hartree-plus-Bogoliubov results.

We present a systematic study of quantum phases in a one-dimensional spin-polarized Fermi gas. Three comparative theoretical methods are used to explore the phase diagram at zero temperature: the mean-field theory with either an order parameter in a single-plane-wave form or a self-consistently determined order parameter using the Bogoliubov-de Gennes equations, as well as the exact Bethe ansatz method. We find that a spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov phase, which lies between the fully paired Bardeen-Cooper-Schrieffer ͑BCS͒ state and the fully polarized normal state, dominates most of the phase diagram of a uniform gas. The phase transition from the BCS state to the Fulde-Ferrell-Larkin-Ovchinnikov phase is of second order, and therefore there are no phase separation states in one-dimensional homogeneous polarized gases. This is in sharp contrast to the three-dimensional situation, where a phase separation regime is predicted to occupy a very large space in the phase diagram. We conjecture that the prediction of the dominance of the phase separation phases in three dimension could be an artifact of the non-self-consistent meanfield approximation, which is heavily used in the study of three-dimensional polarized Fermi gases. We consider also the effect of a harmonic trapping potential on the phase diagram, and find that in this case the trap generally leads to phase separation, in accord with the experimental observations for a trapped gas in three dimensions. We finally investigate the local fermionic density of states of the Fulde-Ferrell-Larkin-Ovchinnikov ansatz. A two-energy-gap structure appears, which could be used as an experimental probe of the Fulde-Ferrell-Larkin-Ovchinnikov states.

Mean-field Boltzmann machine learning is recognized as a practical method to circumvent the difficulty that Boltzmann machine learning is very time-consuming. However, its theoretical meaning is still not clear. In this paper, based on information geometry, we give an information-theoretic interpretation of mean-field Boltzmann machine learning and a clear geometrical explanation of the approximation used there. Based on this interpretation, computer simulations for evaluating the effectiveness of mean-field Boltzmann machine learning are carried out for two-unit Boltzmann machines. The necessity of geometrical analysis in demonstrating the effectiveness of meanfield Boltzmann machine learning is discussed.

The electron-mediated coupling of external electromagnetic fields and Raman-active oscillations is derived for a general electronic model with multiple bands using the adiabatic approach and the explicit diagrammatic approach. The theory is illustrated on the quasi-one-dimensional (Q1D) quarter-filled charge-density-wave (CDW) model. It is shown how the long-range Coulomb forces and the single-electron relaxation processes affect the Raman spectroscopy of the amplitude-oscillation mode in clean CDW systems. It is also argued that the adiabatic treatment of the photon-phonon coupling functions can be safely used in this case. is the energy of the electron-hole pair and E LL (k) ≡ E LL (k, k) is a useful abbreviation. All these density operators are defined in the usual way; for example, the current density operator iŝ

We study the model of interacting agents proposed by Chatterjee (2003) that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined approximations defined in the text, these equations yield the conjecture proposed by Chatterjee (2003) for the form of the stationary agent wealth distribution. If the savings propensity for the equations is chosen according to some random distribution we show further that the wealth distribution for large values of wealth displays a Pareto like power law tail, ie P (w) ∼ w 1+a. However the value of a for the model is exactly 1. Exact numerical simulations for the model illustrate how, as the savings distribution function narrows to zero, the wealth distribution changes from a Pareto form to to an exponential function. Intermediate regions of wealth may be approximately described by a power law with a > 1. However the value never reaches values of ∼ 1.6 − 1.7 that characterise empirical wealth data. This conclusion is not changed if three body agent exchange processes are allowed. We conclude that other mechanisms are required if the model is to agree with empirical wealth data.

In this work we revisit the concept of chemical potential µ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state µ = µ(n, T), where n is the particle density and T the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of µ(n, T) for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on µ(n, T) for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most j particles with 0 j N with N the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state µ = µ(n, T) in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.

In this article the Ginzburg-Landau theory ideas are considered in their application to the description of fluctuations influence on the superfluid density in superconductor. The conclusion about the availability of two incompatible mathematical definitions of the superfluid density is made. That is why, it is suggested not to consider the fluctuating part of order parameter, while calculating the superconductor thermodynamical characteristics in the Mean Field Approximation. Comment: 5 pages, 0 figures

A formalism is developed for evaluating probabilities and cross sections for multiple-electron transitions in scattering of molecules and clusters by charged collision partners. First, the molecule is divided into subclusters each made up of identical centers ͑atoms͒. Within each subcluster coherent scattering from identical centers may lead to observable phase terms and a geometrical structure factor. Then, using a mean field approximation to describe the interactions between centers we obtain A I ϳ ͚ k ͟ k e i␦ I k A Ik . Second, the independent electron approximation for each center may be obtained by neglecting the correlation between electrons in each center. The probability amplitude for each center is then a product of single electron transition probability amplitudes, a Ik i , i.e. A Ik Ϸ͟ i a ik i . Finally, the independent subcluster approximation is introduced by neglecting the interactions between different subclusters in the molecule or cluster. The total probability amplitude then reduces to a simple product of amplitudes for each subcluster, AϷ͟ I A I . Limitations of this simple approximation are discussed.

We study the quantum phase diagram and excitation spectrum of the frustrated J1-J2 spin-1/2 Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea of identifying relevant degrees of freedom, is developed. Thus, by performing educated, manifestly symmetry preserving mean-field approximations, we unveil fundamental properties of the system. We then compare various coverings of the square lattice with plaquettes, dimers and other degrees of freedom, and show that only the symmetric plaquette covering, which reproduces the original Bravais lattice, leads to the known phase diagram. The intermediate quantum paramagnetic phase is shown to be a (singlet) plaquette crystal, connected with the neighbouring Néel phase by a continuous phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and demonstrate that in the paramagnetic phase the ground and first excited states are separated by a finite gap, which closes in the Néel and columnar phases. Our results suggest that the quantum phase transition between Néel and paramagnetic phases can be properly described within the Ginzburg-Landau-Wilson paradigm.

We study both theoretically and experimentally the anchoring properties of photoaligning azo-dye films in contact with a nematic liquid crystal depending on the photoinduced ordering of azo-dye molecules. In the mean field approximation, we found that the bare surface anchoring energy depends linearly on the azo-dye order parameter and the azimuthal anchoring strength decays to zero in the limit of vanishing photoinduced ordering. From the absorption dichroism spectra measured in azo-dye films that are prepared from an azo-dye derivative with polymerizable terminal groups we obtain the dependence of the dichroic ratio on the irradiation dose. We also measure the polar and azimuthal anchoring strengths in nematic liquid crystal ͑NLC͒ cells aligned by the azo-dye films and derive the anchoring strengths as functions of the dichroic ratio, which is proportional to the photoinduced order parameter. Although linear fitting of the experimental data for both anchoring strengths gives reasonable results, it, predicts vanishing of the azimuthal anchoring strength at some nonzero value of the azo-dye order parameter, in contradiction with theory. By using a simple phenomenological model we show that this discrepancy can be attributed to the difference between the surface and bulk order parameters in the films. The measured polar anchoring energy is found to be an order of magnitude higher than the azimuthal strength. Our theory suggests that the quadrupole term of the spherical harmonics expansion for the azo-dye-NLC intermolecular potential might be of importance for the understanding of this difference.

The specific heat of SrRuOs was measured in the temperature range of loo-250 K. The magnetic transition at 165 K is clearly seen, and a magnetic moment of 0.5 /JB is obtained from the magnetic specificheat jump at Tc calculated from the mean-field approximation. @ 1997 Elsevier Science Ltd. All rights reserved 1. Randall, J.

The phase transition to superfluidity and the BCS-BEC crossover for an ultracold gas of fermionic atoms is discussed within a functional renormalization group approach. Non-perturbative flow equations, based on an exact renormalization group equation, describe the scale dependence of the flowing or average action. They interpolate continuously from the microphysics at atomic or molecular distance scales to the macroscopic physics at much larger length scales, as given by the interparticle distance, the correlation length, or the size of the experimental probe. We discuss the phase diagram as a function of the scattering length and the temperature and compute the gap, the correlation length and the scattering length for molecules. Close to the critical temperature, we find the expected universal behavior. Our approach allows for a description of the few-body physics (scattering and molecular binding) and the many-body physics within the same formalism.

We report the results of a Monte Carlo study of a model of (III,Mn)V diluted magnetic semiconductors which uses an impurity band description of carriers coupled to localized Mn spins and is applicable for carrier densities below and around the metal-insulator transition. In agreement with mean field studies, we find a transition to a ferromagnetic phase at low temperatures. We compare our results for the magnetic properties with the mean field approximation, as well as with experiments, and find favorable qualitative agreement with the latter. The local Mn magnetization below the Curie temperature is found to be spatially inhomogeneous, and strongly correlated with the local carrier charge density at the Mn sites. The model contains fermions and classical spins and hence we introduce a perturbative Monte Carlo scheme to increase the speed of our simulations.

We study the magnetic properties of nanometer-sized graphene structures with triangular and hexagonal shapes terminated by zig-zag edges. We discuss how the shape of the island, the imbalance in the number of atoms belonging to the two graphene sublattices, the existence of zero-energy states, and the total and local magnetic moment are intimately related. We consider electronic interactions both in a mean-field approximation of the one-orbital Hubbard model and with density functional calculations. Both descriptions yield values for the ground state total spin, $S$, consistent with Lieb's theorem for bipartite lattices. Triangles have a finite $S$ for all sizes whereas hexagons have S=0 and develop local moments above a critical size of $\\approx 1.5$ nm.

The primary crystallisation of a highly undercooled/supersaturated liquid is considered, and the application to nanocrystallisation by heat treatment of metallic glasses is studied from the thermodynamic, kinetic and microstructural point of view. The thermodynamic evolution is modelled assuming transformation rates low enough to ensure thermal equilibrium to be almost achieved. A mean field approximation is used, which allows us to determine the time evolution of the kinetic variables governing the transformation. The interplay between interface and diffusion controlled growth rate is studied, and both nucleation and crystal growth changes within the transformation are considered as soft mechanisms. The kinetics of the transformation is described in the framework of the Kolmogorov, Johnson and Mehl and Avrami (KJMA) model, which is adequately generalized for primary transformations. The microstructural evolution is described by a populational model, also based on KJMA. The predicted kinetic evolution results are compared to the experimental data on the primary nanocrystallisation of a FINEMET alloy. #

The equation of state for quark matter is derived for a nonlocal, chiral quark model within the mean field approximation. We investigate the effects of a variation of the form factors of the interaction on the phase diagram of quark matter under the condition of β -equilibrium and charge neutrality. Special emphasis is on the occurrence of a diquark condensate which signals a phase transition to color superconductivity and its effects on the equation of state. We calculate the quark star configurations by solving the Tolman-Oppenheimer-Volkoff equations and obtain for the transition from a hot, normal quark matter core of a protoneutron star to a cool diquark condensed one a release of binding energy of the order of ∆Mc 2 ∼ 10 53 erg. We study the consequences of antineutrino trapping in hot quark matter for quark star configurations with possible diquark condensation and discuss the claim that this energy could serve as an engine for explosive phenomena. A "phase diagram" for rotating compact stars (angular velocity-baryon mass plane) is suggested as a heuristic tool for obtaining constraints on the equation of state of QCD at high densities. It has a critical line dividing hadronic from quark core stars which is correlated with a local maximum of the moment of inertia and can thus be subject to experimental verification by observation of the rotational behavior of accreting compact stars.

A theoretical description for the radial density profile of a finite number of identical charged particles confined in a harmonic trap is developed for application over a wide range of Coulomb coupling (or, equivalently, temperatures) and particle numbers. A simple mean field approximation neglecting correlations yields a density profile which is monotonically decreasing with radius for all temperatures, in contrast to molecular dynamics simulations and experiments showing shell structure at lower temperatures. A more complete theoretical description including charge correlations is developed here by an extension of the hypernetted chain approximation, developed for bulk fluids, to the confined charges. The results reproduce all of the qualitative features observed in molecular dynamics simulations and experiments. These predictions are then tested quantitatively by comparison with new benchmark Monte Carlo simulations. Quantitative accuracy of the theory is obtained for the selected conditions by correcting the hypernetted chain approximation with a representation for the associated bridge functions.

with repulsive forces and confined in a harmonic anisotropic trap. We develop the formalism of mean field theory for non uniform systems at finite temperature, based on the generalization of Bogoliubov theory for uniform gases. By employing the WKB semiclassical approximation for the excited states we derive systematic results for the temperature dependence of various thermodynamic quantities: condensate fraction, density profiles, thermal energy, specific heat and moment of inertia. Our analysis points out important differences with respect to the thermodynamic behaviour of uniform Bose gases. This is mainly the consequence of a major role played by single particle states at the boundary of the condensate. We find that the thermal depletion of the condensate is strongly enhanced by the presence of repulsive interactions and that the critical temperature is decreased with respect to the predictions of the non-interacting model. Our work points out an important scaling behaviour exhibited by the system in large N limit. Scaling permits to express all the relevant thermodynamic quantities in terms of only two parameters: the

The transport of particles moving out of equilibrium in an asymmetric substrate potential has been studied intensively for a variety of different systems [1, 2] in order to achieve an efficient control of the net particle flow. Various realizations of rachet systems working out of equilibrium have been proposed involving different rectification mechanisms, like temporal temperature oscillations (temperature ratchet [3]), zeroaverage sinusoidal ac forces (ac tilted or rocked ratchet [4]), stochastic and deterministic fluctuations of the ratchet potentials [5], ...

The Heisenberg nearest-neighbor antiferromagnet on the pyrochlore (three-dimensional) lattice is highly frustrated and does not order at low temperature where spin-spin correlations remain short ranged. Dzyaloshinsky-Moriya interactions (DMI&#39;s) may be present in pyrochlore compounds as is shown, and the consequences of such interactions on the magnetic properties are investigated through mean-field approximation and Monte Carlo simulations. It is found that

Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this paper we develop a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow, and analyze it using simulation, dimensional analysis, and theoretical tools based on mean field approximations. The model makes testable predictions for basic properties of markets, such as price volatility, the depth of stored supply and demand vs. price, the bid-ask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow and the limit order book, such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly there are no free parameters. We show that the order size, which can be cast as a nondimensional granularity parameter, is in most cases a more significant determinant of market behavior than tick size. We also provide an explanation for the observed highly concave nature of the price impact function. On a broader level, this work suggests how stochastic models based on zero-intelligence agents may be useful to probe the structure of market institutions. Like the model of perfect rationality, a stochastic-zero intelligence model can be used to make strong predictions based on a compact set of assumptions, even if these assumptions are not fully believable.