Class Of Mean-field Models Research Articles

Semiclassical Equivalence of Two White Dwarf Models as Ground States of the Relativistic Hartree–Fock and Vlasov–Poisson energies

AbstractWe are concerned with the semi-classical limit for ground states of the relativistic Hartree–Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs (Lenzmann and Lewin in Duke Math J 152:257–315, 2010). In Jang and Seok (Kinet Relat Models 15:605–620, 2022), fermionic ground states of the relativistic Vlasov–Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $$\hbar $$ ħ goes to the zero, the $$\hbar $$ ħ -parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.

Comparison of Grain-Growth Mean-Field Models Regarding Predicted Grain Size Distributions.

Mean-field models have the ability to predict the evolution of grain size distribution that occurs through thermomechanical solicitations. This article focuses on a comparison of mean-field models under grain-growth conditions. Different microstructure representations are considered and discussed, especially regarding the consideration of topology in the neighborhood construction. Experimental data obtained with a heat treatment campaign on 316L austenitic stainless steel are used for the identification of material parameters and as a reference for model comparisons. Mean-field models are also applied to both mono- and bimodal initial grain size distributions to investigate the potential benefits of introducing neighborhood topology in microstructure prediction models. This article demonstrates that improvements in the predictions can be obtained in monomodal cases for topological models. In the bimodal test, no comparison with experimental data was performed as no data were available. But relative comparisons between models indicated few differences in the predictions. Although of interest, the consideration of neighborhood topology in grain-growth mean-field models generally results in only small improvements compared to classical mean-field models, especially in terms of implementation complexity.

Migration–contagion processes

AbstractConsider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $\cdot$ /M/ $\infty$ queue with service (or migration) rate $\mu$ . Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $\alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $\beta$ . We let N tend to infinity and assume that $\lim_{N\to \infty} K_N/N= \eta $ , where $\eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $\eta$ . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.

Magneto transport and critical behavior in metastable Fe15Cu85

FexCu[Formula: see text] binary alloys attracted much attention owing to their rich magnetic phase diagram and the challenges of achieving single phases in these systems. Despite early efforts put toward studying the physical properties of low-Cu doping in Fe–Cu alloys, a detailed understanding of the magnetic, structural and electrical properties of high Cu doping in Fe–Cu alloys remains not clear. In this report, we use X-ray diffraction (XRD), scanning electron microscopy, electrical transport and magnetic measurements to gain more insight into Fe[Formula: see text]Cu[Formula: see text] single-phase alloy system. Experimental findings show that the samples exhibit a second-order phase transition, which is consistent with the classical mean-field model. Moreover, negative magnetoresistance is observed at all temperatures and explained through classical models.

Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the point vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher-dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding “microscopic” concavity result at finite N, shown by leveraging an unexpected link to Kähler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application, it is shown that the critical inverse negative temperatures—in the macroscopic as well as the microscopic setting—coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way, we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.

Tricritical point in the quantum Hamiltonian mean-field model.

Engineering long-range interactions in experimental platforms has been achieved with great success in a large variety of quantum systems in recent years. Inspired by this progress, we propose a generalization of the classical Hamiltonian mean-field model to fermionic particles. We study the phase diagram and thermodynamic properties of the model in the canonical ensemble for ferromagnetic interactions as a function of temperature and hopping. At zero temperature, small charge fluctuations drive the many-body system through a first-order quantum phase transition from an ordered to a disordered phase. At higher temperatures, the fluctuation-induced phase transition remains first order initially and switches to second-order only at a tricritical point. Our results offer an intriguing example of tricriticality in a quantum system with long-range couplings, which bears direct experimental relevance. The analysis is performed by exact diagonalization and mean-field theory.

Classical and quantum harmonic mean-field models coupled intensively and extensively with external baths

We study the nonequilibrium steady-state of a fully-coupled network of N quantum harmonic oscillators, interacting with two thermal reservoirs. Given the long-range nature of the couplings, we consider two setups: one in which the number of particles coupled to the baths is fixed (intensive coupling) and one in which it is proportional to the size N (extensive coupling). In both cases, we compute analytically the heat fluxes and the kinetic temperature distributions using the nonequilibrium Green's function approach, both in the classical and quantum regimes. In the large N limit, we derive the asymptotic expressions of both quantities as a function of N and the temperature difference between the baths. We discuss a peculiar feature of the model, namely that the bulk temperature vanishes in the thermodynamic limit, due to a decoupling of the dynamics of the inner part of the system from the baths. At variance with the usual case, this implies that the steady-state depends on the initial state of the bulk particles. We also show that quantum effects are relevant only below a characteristic temperature that vanishes as 1/N. In the quantum low-temperature regime the energy flux is proportional to the universal quantum of thermal conductance.

Q-Learning in Regularized Mean-field Games

In this paper, we introduce a regularized mean-field game and study learning of this game under an infinite-horizon discounted reward function. Regularization is introduced by adding a strongly concave regularization function to the one-stage reward function in the classical mean-field game model. We establish a value iteration based learning algorithm to this regularized mean-field game using fitted Q-learning. The regularization term in general makes reinforcement learning algorithm more robust to the system components. Moreover, it enables us to establish error analysis of the learning algorithm without imposing restrictive convexity assumptions on the system components, which are needed in the absence of a regularization term.

Optically controlled polariton condensate molecules

A condensed matter platform for analogue simulation of complex two-dimensional molecular bonding configurations, based on optically trapped exciton-polariton condensates is proposed. The stable occupation of polariton condensates in the excited states of their optically configurable potential traps permits emulation of excited atomic orbitals. A classical mean field model describing the dissipative coupling mechanism between p-orbital condensates is derived, identifying lowest threshold condensation solutions as a function of trap parameters corresponding to bound and antibound $\pi$ and $\sigma$ bonding configurations, similar to those in quantum chemistry.

Magnetic anisotropy and exchange paths for octahedrally and tetrahedrally coordinated Mn2+ ions in the honeycomb multiferroic Mn2Mo3O8

We investigated the static and dynamic magnetic properties of the polar ferrimagnet Mn$_2$Mo$_3$O$_8$ in three magnetically ordered phases via magnetization, magnetic torque, and THz absorption spectroscopy measurements. The observed magnetic field dependence of the spin-wave resonances, including Brillouin zone-center and zone-boundary excitations, magnetization, and torque, are well described by an extended two-sublattice antiferromagnetic classical mean-field model. In this orbitally quenched system, the competing weak easy-plane and easy-axis single-ion anisotropies of the two crystallographic sites are determined from the model and assigned to the tetra- and octahedral sites, respectively, by ab initio calculations.

Thermo-mechanical Evaluation of Slurry-Sprayed Multi-layered Coatings

Thermal barrier coatings (TBCs) represent a relatively thin layer of ceramic with the favourable insulating properties, which are generally used to improve the temperature stability of the engineering component such as turbine blades. One of the prime prerequisites of TBCs is to determine the optimised coating configuration with the desired thermo-mechanical properties and enhanced service life. In a typical functionally graded coating structure, this could be achieved by having a trade-off between the thermal insulation and fatigue toughness offered by ceramic in top coat and metal towards the substrate, respectively. In this work, a computational method was used to analyse and optimise the parameters pertaining to thermo-mechanical evaluation of the slurry spray-coated (SST) mullite–nickel ASTM 1018 steel. The composite material properties have been predicted using the classical mean-field micromechanics model and rule of mixtures and the finite element simulation package ANSYS. Experimental validation has been performed to analyse the relative thermal and structural properties of the composite layers of the coatings using transient plane source (TSP)-based thermal constants analyser. The predicted and experimental results were further analysed and optimised for various process parameters using response surface optimisation and multi-objective genetic algorithm, which evaluated the optimum results considering the boundary conditions with a temperature reduction of nearly 306 °C. Further, the research results indicate that suitable thickness of the coating configuration of the slurry-sprayed mullite–nickel TBC system included coating thickness of 57.2 μm, 147.1 μm, and 143.9 μm for bond, intermediate, and top coats, respectively. The material properties were found dependent on the coating composition across the FG structure, and the predicted, computed, and experimental results were in reasonable agreement.

Small-scale spatial structure affects predator-prey dynamics and coexistence

Small-scale spatial variability can affect community dynamics in many ecological and biological processes, such as predator-prey dynamics and immune responses. Spatial variability includes short-range neighbour-dependent interactions and small-scale spatial structure, such as clustering where individuals aggregate together, and segregation where individuals are spaced apart from one another. Yet, a large class of mathematical models aimed at representing these processes ignores these factors by making a classical mean-field approximation, where interactions between individuals are assumed to occur in proportion to their average density. Such mean-field approximations amount to ignoring spatial structure. In this work, we consider an individual-based model of a two-species community that is composed of consumers and resources. The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the individual-based model with the solution of a classical mean-field approximation, and this comparison provides insight into how spatial structure can drive the system away from mean-field dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the mean-field prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical mean-field model predicts the coexistence of both species.

Small-Scale Analysis of Hydrodynamical Helicity Suppression in the Mean-Field Dynamo-Model

We compare the classical mean-field dynamo model proposed by Steenbeck, Krause, and Radler to describe the generation of large-scale magnetic fields and the Kazantsev model that describes the small-scale dynamo in an unbounded homogeneous and isotropic flow. We consider the subcritical regime of small magnetic Reynolds numbers whereby there is no rapid generation. The same regime can also be understood as a process in which the small-scale generation is stopped due to its intrinsic mechanisms. Within both approaches we examine what distinguishes the spectra of the linear and nonlinear processes under the suppression of hydrodynamic (kinetic) helicity or, in other words, compare the alpha-quenchings. We check whether averaging the induction equation over scales larger than the velocity field correlation length leads to the loss of any features in the spectrum near the dissipative scale. We study the various types of dynamo stabilization using which seems more justified physically than the standard alpha-quenching, but more difficult within large-scale models containing limited information about the random velocity field. In particular, we compare the integral suppression whereby the total energy is conserved and the spectral suppression that suggests the conservation of energy and helicity in each spectral shell without assuming their redistribution over the spectrum.

On the modelling of precipitation kinetics in a turbine disc nickel based superalloy

The precipitation kinetics of gamma prime in the nickel based superalloy RR1000 has been characterised after solid-solution heat treatments and isothermal aging conditions relevant to service conditions. Multi-modal precipitate dispersions are formed within the alloy. Numerical methods are presented for determining the three dimensional size of the particle populations combining information obtained from Scanning Electron microscopy and Transmission Electron microscopy. This information has been used to develop a multi-component mean-field model descriptive of precipitation kinetics. The smallest particle population increases in mean size during isothermal aging at 700 ∘C where classical mean-field models of coarsening kinetics suggest that these particles should dissolve. A phenomenological model has been proposed to capture this behaviour within a statistical formulation that is applicable to both processing and service conditions.

Classical glasses, black holes, and strange quantum liquids

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a time-reparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.

A conservative scheme for Vlasov Poisson Landau modeling collisional plasmas

We have developed a deterministic conservative solver for the inhomogeneous Fokker–Planck–Landau equation coupled with the Poisson equation, which is a classical mean-field primary model for collisional plasmas. Two subproblems, i.e. the Vlasov–Poisson problem and homogeneous Landau problem, are obtained through time-splitting methods, and treated separately by the Runge–Kutta Discontinuous Galerkin method and a conservative spectral method, respectively. To ensure conservation when projecting between the two different computing grids, a special conservation routine is designed to link the solutions of these two subproblems. This conservation routine accurately enforces conservation of moments in Fourier space. The entire numerical scheme is implemented with parallelization with hybrid MPI and OpenMP. Numerical experiments are provided to study linear and nonlinear Landau Damping problems and two-stream flow problem as well.

Fluctuation effects in grain growth

In this study, we attempted to clarify the roles of fluctuation effects in grain growth. To capture the persistent nature in both space and time of fluctuations due to variations in the local surroundings of individual grains, we developed a local mean-field model. The fluctuation strength in this model is arbitrarily controlled by employing an artificial number, , of nearest neighbor grains. Large-scale numerical computations of the model for various values and initial GSDs were carried out to follow transient behaviors and determine the steady states. This study reveals that, in the classical mean-field model with no fluctuation effects, the steady state is not unique but is strongly dependent upon the initial GSD. However, a small fluctuation drives the mean-field model to reach the Hillert solution, independent of the fluctuation strength and initial GSD, as long as the fluctuation strength is sufficiently small. On the other hand, when the fluctuation is sufficiently strong, the fluctuation pushes the steady state of the mean-field model out of the Hillert solution, and its strength determines a unique steady state independent of the initial GSD. The strong fluctuation makes the GSD more symmetric than the Hillert distribution. Computations designed to mimic actual 2 and 3D grain growth were carried out by taking the number of nearest neighbors of each grain as a function of the scaled grain size. The resultant GSDs in two and three dimensions were compared with the direct simulations of ideal grain growth.

Shared information in classical mean-field models

Universal scaling of entanglement estimators of critical quantum systems has drawn a lot of attention in the past. Recent studies indicate that similar universal properties can be found for bipartite information estimators of classical systems near phase transition, opening a new direction in the study of critical phenomena. We explore this subject by studying the information estimators of classical spin chains with general mean-field interactions. In our explicit analysis of two different bipartite information estimators in the canonical ensemble we find that, away from criticality, both the estimators remain finite in the thermodynamic limit. On the other hand, along the critical line there is a logarithmic divergence with increasing system size. The coefficient of the logarithm is fully determined by the mean-field interaction and it is the same for the class of models we consider. The scaling function, however, depends on the details of each model. In addition, we study the information estimators in the micro-canonical ensemble, where they are shown to exhibit a different universal behavior. We verify our results using numerical calculation for two specific cases of the general Hamiltonian.

On quantum and relativistic mechanical analogues in mean-field spin models

Conceptual analogies among statistical mechanics and classical or quantum mechanics have often appeared in the literature. For classical two-body mean-field models, such an analogy is based on the identification between the free energy of Curie–Weiss-type magnetic models and the Hamilton–Jacobi action for a one-dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrödinger equation. We show that this identification can be remarkably extended to include a wider family of magnetic models that are classified by normal forms of suitable real algebraic dispersion curves . In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton–Jacobi type. We observe that the mechanical analogue of these models can be viewed as the relativistic analogue of the Curie–Weiss model and this helps to clarify the connection between generalized self-averaging in statistical thermodynamics and the semiclassical dynamics of viscous conservation laws.

Kinetics of Ga droplet decay on thin carbon films

Using in situ transmission electron microscopy, we investigated the kinetics of liquid Ga droplet decay on thin amorphous carbon films during annealing at 773 K. The transmission electron microscopy images reveal that liquid Ga forms spherical droplets and undergo coarsening/decay with increasing time. We find that the droplet volumes change non-linearly with time and the volume decay rates depend on their local environment. By comparing the late-stage decay behavior of the droplets with the classical mean-field theory model for Ostwald ripening, we determine that the decay of Ga droplets occurs in the surface diffusion limited regime.