Mean-field Interaction Research Articles

Physical interpretation of the canonical ensemble for long-range interacting systems in the absence of ensemble equivalence.

In systems with long-range interactions, since energy is a nonadditive quantity, ensemble inequivalence can arise: it is possible that different statistical ensembles lead to different equilibrium descriptions, even in the thermodynamic limit. The microcanonical ensemble should be considered the physically correct equilibrium distribution as long as the system is isolated. The canonical ensemble, on the other hand, can always be defined mathematically, but it is quite natural to wonder to which physical situations it does correspond. We show numerically and, in some cases, analytically that the equilibrium properties of a generalized Hamiltonian mean-field model in which ensemble inequivalence is present are correctly described by the canonical distribution in (at least) two different scenarios: (a) when the system is coupled via local interactions to a large reservoir (even if the reservoir shows, in turn, ensemble inequivalence), and (b) when the mean-field interaction between a small part of a system and the rest of it is weakened by some kind of screening.

The Hydrodynamic Limit for Local Mean-Field Dynamics with Unbounded Spins

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean field interactions. We proof convergence of the empirical measure to the solution of a McKean-Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of G\"artner, Comets and others for bounded spins or strict mean field interactions.

Analytical approximations of non-linear SDEs of McKean–Vlasov type

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean–Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. “Propagation of chaos“ results ([15]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean–Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. Concise numerical tests are presented to illustrate the accuracy of the resulting approximation formulas. The latter are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

Mean field dynamics of some open quantum systems.

We consider a large number N of quantum particles coupled via a mean field interaction to another quantum system (reservoir). Our main result is an expansion for the averages of observables, both of the particles and of the reservoir, in inverse powers of [Formula: see text]. The analysis is based directly on the Dyson series expansion of the propagator. We analyse the dynamics, in the limit [Formula: see text], of observables of a fixed number n of particles, of extensive particle observables and their fluctuations, as well as of reservoir observables. We illustrate our results on the infinite mode Dicke model and on various energy-conserving models.

Viability Theorem for Deterministic Mean Field Type Control Systems

A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.

Solitons and rogue waves in spinor Bose-Einstein condensates.

We present a general classification of one-soliton solutions as well as families of rogue-wave solutions for F=1 spinor Bose-Einstein condensates (BECs). These solutions are obtained from the inverse scattering transform for a focusing matrix nonlinear Schrödinger equation which models condensates in the case of attractive mean-field interactions and ferromagnetic spin-exchange interactions. In particular, we show that when no background is present, all one-soliton solutions are reducible via unitary transformations to a combination of oppositely polarized solitonic solutions of single-component BECs. On the other hand, we show that when a nonzero background is present, not all matrix one-soliton solutions are reducible to a simple combination of scalar solutions. Finally, by taking suitable limits of all the solutions on a nonzero background we also obtain three families of rogue-wave (i.e., rational) solutions.

Spin-chain model of a many-body quantum battery

Recently, it has been shown that energy can be deposited on a collection of quantum systems at a rate that scales super-extensively. Some of these schemes for `quantum batteries' rely on the use of global many-body interactions that take the batteries through a correlated short cut in state space. Here, we extend the notion of a quantum battery from a collection of a priori isolated systems to a many-body quantum system with intrinsic interactions. Specifically, we consider a one-dimensional spin chain with physically realistic two-body interactions. We find that the spin-spin interactions can yield an advantage in charging power over the non-interacting case, and we demonstrate that this advantage can grow super-extensively when the interactions are long ranged. However, we show that, unlike in previous work, this advantage is a mean-field interaction effect that does not involve correlations and that relies on the interactions being intrinsic to the battery.

Magnetar crust electron capture for ^{55}Co and ^{56}Ni

Based on the relativistic mean-field effective interaction principle and random phase approximation theory in superstrong magnetic fields (SMFs), we present an analysis of the influence of SMFs on the electron Fermi energy, nuclear blinding energy, single-particle level structure and electron capture for ^{55}Co, and ^{56}Ni by the shell-model Monte Carlo method in the magnetar’s crust. The electron capture rates increase by two orders of magnitude due to an increase in the electron Fermi energy and a change in single-particle level structure by SMFs. Then the rates decrease by more than two orders of magnitude due to an increase in the nuclear binding energy and a reduction in the electron Fermi energy by SMFs.

Linear quadratic mean-field-game of backward stochastic differential systems

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Equilibrium large deviations for mean-field systems with translation invariance

We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean-Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of appropriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.

Self-Consistent Field Lattice Model for Polymer Networks.

A lattice model based on polymer self-consistent field theory is developed to predict the equilibrium statistics of arbitrary polymer networks. For a given network topology, our approach uses moment propagators on a lattice to self-consistently construct the ensemble of polymer conformations and cross-link spatial probability distributions. Remarkably, the calculation can be performed “in the dark”, without any prior knowledge on preferred chain conformations or cross-link positions. Numerical results from the model for a test network exhibit close agreement with molecular dynamics simulations, including when the network is strongly sheared. Our model captures nonaffine deformation, mean-field monomer interactions, cross-link fluctuations, and finite extensibility of chains, yielding predictions that differ markedly from classical rubber elasticity theory for polymer networks. By examining polymer networks with different degrees of interconnectivity, we gain insight into cross-link entropy, an important quantity in the macroscopic behavior of gels and self-healing materials as they are deformed.

Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

Nous etudions la mesure de trajectoire transformee engendree par l’auto-interaction d’un mouvement Brownien tridimensionnel en utilisant un biais exponentiel par l’energie de Coulomb des mesures d’occupation de ce mouvement au temps $t$. Les asymptotes logarithmiques de la fonction de partition ont ete identifiees dans les annees 1980 par Donsker et Varadhan [Comm. Pure Appl. Math. 505 (1983) 505–528] au moyen d’une formule variationnelle. Recemment, dans (Brownian occupations measures, compactness and large deviations (2014) Preprint), une nouvelle technique pour etudier la mesure de chemins elle-meme a ete introduite. Elle permet de prouver que la mesure d’occupation normalisee se concentre asymptotiquement autour de l’ensemble des maximums de la formule. Dans le present article, nous prouvons que la fonctionnelle de Coulomb de la mesure d’occupation se concentre elle aussi, dans la topologie uniforme, autour de l’ensemble des fonctionnelles de Coulomb correspondant aux maximums. Ceci represente une etape decisive vers une preuve rigoureuse de la convergence des mesures d’occupation normalisees vers un melange explicite des maximums, derivee dans (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Nos methodes reposent sur l’obtention de la continuite holderienne de la fonctionnelle de Coulomb de la mesure d’occupation avec des probabilites de deviations exponentiellement petites, en invoquant la theorie des grandes deviations developpee dans (Brownian occupations measures, compactness and large deviations (2014) Preprint) pour une certaine fonctionnelle, invariante par dcalage, des mesures d’occupation.

Composite Boson Description of a Low-Density Gas of Excitons

Ground state properties of a fermionic Coulomb gas are calculated using the fixed-node diffusion Monte Carlo method. The validity of the composite boson description is tested for different densities. We extract the exciton-exciton $s$-wave scattering length by solving the four-body problem in a harmonic trap and mapping the energy to that of two trapped bosons. The equation of state is consistent with the Bogoliubov theory for composite bosons interacting with the obtained $s$-wave scattering length. The perturbative expansion at low density has contributions physically coming from (a) exciton binding energy, (b) mean-field Gross-Pitaevskii interaction between excitons, (c) quantum depletion of the excitonic condensate (Lee-Huang-Yang terms for composite bosons). In addition, for low densities we find a good agreement with the Bogoliubov bosonic theory for the condensate fraction of excitons. The equation of state in the opposite limit of large density is found to be well described by the perturbative theory including (a) mixture of two ideal Fermi gases (b) exchange energy. We find that for low densities both energetic and coherent properties are correctly described by the picture of composite bosons (excitons).

Finite system scheme for mutually catalytic branching with infinite branching rate

For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here, we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments, the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, where the latter converge at a different time scale.

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents with mean-field interactions. The individual admissible controls are constrained in closed convex subsets $Γ k$ of full space $R m$. The decentralized strategies for individual agents and consistency condition system are represented in an unified manner via a class of mean-field forward-backward stochastic differential equations involving projection operators on $Γ k$. The well-posedness of consistency system is established in the local and global cases both by the contraction mapping and discounting method respectively. The related $e$−Nash equilibrium property is also verified.

Central limit results for jump diffusions with mean field interaction and a common factor

ABSTRACTA system of N weakly interacting particles whose dynamics is given in terms of jump-diffusions with a common factor is considered. The common factor is described through another jump-diffusion and the coefficients of the evolution equation for each particle depend, in addition to its own state value, on the empirical measure of the states of the N particles and the common factor. A Central Limit Theorem, as N → ∞, is established. The limit law is described in terms of a certain Gaussian mixture. An application to models in mathematical finance of self-excited correlated defaults is described.

Microscopic statistical description of incompressible Navier-Stokes granular fluids

Based on the recently established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1-body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the same 1-body PDF asymptotically relaxes to a stationary and spatially uniform Maxwellian PDF.

Single-ion properties of theSeff=12XY antiferromagnetic pyrochloresNaA′Co2F7(A′=Ca2+, Sr2+)

The antiferromagnetic pyrochlore material NaCaCo$_2$F$_7$ is a thermal spin liquid over a broad temperature range (~ 140 K down to $T_F = 2.4 $K), in which magnetic correlations between Co$^{2+}$ dipole moments explore a continuous manifold of antiferromagnetic XY states. The thermal spin liquid is interrupted by spin freezing at a temperature that is ~ 2 % of the mean field interaction strength, leading to short range static XY clusters with distinctive relaxation dynamics. Here we report the low energy inelastic neutron scattering response from the related compound NaSrCo$_2$F$_7$, confirming that it hosts the same static and dynamic spin correlations as NaCaCo$_2$F$_7$. We then present the single-ion levels of Co$^{2+}$ in these materials as measured by inelastic neutron scattering. An intermediate spin orbit coupling model applied to an ensemble of trigonally distorted octahedral crystal fields accounts for the observed transitions. The single-ion ground state of Co$^{2+}$ is a Kramers doublet with a strongly XY-like $g$-tensor ($g_{xy}/g_{z}$ ~ 3). The local disorder inherent from the mixed pyrochlore $A$ sites (Na$^{+}$/Ca$^{2+}$ and Na$^{+}$/Sr$^{2+}$) is evident in these measurements as exaggerated broadening of some of the levels. A simple model that reproduces the salient features of the single-ion spectrum produces approximately 8.4 % and 4.1 % variation in the $z$ and $xy$ components of the $g$-tensor, respectively. This study confirms that an $S_{eff} =1/2$ model with XY antiferromagnetic exchange and weak exchange disorder serves as a basic starting point in understanding the low temperature magnetic behavior of these strongly frustrated magnets.

Fluctuating hydrodynamics for ionic liquids

We present a mean-field fluctuating hydrodynamics (FHD) method for studying the structural and transport properties of ionic liquids in bulk and near electrified surfaces. The free energy of the system consists of two competing terms: (1) a Landau–Lifshitz functional that models the spontaneous separation of the ionic groups, and (2) the standard mean-field electrostatic interaction between the ions in the liquid. The numerical approach used to solve the resulting FHD-Poisson equations is very efficient and models thermal fluctuations with remarkable accuracy. Such density fluctuations are sufficiently strong to excite the experimentally observed spontaneous formation of liquid nano-domains. Statistical analysis of our simulations provides quantitative information about the properties of ionic liquids, such as the mixing quality, stability, and the size of the nano-domains. Our model, thus, can be adequately parameterized by directly comparing our prediction with experimental measurements and all-atom simulations. Conclusively, this work can serve as a practical mathematical tool for testing various theories and designing more efficient mixtures of ionic liquids.

Multicluster oscillation death and chimeralike states in globally coupled Josephson Junctions.

We observe the multiclustered oscillation death and chimeralike states in an array of Josephson junctions under a combination of self-repulsive and cross-attractive mean-field interaction when each isolated junction is in a bistable state, a coexisting fixed point and an oscillatory state. We locate the parameter landscape of the multiclustered oscillation death and chimeralike states. Alternatively, a purely repulsive mean-field interaction in an array of all oscillatory junctions produces chimeralike states with signatures of metastability in the incoherent subpopulation of junctions.