Mean-field Term Research Articles

Maximum principle for discrete-time stochastic control problem of mean-field type

In this work, a discrete-time mean-field type stochastic optimal control problem is studied. The goal is to derive the stochastic maximum principle with convex control domains. L-derivative is applied to handle the mean-field term and a technique of adjoint operator is used to overcome the difficulties of obtaining adjoint equations and duality relation. Then, the stochastic maximum principle for discrete-time mean-field type stochastic optimal control problem is established. Finally, as an illustration of our stochastic maximum principle, a discrete-time mean–variance portfolio selection problem is solved with the decoupling technique which is different from the continuous-time case.

A Mean-Field Game Control for Large-Scale Swarm Formation Flight in Dense Environments.

As an important part of cyberphysical systems (CPSs), multiple aerial drone systems are widely used in various scenarios, and research scenarios are becoming increasingly complex. However, planning strategies for the formation flying of aerial swarms in dense environments typically lack the capability of large-scale breakthrough because the amount of communication and computation required for swarm control grows exponentially with scale. To address this deficiency, we present a mean-field game (MFG) control-based method that ensures collision-free trajectory generation for the formation flight of a large-scale swarm. In this paper, two types of differentiable mean-field terms are proposed to quantify the overall similarity distance between large-scale 3-D formations and the potential energy value of dense 3-D obstacles, respectively. We then formulate these two terms into a mean-field game control framework, which minimizes energy cost, formation similarity error, and collision penalty under the dynamical constraints, so as to achieve spatiotemporal planning for the desired trajectory. The classical task of a distributed large-scale aerial swarm system is simulated by numerical examples, and the feasibility and effectiveness of our method are verified and analyzed. The comparison with baseline methods shows the advanced nature of our method.

Transitional patterns in the spherical mean-field plus quadrupole-quadrupole and pairing model within two-j shells

Patterns of transition between several terms in the spherical mean-field plus quadrupole-quadrupole and pairing model within two-j shells are studied. By analyzing the variational features of the quantities in the quadrupole-quadrupole term to pairing term, mean-field term to quadrupole-quadrupole term, and mean-field term to pairing term, we find that with the even-n cases, some observables such as low-lying energy level ratio E(41+)/E(21+), the overlap |〈41+;x0|41+;x〉| with x0=0 and x0=1, B(E2;41+→21+) and electric quadrupole moment value Q(41+), are sensitive with respect to a control parameter or the number of valence nucleons. The investigation also indicates that increasing the spacing of single-particle energy level can produce the significant transitional behavior.

Toward a Lattice Boltzmann Method for Solids—Application to Static Equilibrium of Isotropic Materials

This work presents a novel method for simulating the behavior of solid objects with the Lattice Boltzmann Method (LBM). To introduce and validate our proposed framework, comparative studies are performed for computing the static equilibrium of isotropic materials. Remembering that the LBM has strong theoretical foundations in the Boltzmann equation; this latter is firstly adjusted to solid motions, through its Boltzmann-Vlasov special case. This is indeed the case when combined with a suitable mean-field external force term to set a reliable solid framework. Secondly, a library is built and plugged on the top of the well-known Parallel Lattice Boltzmann Solver (PaLaBoS) library. Numerical implementations based on the previous equation of motion for solids are led in a non-intrusive manner so as to present results with an easy and flawless reproducibility. A newly designed Lattice Boltzmann Method for Solids (LBMS) is exhibited through a few key algorithms, showing the overall operation plus the major improvements. Efficiency, robustness and accuracy of the proposed approach are illustrated and contrasted with a commercial Finite Element Analysis (FEA) software. The obtained results reveal considerable potential concerning static and further dynamic simulations involving solid constitutive laws within the LBM formalism.

The primitive model in classical density functional theory: beyond the standard mean-field approximation

The primitive model describes ions by point charges with an additional hard-core interaction. In classical density-functional theory (DFT) the mean-field electrostatic contribution can be obtained from the first order of a functional perturbation of the pair potential for an uncharged reference system of hard spheres. This mean-field electrostatic term particularly contributes at particle separations that are forbidden due to hard-core overlap. In this work we modify the mean-field contribution such that the pair potential is constant for distances smaller than the contact distance of the ions. We motivate our modification by the underlying splitting of the potential, which is similar to the splitting of the Weeks–Chandler–Andersen potential and leads to higher-order terms in the respective expansion of the functional around the reference system. The resulting formalism involves weighted densities similar to the ones found in fundamental measure theory. To test our modifications, we analyze and compare density profiles, direct and total correlation functions, and the thermodynamic consistency of the functional via a widely established sum rule and the virial pressure formula for our modified functional, for established functionals, and for data from computer simulations. We found that our modifications clearly show improvements compared to the standard mean-field functional, especially when predicting layering effects and direct correlation functions in high concentration scenarios; for the latter we also find improved consistency when calculated via different thermodynamic routes. In conclusion, we demonstrate how modifications toward higher order corrections beyond mean-field functionals can be made and how they perform, by this providing a basis for systematic future improvements in classical DFT for the description of electrostatic interactions.

Infinite Horizon Stackelberg Games With a Large Follower Population for Stochastic LPV Systems

In this letter, infinite horizon Stackelberg games with a large follower population for stochastic linear parameter-varying (LPV) systems is studied. Players are assumed to have their own local dynamics and they are coupled with each other through a mean field term in the cost functionals. It is assumed that followers adopt Nash equilibrium strategies and can only access their own local state information. First, the conditions for the existence of a centralized Stackelberg strategy set and its solutions are obtained by using the cross-coupled matrix inequalities (CCMIs) and cross-coupled matrix equations (CCMEs), respectively. Then, the conditions for the existence of a decentralized Stackelberg strategy set are obtained by using the reduced-order CCMIs and CCMEs respectively. It is proved that, when the follower population size is sufficiently large, the centralized Stackelberg strategy set approximates to the decentralized strategy set and the approximation is evaluated by using the weakly-coupled stochastic systems theory. Finally, a simple example is computed to demonstrate the validity and usefulness of the proposed approach.

Vehicle-to-Grid Coordination via Mean Field Game

The electrification of the transportation sector is urgently needed to combat global warming. However, the rapidly increasing number of electric vehicles (EVs) on the road is threatening the reliable operation of the power grid. Specifically, the simultaneous charging of many EVs often creates a sudden increase in demand, which needs to be handled by scarce flexibility in the system. Despite this, EVs can become part of the scarce flexibility through vehicle-to-grid (V2G) services if well-coordinated. This letter casts the EV coordination problem in the game-theoretic framework to study the equilibrium charging strategy. Specifically, we assume each EV aims to minimize the charging cost yet maximize the benefit of charging. Both terms couple the EVs’ decisions, yielding the EV coordination game (EVCG). Note that the detailed charging behaviors of a large population of EVs can be rather hard to analyze accurately. We resort to the notion of the mean field game (MFG). By introducing the mean field term, we can analyze the MFG corresponding to EVCG. Specifically, we prove that the MFG admits a Nash equilibrium, which is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -Nash equilibrium of EVCG. Besides showing the existence of the equilibrium, we propose an efficient algorithm to enable the distributed control based on the equilibrium strategy.

Task Selection and Collision-Free Route Planning for Mobile Crowdsensing Using Multi-Population Mean-Field Games

With the increasing deployment of mobile vehicles, such as mobile robots and unmanned aerial vehicles (UAVs), it is foreseen that they will play an important role in mobile crowd sensing (MCS). Specifically, mobile vehicles equipped with sensors and communication modules are able to execute large scale tasks due to their fast and flexible mobility in MCS systems. However, the battery capacity of mobile vehicles imposes a significant limitation on their performance, and so energy efficiency is an important metric especially when a large number of mobile vehicles collect sensing data. In this paper, we consider a mobile-vehicle-assisted MCS system where vehicles are owned by different operators or individuals who compete against others for limited sensing resources. We investigate the joint task selection and route planning problem for such an MCS system from an energy-efficiency perspective. However, since the computational complexity of the original problem is very high due to the large number of vehicles, we propose a multi-population Mean-Field Game (MPMFG) problem by simplifying the interaction between vehicles as a distribution over their strategy space, known as the mean-field term. To solve the MPMFG problem efficiently, we propose a G-prox primal-dual hybrid gradient method (PDHG) algorithm whose computational complexity is independent of the number of vehicles. Numerical results verify the effectiveness and efficiency of the proposed MPMFG scheme and G-prox PDHG algorithm.

Second order relativistic viscous hydrodynamics within an effective description of hot QCD medium

The second-order hydrodynamic equations for evolution of shear and bulk viscous pressure have been derived within the framework of covariant kinetic theory based on the effective fugacity quasiparticle model. The temperature-dependent fugacity parameter in the equilibrium distribution function leads to a mean field term in the Boltzmann equation which affects the interactions in the hot QCD matter. The viscous corrections to distribution function, up to second-order in gradient expansion, have been obtained by employing a Chapman–Enskog like iterative solution of the effective Boltzmann equation within the relaxation time approximation. The effect of mean field contributions to transport coefficients as well as entropy current has been studied up to second-order in gradients. In contrast to the previous calculations, we find non-vanishing entropy flux at second order. The effective description of relativistic second-order viscous hydrodynamics, for a system of interacting quarks and gluons, has been quantitatively analyzed in the case of the 1 + 1-dimensional boost invariant longitudinal expansion. We study the proper time evolution of temperature, pressure anisotropy, and viscous corrections to entropy density for this simplified expansion. The second order evolution of quark–gluon plasma is seen to be affected significantly with the inclusion of mean field contributions and the realistic equation of state.

Long-term evolution of neutron-star merger remnants in general relativistic resistive magnetohydrodynamics with a mean-field dynamo term

Long-term neutrino-radiation resistive-magnetohydrodynamics simulations in full general relativity are performed for a system composed of a massive neutron star and a torus formed as a remnant of binary neutron-star mergers. The simulation is performed in axial symmetry incorporating a mean-field dynamo term for a hypothetical amplification of the magnetic-field strength. We first calibrate the mean-field dynamo parameters by comparing the results for the evolution of black hole--disk systems with viscous hydrodynamics results. We then perform simulations for the system of a remnant massive neutron star and a torus. As in the viscous hydrodynamics case, the mass ejection occurs primarily from the torus surrounding the massive neutron star. The total ejecta mass and electron fraction in the new simulation are similar to those in the viscous hydrodynamics case. However, the velocity of the ejecta can be significantly enhanced by magnetohydrodynamics effects caused by global magnetic fields.

Machine-Learned Corrections to Mean-Field Microkinetic Models at the Fast Diffusion Limit

We present a machine learning-based formalism to correct the mean-field assumption in microkinetic models to incorporate adsorbate interactions and surface inhomogeneity at the fast diffusion limit. Lattice Monte Carlo simulations are used to compute the macroscopic reaction rate in the presence of adsorbate interaction at different values of surface coverage. This dataset is then used to train an artificial neural network to compute precise reaction rates as a function of surface coverage of intermediates, and the underlying microkinetic model of the reaction system is modified by correcting the typical mean-field rate terms with these data-driven functions. An example of CO oxidation on the square ordered lattice is used to illustrate the speed, accuracy, and robustness of this approach, vis-à-vis a full-fledged kinetic Monte Carlo simulation. In particular, we show that while the traditional mean-field model completely misses the bistability of this system under certain conditions, the neural network-modified formalism correctly captures this phenomenon. We posit that this method scales well to larger reaction systems and is a cost-effective means to improve the accuracy of differential equation-based microkinetic models.

Magneto-structural studies of a one dimensional Mn(II) coordination polymer derived from a flexible polydentate ligand

The polydentate ligand LHμ contains bipyridine, hydrazone and oxime functionalities and reacts with Mn(CH3COO)2·4H2O to afford an alternating one dimensional chain comprising Mn(II) ions, [{Mn2(LH)(CH3COO)3(CH3OH)2}·CH3OH]n(1). The structure of 1 contains two Mn(II) ions in the asymmetric unit; Mn1 adopts an unusual pentagonal bipyramidal coordination geometry comprising an [N3O4] donor set, while Mn2 adopts a more conventional octahedral geometry coordinated by an [NO5] donor set. The two manganese ions within the dinuclear unit are connected to each either via Ohydrazone and syn-syn Oacetate bridges. These dinuclear units are then linked by anti-anti Oacetate bridges, affording a magnetic 1-D alternating chain topology. Magnetic studies on 1 reveal the presence of weak antiferromagnetic interactions via Ohydrazone and acetate bridges. The magnetism of 1 was modelled using a dimer model [Ĥ = −2JŜ1Ŝ2], with inter-dimer exchange modelled using a mean field term (zJ’) to afford g = 2.046 ± 0.002, J/k = -1.03 ± 0.02 cm−1 and zJ’ = −0.06 ± 0.01 cm−1.

Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation

Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of a mean-field term to account for intermolecular forces in dilute polymer solutions for background profiles of weak shear and elongation. These investigations involved the solution of the Fokker–Planck equation incorporating a constitutive equation for the second moment. In this paper, we extend this analysis to include the effects of large background shear and elongation beyond the Hookean regime. Further, the constitutive equation is solved for the probability density function which permits the computation of any macroscopic variable, allowing direct comparison of the model predictions with molecular dynamics simulations. It was found that if the concentration effects at equilibrium are taken into account, the FENE-P model gives qualitatively the correct predictions, although the over-shoot in extension in comparison to the infinitely dilute case is significantly underpredicted.

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

Numerical study of the stock market crises based on mean field games approach

Abstract We present an approach to describe the stock market crises based on Mean Field Games (MFGs) and Optimal Control theory with a turnpike effect. The impact of the large amount of high-frequency traders (HFTs) can be modeled via a mean field term. We introduce the turnpike as a function that relies on the changes of the asset share price. An MFG is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time, and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The ill-posedness of this system comes from a turnpike effect. The numerical solution of an extremal problem that is dual to a PDE system is presented. We apply this approach to describe the Chinese stock market crash in 2015 considering the representative stock of CITIC Securities (ticker 600030). We consider HFTs that form a dominating bull and bear market. As a result, the bull strategy imitators do not make any profit.

Delay Optimal Random Access With Heterogeneous Device Capabilities in Energy Harvesting Networks Using Mean Field Game

In distributed random access (RA), each device should consider its own information as well as the influence from others, which is difficult to obtain with diverse capabilities of devices. In this paper, we study the RA problem for massive devices with heterogeneous device capabilities in large-scale energy harvesting IoT networks. To deal with the overload issue for massive devices with different capabilities, we propose an optimal RA policy by improving the conventional mean field games (MFG) via exchanging the mean field terms (MFT) among devices. Specifically, we formulate the delay optimal problem as a two-dimensional Markov Decision Process (MDP) problem involving both energy and data states. For distributed deployment of massive RA, we divide the MDP into multiple per-device subproblems, and propose the distributed RA scheme by solving the Hamilton-Jacobi-Bellman (HJB) equation using stochastic learning. Considering the deviation of MFT estimation induced by heterogeneous device capabilities, we design an MFT consensus scheme based on stochastic approximation by information exchange among neighbor devices. For reducing the state space and exchanging the MFT efficiently, we adopt the number of simultaneous access devices instead of the conventional MFT. Furthermore, we prove the convergence of the proposed scheme with coupling MFT and Q-factor. Finally, simulation results demonstrate that the proposed RA scheme outperforms baseline schemes on delay performance, especially in the heavy traffic load regime.

Mixed Bubbles in Bose-Bose Mixtures.

Repulsive Bose-Bose mixtures are known to either mix or phase separate into pure components. Here we predict a mixed-bubble regime in which bubbles of the mixed phase coexist with a pure phase of one of the components. This is a beyond-mean-field effect that occurs for unequal masses or unequal intraspecies coupling constants and is due to a competition between the mean-field term, quadratic in densities, and a nonquadratic beyond-mean-field correction. We find parameters of the mixed-bubble regime in all dimensions and discuss implications for current experiments.

Phase diagram of α matter with a Skyrme-like scalar interaction

The equation of state and phase diagram of strongly interacting matter composed of $\alpha$ particles are studied in the mean-field approximation. The particle interactions are included via a Skyrme-like mean field, containing both attractive and repulsive terms. The model parameters are found by fitting known values of binding energy and baryon density in the ground state of $\alpha$ matter, obtained from microscopic calculations by Clark and Wang. Thermodynamic quantities of $\alpha$ matter are calculated in the broad domains of temperature and baryon density, which can be reached in heavy-ion collisions at intermediate energies. The model predicts both first-order liquid-gas phase transition and Bose-Einstein condensation of $\alpha$ particles. We present the profiles of scaled variance, sound velocity and isochoric heat capacity along the isentropic trajectories of $\alpha$ matter. Strong density fluctuations are predicted in the vicinity of the critical point at temperature $T_c\approx 14~\textrm{MeV}$ and density $n_c\approx 0.012~\textrm{fm}^{-3}$.

Joint Sensing Task Assignment and Collision-Free Trajectory Optimization for Mobile Vehicle Networks Using Mean-Field Games

With the increasing popularity of mobile vehicles, such as unmanned aerial vehicles (UAVs) and mobile robots, it is foreseen that they will play an important role in Internet-of-Things (IoT) networks due to their high mobility and rapid deployment. Specifically, mobile vehicles equipped with sensors act as IoT devices and can be dispatched to several sensing regions to perform sensing tasks. In this article, we consider mobile vehicles for sensing applications and investigate the corresponding joint task assignment and collision-free trajectory optimization problem. This problem is challenging as the number of involved vehicles can be very large, and to tackle the problem efficiently, we reformulate the original optimization problem into a mean-field-game (MFG) problem by simplifying the interaction between vehicles as a distribution over their state space, known as the mean-field term. To solve the MFG problem efficiently, we propose a G-prox primal-dual hybrid gradient (PDHG) algorithm that transforms the MFG problem into a saddle-point problem by defining a Lagrangian functional with a proximal operator. The complexity of this algorithm is shown to be linear with the total number of grid points in the proposed MFG problem. We provide a comprehensive theoretical analysis of the proposed model and algorithm. Numerical results together with the practical implementation on real mobile robots show that our proposed system model and algorithm are of significant effectiveness and efficiency.

Classical density-functional theory applied to the solid state.

The standard model of classical density-functional theory (cDFT) for pair potentials consists of a hard-sphere functional plus a mean-field term accounting for long ranged attraction. However, most implementations using sophisticated fundamental measure hard-sphere functionals suffer from potential numerical instabilities either due to possible instabilities in the functionals themselves or due to implementations that mix real- and Fourier-space components inconsistently. Here we present a new implementation based on a demonstrably stable hard-sphere functional that is implemented in a completely consistent manner. The present work does not depend on approximate spherical integration schemes and so is much more robust than previous algorithms. The methods are illustrated by calculating phase diagrams for the solid state using the standard Lennard-Jones potential as well as a new class of potentials recently proposed by Wang etal. [Phys. Chem. Chem. Phys. 22, 10624 (2020)PPCPFQ1463-907610.1039/C9CP05445F]. The latter span the range from potentials for small molecules to those appropriate to colloidal systems simply by varying a parameter. We verify that cDFT is able to semiquantitatively reproduce the phase diagram in all cases. We also show that for these problems computationally cheap Gaussian approximations are nearly as good as full minimization based on finite differences.