Found 1683 results, showing the newest relevant preprints. Sort by relevancy only.Update me on new preprints

Existence of weak solutions to time-dependent **mean**-**field** games

Here, we establish the existence of weak solutions to a wide class of time-dependent monotone

**mean**-**field**games (MFGs). Expand abstract. Here, we establish the existence of weak solutions to a wide class of time-dependent monotone

**mean**-**field**games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space-time. Then, using Schaefer's fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty's method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.43 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv

Derivation of the Landau-Pekar equations in a many-body **mean**-**field** limit

We consider the Fr\"ohlich Hamiltonian in a

**mean**-**field**limit where many bosonic particles weakly couple to the quantized phonon field. Expand abstract. We consider the Fr\"ohlich Hamiltonian in a

**mean**-**field**limit where many bosonic particles weakly couple to the quantized phonon**field**. For large particle number and suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.43 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv

Equivalence of ensembles in Curie-Weiss models using coupling techniques

We consider equivalence of ensembles for two

**mean****field**models: the discrete, standard Curie-Weiss model and its continuum version, also called the mean-field spherical model. Expand abstract. We consider equivalence of ensembles for two

**mean****field**models: the discrete, standard Curie-Weiss model and its continuum version, also called the**mean**-**field**spherical model. These systems have two thermodynamically relevant quantities and we consider the three associated standard probability measures: the microcanonical, canonical, and grand canonical ensembles. We prove that there are ranges of parameters where at least two of the ensembles are equivalent. The equivalence is not restricted to proving that the ensembles have the same thermodynamic limit of the specific free energy but we also give classes of observables whose ensemble averages agree in the limit. Moreover, we obtain explicit error estimates for the difference in the ensemble averages. The proof is based on a construction of suitable couplings between the relevant ensemble measures, proving that their Wasserstein fluctuation distance is small enough for the error in the ensemble averages to vanish in the thermodynamic limit. A crucial property for these estimates is permutation invariance of the ensemble measures.46 days ago

5/10 relevant

arXiv

5/10 relevant

arXiv

Mean-**field** density of states of a small-world model and a jammed soft
spheres model

We consider a class of random block matrix models in $d$ dimensions, $d \ge 1$, motivated by the study of the vibrational density of states (DOS) of soft spheres near the isostatic point. Expand abstract.

We consider a class of random block matrix models in $d$ dimensions, $d \ge 1$, motivated by the study of the vibrational density of states (DOS) of soft spheres near the isostatic point. The contact networks of average degree $Z = z_0 + \zeta$ are represented by random $z_0$-regular graphs (only the circle graph in $d=1$ with $z_0=2$) to which Erd\"os-Renyi graphs having a small average degree $\zeta$ are superimposed. In the case $d=1$, for $\zeta$ small the shifted Kesten-McKay DOS with parameter $Z$ is a

**mean**-**field**solution for the DOS. Numerical simulations in the $z_0=2$ model, which is the $k=1$ Newman-Watts small-world model, and in the $z_0=3$ model lead us to conjecture that for $\zeta \to 0$ the cumulative function of the DOS converges uniformly to that of the shifted Kesten-McKay DOS, in an interval $[0, \omega_0]$, with $\omega_0 < \sqrt{z_0-1} + 1$. For $2 \le d \le 4$, we introduce a cutoff parameter $K_d \le 0.5$ modeling sphere repulsion. The case $K_d=0$ is the random elastic network case, with the DOS close to the Marchenko-Pastur DOS with parameter $t=\frac{Z}{d}$. For $K_d$ large the DOS is close for small $\omega$ to the shifted Kesten-McKay DOS with parameter $t=\frac{Z}{d}$; in the isostatic case the DOS has around $\omega=0$ the expected plateau. The boson peak frequency in $d=3$ with $K_3$ large is close to the one found in molecular dynamics simulations for $Z=7$ and $8$.47 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv

The **mean**-**field** limit of quantum Bose gases at positive temperature

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\"odinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Expand abstract.

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\"odinger equation in the

**mean**-**field**limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the**mean**-**field**limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.49 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv

**Mean** **field** equations and domains of first kind

Finally, we show that the set of simply connected domains of first kind is contractible. Expand abstract.

In this paper we are interested in understanding the structure of domains of first and second kind, a concept motivated by problems in statistical mechanics. We prove some openness property for domains of first kind with respect to a suitable topology, as well as some sufficient condition for a simply connected domain to be of first kind in terms of the Fourier coefficients of the Riemann map. Finally, we show that the set of simply connected domains of first kind is contractible.

49 days ago

7/10 relevant

arXiv

7/10 relevant

arXiv

Relativistic **Mean**-**Field** Approach in Nuclear Systems

Basic features of more sophisticated DBHF calculations for finite nuclei are reproduced. Expand abstract.

A new scheme to study the properties of finite nuclei is proposed based on the Dirac-Brueckner-Hartree-Fock (DBHF) approach starting from a bare nucleon-nucleon interaction. The relativistic structure of the nucleon self-energies in nuclear matter depending on density, momentum and isospin asymmetry are determined through a subtracted T-matrix technique and parameterized, which makes them easily accessible for general use. The scalar and vector potentials of a single particle in nuclei are generated via a local density approximation (LDA). The surface effect of finite nuclei can be taken into account by an improved LDA (ILDA), which has successfully been applied in microscopic derivations of the optical model potential for nucleon-nucleus scattering. The bulk properties of nuclei can be determined in a self-consistent scheme for nuclei all over the nuclear mass table. Calculated binding energies agree very well with the empirical data, while the predicted values for radii and spin-orbit splitting of single-particle energies are about 10 \% smaller than the experimental data. Basic features of more sophisticated DBHF calculations for finite nuclei are reproduced.

53 days ago

8/10 relevant

arXiv

8/10 relevant

arXiv

Rigorous results on topological superconductivity with particle number conservation

These results show that many of the remarkable properties of

**mean**-**field**models of topological superconductivity persist in more realistic models with number-conserving dynamics. Expand abstract. Most theoretical studies of topological superconductors and Majorana-based quantum computation rely on a

**mean**-**field**approach to describe superconductivity. A potential problem with this approach is that real superconductors are described by number-conserving Hamiltonians with long-range interactions, so their topological properties may not be correctly captured by**mean**-**field**models that violate number conservation and have short-range interactions. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the**mean**-**field**models, including a finite energy gap in a sector of fixed total particle number, the existence of long range "Majorana-like" correlations between the ends of an open wire, and a change in the ground state fermion parity for periodic vs. anti-periodic boundary conditions. These results show that many of the remarkable properties of**mean**-**field**models of topological superconductivity persist in more realistic models with number-conserving dynamics.59 days ago

7/10 relevant

arXiv

7/10 relevant

arXiv

**Mean** **field** games with branching

For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the

**mean****field**game problem. Expand abstract.**Mean**

**field**games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the

**mean**

**field**game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear--quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.

60 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv

Some aspects of the Markovian SIRS epidemics on networks and its
**mean**-**field** approximation

Then, we pass to consider a first-order

**mean**-**field**approximation of the exact model. Expand abstract. We study the spread of an SIRS-type epidemics with vaccination on network. Starting from an exact Markov description of the model we investigate the

**mean**epidemic lifetime by providing a sufficient condition for the fast extinction, depending on the topology of the network. Then, we pass to consider a first-order**mean**-**field**approximation of the exact model. At this point, we dwell on the stability properties of the system by relying on the graph-theoretical notion of equitable partition. In the case of graphs possessing this kind of partition, we find a positively invariant set which contain the endemic equilibrium, that can be computed by using a lower-dimensional dynamical system. Finally, in the special case of regular graphs, we show that when the recovery rate is higher than the vaccination rate, the aforementioned invariant set is contained in the domain of attraction of the endemic equilibrium.62 days ago

10/10 relevant

arXiv

10/10 relevant

arXiv