Mean-**field** electrodynamics of fluids with fluctuating electric
conductivity

**mean**-

**field**electrodynamics is considered. If the conductivity fluctuations are assumed as uncorrelated with the turbulent velocity

**field**then only the effective magnetic diffusivity of the fluid is reduced and the decay time of a large-scale magnetic

**field**is increased. If the fluctuations of conductivity and flow are correlated in a certain direction then an additional diamagnetic pumping effect results transporting magnetic

**field**in opposite direction to the resistivity flux vector $\langle \eta'\vec{u}'\rangle$. Even for homogeneous turbulence

**fields**in the presence of rotation an $\alpha$ effect appears. With the characteristic values of the outer Earth core or the solar convection zone, however, the dynamo number of the new $\alpha$ effect never reaches supercritical values to operate as an $\alpha^2$-dynamo.

10/10 relevant

arXiv

Microscopic Derivation of **Mean** **Field** Game Models

**mean**

**field**limit equations and we study the scaling behavior of the system as the number of agents tends to infinity and find several mean field game limits. Expand abstract.

**Mean**

**field**game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive

**mean**

**field**game partial differential equation systems from deterministic microscopic agent dynamics. The dynamics are given by a particular class of ordinary differential equations, for which an optimal strategy can be computed (Bressan, Milan Journal of Mathematics, 2011). We use the concept of Nash equilibria and apply the dynamic programming principle to derive the

**mean**

**field**limit equations and we study the scaling behavior of the system as the number of agents tends to infinity and find several

**mean**

**field**game limits. Especially we avoid in our derivation the notion of measure derivatives. Novel scales are motivated by an example of an agent-based financial market model.

10/10 relevant

arXiv

Ensemble Kalman Sampling: **mean**-**field** limit and convergence analysis

**mean**-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution. Expand abstract.

**mean**-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution. This proves that in long time, the samples generated by EKS indeed are approximately i.i.d. samples from the target distribution. We further show the ensemble distribution of EKS converges, in Wasserstein-2 sense, to the target distribution with a near-optimal rate. We emphasize that even when the forward map is linear, due to the ensemble nature of the method, the SDE system and the corresponding Fokker-Planck equation are still nonlinear.

8/10 relevant

arXiv

Metastability in a continuous **mean**-**field** model at low temperature and
strong interaction

**mean**-

**field**interacting stochastic differential equations that are driven by a single-site potential of double-well form and by Brownian noise. The strength of the noise is measured by a small parameter $ \varepsilon >0$ (which we interpret as the \emph{temperature}), and we suppose that the strength of the interaction is given by $ J>0 $. Choosing the \emph{empirical mean} ($ P:\mathbb{R}^N \rightarrow \mathbb{R} $, $ Px =1/N \sum_i x_i $) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two global minima, one at $ -m^\star_\varepsilon 0 $. Following this observation, we are interested in the average transition time of the system to $ P^{-1}(m^\star_\varepsilon) $, when the initial configuration is drawn according to a probability measure (the so-called \emph{last-exit distribution}), which is supported around the hyperplane $ P^{-1}(-m^\star_\varepsilon) $. Under the assumption of strong interaction, $ J>1 $, the main result is a formula for this transition time, which is reminiscent of the celebrated Eyring-Kramers formula up to a multiplicative error term that tends to $ 1 $ as $ N \rightarrow \infty $ and $ \varepsilon \rightarrow 0 $. The proof is based on the \emph{potential-theoretic approach to metastability.} In the last chapter we add some estimates on the metastable transition time in the high temperature regime, where $ \varepsilon =1 $, and for a large class of single-site potentials.

10/10 relevant

arXiv

Local minima in disordered **mean**-**field** ferromagnets

8/10 relevant

arXiv

Weak and strong error analysis for **mean**-**field** rank based particle
approximations of one dimensional viscous scalar conservation law

**mean**-

**field**rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov to check trajectorial propagation of chaos with optimal rate $N^{-1/2}$ to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy to check convergence in $L^1(\mathbb{R})$ with rate ${\mathcal O}(\frac{1}{\sqrt N} + h)$ of the empirical cumulative distribution function of the Euler discretization with step $h$ of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in ${\mathcal O}(\frac{1}{N} + h)$. We provide numerical results which confirm our theoretical estimates.

10/10 relevant

arXiv

Small-scale spatial structure influences large-scale invasion rates

**mean**-

**field**models, even when they account for non-local processes such as dispersal and competition, can give misleading predictions for the speed of a moving invasion front. Expand abstract.

**mean**-

**field**model. We conclude that

**mean**-

**field**models, even when they account for non-local processes such as dispersal and competition, can give misleading predictions for the speed of a moving invasion front.

4/10 relevant

bioRxiv

Convergence of a first-order consensus-based global optimization algorithm

**mean**-

**field**model. Expand abstract.

**mean**-

**field**limit, a Fokker-Planck equation, which does not imply the convergence of the CBO method {\it per se}. Based on the consensus estimate directly on the first-order CBO model, we provide a convergence analysis of the first-order CBO method \cite{C-J-L-Z} without resorting to the corresponding

**mean**-

**field**model. Our convergence analysis consists of two steps. In the first step, we show that the CBO model exhibits a global consensus time asymptotically for any initial data, and in the second step, we provide a sufficient condition on system parameters--which is dimension independent-- and initial data which guarantee that the converged consensus state lies in a small neighborhood of the global minimum almost surely.

4/10 relevant

arXiv

Actor-Critic Provably Finds Nash Equilibria of Linear-Quadratic
**Mean**-**Field** Games

**mean**-

**field**Markov games with provable non-asymptotic global convergence guarantees. Expand abstract.

**mean**-

**field**Markov games with infinite numbers of agents where each agent aims to minimize its ergodic cost. We consider the setting where the agents have identical linear state transitions and quadratic cost functions, while the aggregated effect of the agents is captured by the population

**mean**of their states, namely, the

**mean**-

**field**state. For such a game, based on the Nash certainty equivalence principle, we provide sufficient conditions for the existence and uniqueness of its Nash equilibrium. Moreover, to find the Nash equilibrium, we propose a

**mean**-

**field**actor-critic algorithm with linear function approximation, which does not require knowing the model of dynamics. Specifically, at each iteration of our algorithm, we use the single-agent actor-critic algorithm to approximately obtain the optimal policy of the each agent given the current

**mean**-

**field**state, and then update the

**mean**-

**field**state. In particular, we prove that our algorithm converges to the Nash equilibrium at a linear rate. To the best of our knowledge, this is the first success of applying model-free reinforcement learning with function approximation to discrete-time

**mean**-

**field**Markov games with provable non-asymptotic global convergence guarantees.

10/10 relevant

arXiv

Wasserstein stability estimates for covariance-preconditioned Fokker-Planck equations

**mean**

**field**PDE by using optimal transport arguments. Expand abstract.

**mean**

**field**PDE associated with the derivative-free methodologies for solving inverse problems. We show stability estimates in the euclidean Wasserstein distance for the

**mean**

**field**PDE by using optimal transport arguments. As a consequence, this recovers the known convergence towards equilibrium estimates in the case of a linear forward model.

4/10 relevant

arXiv