Mean-field Diffusion Research Articles

Time-inconsistent mean-field optimal stopping: A limit approach

We provide a characterization of an optimal stopping time for a class of finite horizon time-inconsistent optimal stopping problems (OSPs) of mean-field type, adapted to the Brownian filtration, including those related to mean-field diffusion processes and recursive utility functions. Despite the time-inconsistency of the OSP, we show that it is optimal to stop when the value-process hits the reward process for the first time, as is the case for the standard time-consistent OSP. We solve the problem by approximating the corresponding value-process with a sequence of Snell envelopes of processes, for which a sequence of optimal stopping times is constituted of the hitting times of each of the reward processes by the associated value-process. Then, under mild assumptions, we show that this sequence of hitting times converges in probability to the hitting time for the mean-field OSP and that the limit is optimal.

Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions

Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions

Linear quadratic leader–follower stochastic differential games for mean-field switching diffusions

In this paper, we consider a linear quadratic (LQ) leader–follower stochastic differential game for regime switching diffusions with mean-field interactions. One of the salient features of this paper is that conditional mean-field terms are included in the state equation and cost functionals. Based on stochastic maximum principles (SMPs), the follower’s problem and the leader’s problem are solved sequentially and an open-loop Stackelberg equilibrium is obtained. Further, with the help of the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and then the open-loop Stackelberg equilibrium admits a state feedback representation if some new-type Riccati equations are solvable.

Explosive death transitions in complex networks of limit cycle and chaotic systems

The conditions for explosive death transitions in complex networks of oscillators having generalized network topology, coupled via mean-field diffusion, are derived analytically. The network behaviour is characterized using three order parameters that define the average amplitude of oscillations, the mean state and the fraction of dead oscillators. As the mean field coupling is changed adiabatically, the amplitude order parameter undergoes explosive death transitions and the nodes in the network collectively cease to oscillate. The transition points in the parameter space and the boundaries of amplitude death regime are derived analytically. Sub-critical Hopf bifurcations are shown to be responsible for amplitude death transition. Explosive death transitions are characterized by hysteresis on adiabatic change of the bifurcation parameter and surprisingly, the backward transition point is shown to be independent of network topology. The theoretical developments have been validated through numerical examples, involving networks of limit cycle systems — such as the Van der Pol oscillator and chaotic systems, such as the Rossler attractor for both random and small world networks.

Logarithmic expansion of many-body wave packets in random potentials

Anderson localization confines the wave function of a quantum particle in a one-dimensional random potential to a volume of the order of the localization length $\xi$. Nonlinear add-ons to the wave dynamics mimic many-body interactions on a mean field level, and result in escape from the Anderson cage and in unlimited subdiffusion of the interacting cloud. We address quantum corrections to that subdiffusion by (i) using the ultrafast unitary Floquet dynamics of discrete-time quantum walks, (ii) an interaction strength ramping to speed up the subdiffusion, and (iii) an action discretization of the nonlinear terms. We observe the saturation of the cloud expansion of $N$ particles to a volume $\sim N\xi$. We predict and observe a universal intermediate logarithmic expansion regime which connects the mean-field diffusion with the final saturation regime and is entirely controlled by particle number $N$. The temporal window of that regime grows exponentially with the localization length $\xi$.

Perpendicular diffusion of solar energetic particles: When is the diffusion approximation valid?

Multi-spacecraft observations of widespread solar energetic particle (SEP) events indicate that perpendicular (to the mean field) diffusion is an important SEP transport mechanism. However, this is in direct contrast to so-called spike and drop-out events, which indicate very little lateral transport. To better understand these seemingly incongruous observations, we discuss the recent progress made towards understanding and implementing perpendicular diffusion in transport models of SEP electrons. This includes a re-derivation of the relevant focused transport equation, a discussion surrounding the correct form of the pitch-angle dependent perpendicular diffusion coefficient and what turbulence quantities are needed as input, and how models lead to degenerate solutions of the particle intensity. Lastly, we evaluate the validity of a diffusion approach to SEP transport and conclude that it is valid when examining a large number of (an ensemble of) events, but that individual SEP events may exhibit coherent structures related to the magnetic field turbulence at short timescales that cannot be accounted for in this modelling approach.

Mean-field effects on the aging transitions in the damaged networks

We investigate the dynamical robustness property of the damaged network of active and inactive oscillators under the influence of the mean-field diffusion. The tolerance of dynamical activity of the entire coupled network has realized through the aging transition in the coupled dynamical network. We analytically derived the critical threshold of mean-field density and coupling values for the appearance of the aging transition in the damaged network. By using the critical values as a quantifiable measure of dynamical robustness of the damaged network, we showed that higher mean-field value is favorable to increase the dynamical robustness of the entire network. We also perform the numerical experiment on the network of Stuart-Landau oscillators and the obtained numerical results have an excellent agreement with the analytical findings. Finally, we extend our investigation into the coupled time-delayed network and discussed the affirmative influence of the mean-field parameter on the dynamical robustness of the network.

Tunable chemical complexity to control atomic diffusion in alloys

In this paper we report a new fundamental understanding of chemically-biased diffusion in Ni–Fe random alloys that is tuned/controlled by the intrinsic quantifiable chemical complexity. Development of radiation-tolerant alloys has been a long-standing challenge. Here we show how intrinsic chemical complexity can be utilized to guide the atomic diffusion and suppress radiation damage. The influence of chemical complexity is shown by the example of interstitial atom (IA) diffusion that is the most important defect in radiation effects. We use μs-scale molecular dynamics to reveal sluggish diffusion and percolation of IAs in concentrated Ni–Fe alloys. We develop a mean field diffusion model to take into account the effect of migrating defect energy properties on diffusion percolation, which is verified by a new kinetic Monte Carlo approach addressing detailed processes. We demonstrate that the local variations in the ground state energy of IA configurations in alloys, reflecting the chemical difference between alloying components, drives the percolation effects for atomic diffusion. Percolation, chemically-biased and sluggish diffusion are phenomena that are directly related to the chemical complexity intrinsically to multicomponent alloys.

Chimera States in Ecological Network Under Weighted Mean-Field Dispersal of Species

In ecological landscapes, species tend to migrate between patches in search of a better survivability condition. By this dispersal process, they form connectivity between the patches and thereby may develop various correlated or partially correlated population dynamics among species living in the patches. We explore various possible emergent collective population patterns using a simple ecological network model of all-to-all connected patches where we use a particular type of dispersal process that is controlled by a weighted mean-field diffusion to include the failed migration between the interacting patches. We represent the population dynamics of both the predator and prey in each patch by a modified Rosenzweig-MacArthur (mRM) model that incorporates an additional effect of habitat complexity. Our investigation on the network dynamics shows various interesting collective patterns, namely, clustered states and chimera states, besides synchrony and homogeneous steady states (HSS) of species. An important observation is that habitat complexity enhances survival probabilities of interacting species and thus increases population persistence in a natural ecosystem.

Explosive death induced by mean\u2013field diffusion in identical oscillators

We report the occurrence of an explosive death transition for the first time in an ensemble of identical limit cycle and chaotic oscillators coupled via mean–field diffusion. In both systems, the variation of the normalized amplitude with the coupling strength exhibits an abrupt and irreversible transition to death state from an oscillatory state and this first order phase transition to death state is independent of the size of the system. This transition is quite general and has been found in all the coupled systems where in–phase oscillations co–exist with a coupling dependent homogeneous steady state. The backward transition point for this phase transition has been calculated using linear stability analysis which is in complete agreement with the numerics.

Conserving approximations for response functions of the Fermi gas in a random potential

We scrutinize the diagrammatic perturbation theory of noninteracting electrons in a random potential with the aim to accomplish a consistent comprehensive theory of quantum diffusion. Ward identity between the one-electron self-energy and the two-particle irreducible vertex is generally not guaranteed in the perturbation theory with only elastic scatterings. We show how the Ward identity can be established in practical approximations and how the functions from the perturbation expansion should be used to obtain a fully consistent conserving theory. We derive the low-energy asymptotics of the conserving full two-particle vertex from which we find an exact representation of the diffusion pole and of the static diffusion constant in terms of Green functions of the perturbation expansion. We illustrate the construction on the leading vertex corrections to the mean-field diffusion due to maximally-crossed diagrams responsible for weak localization.

Perturbative Calculation of Quasi-Potential in Non-equilibrium Diffusions: A Mean-Field Example

In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\cite{bertini2014},which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$, where $N$ is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter $\lambda$, starting from a known quasi-potential for $\lambda=0$. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively.

Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $\theta_i$ is attached to one site $x_i$ of a periodic lattice and the interaction between particles $\theta_i$ and $\theta_j$ decreases as $| x_i-x_j|^{-\alpha}$ for $\alpha\in[0,1)$. In a previous work, it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean-Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $\alpha\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-\alpha}$ in the case $\alpha\in(\frac{1}{2},1)$.

Suppression of oscillations in mean-field diffusion

We study the role of mean-field diffusive coupling on suppression of oscillations for systems of limit cycle oscillators. We show that this coupling scheme not only induces amplitude death (AD) but also oscillation death (OD) in coupled identical systems. The suppression of oscillations in the parameter space crucially depends on the value of mean-field diffusion parameter. It is also found that the transition from oscillatory solutions to OD in conjugate coupling case is different from the case when the coupling is through similar variable. We rationalize our study using linear stability analysis.

Experimental observation of a transition from amplitude to oscillation death in coupled oscillators.

We report the experimental evidence of an important transition scenario, namely the transition from amplitude death (AD) to oscillation death (OD) state in coupled limit cycle oscillators. We consider two Van der Pol oscillators coupled through mean-field diffusion and show that this system exhibits a transition from AD to OD, which was earlier shown for Stuart-Landau oscillators under the same coupling scheme [T. Banerjee and D. Ghosh, Phys. Rev. E 89, 052912 (2014)]. We show that the AD-OD transition is governed by the density of mean-field and beyond a critical value this transition is destroyed; further, we show the existence of a nontrivial AD state that coexists with OD. Next, we implement the system in an electronic circuit and experimentally confirm the transition from AD to OD state. We further characterize the experimental parameter zone where this transition occurs. The present study may stimulate the search for the practical systems where this important transition scenario can be observed experimentally.

Transition from amplitude to oscillation death under mean-field diffusive coupling.

We study the transition from the amplitude death (AD) to the oscillation death (OD) state in limit-cycle oscillators coupled through mean-field diffusion. We show that this coupling scheme can induce an important transition from AD to OD even in identical limit cycle oscillators. We identify a parameter region where OD and a nontrivial AD (NTAD) state coexist. This NTAD state is unique in comparison with AD owing to the fact that it is created by a subcritical pitchfork bifurcation and parameter mismatch does not support this state, but destroys it. We extend our study to a network of mean-field coupled oscillators to show that the transition scenario is preserved and the oscillators form a two-cluster state.

Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion

We explore and experimentally demonstrate the phenomena of amplitude death (AD) and the corresponding transitions through synchronized states that lead to AD in coupled intrinsic time-delayed hyperchaotic oscillators interacting through mean-field diffusion. We identify a novel synchronization transition scenario leading to AD, namely transitions among AD, generalized anticipatory synchronization (GAS), complete synchronization (CS), and generalized lag synchronization (GLS). This transition is mediated by variation of the difference of intrinsic time-delays associated with the individual systems and has no analogue in non-delayed systems or coupled oscillators with coupling time-delay. We further show that, for equal intrinsic time-delays, increasing coupling strength results in a transition from the unsynchronized state to AD state via in-phase (complete) synchronized states. Using Krasovskii-Lyapunov theory, we derive the stability conditions that predict the parametric region of occurrence of GAS, GLS, and CS; also, using a linear stability analysis, we derive the condition of occurrence of AD. We use the error function of proper synchronization manifold and a modified form of the similarity function to provide the quantitative support to GLS and GAS. We demonstrate all the scenarios in an electronic circuit experiment; the experimental time-series, phase-plane plots, and generalized autocorrelation function computed from the experimental time series data are used to confirm the occurrence of all the phenomena in the coupled oscillators.

Amplitude death with mean-field diffusion

We study the dynamics of nonlinear oscillators under mean-field diffusive coupling. We observe that this form of coupling leads to amplitude death via a synchronization transition in the parameter space of the coupling strength and mean-field control parameter. A general criterion for amplitude death for any given dynamical system with mean-field diffusion is obtained, and these dynamical transitions are characterized using various indices such as average phase difference, Lyapunov exponents, and average amplitude. This behavior is analyzed in the parameter plane by numerical studies of specific cases of the Landau-Stuart limit-cycle oscillator, and Rössler, Lorenz, FitzHugh-Nagumo excitable, and Chua systems.

A mean-field anisotropic diffusion model for unentangled polymeric liquids and semi-dilute solutions: Model development and comparison with experimental and simulation data

A coarse-grained mesoscopic model was developed based on the ansatz that a specific polymer molecule diffuses through the nearby neighboring chains more easily in the direction parallel to its molecular axis than perpendicular to it. This idea is modeled using a mean-field approach in terms of an anisotropic diffusion matrix, which represents enhanced diffusion along the chain background once a significant degree of molecular extension and orientation has developed in response to an applied flow field. The rheological and microstructural characteristics of this model are examined and compared with atomistic nonequilibrium molecular dynamics (NEMD) simulation data of short-chain polyethylene liquids and experiments of semi-dilute DNA solutions under shear flow. Rheological and microstructural properties examined include the viscosity, normal stress coefficients, conformation tensor, etc., to gauge the usefulness of the model. In addition, this model was further coarse-grained to the continuum level through pre-averaging, and was also compared with the simulation and experimental data to examine the relationships between different levels of description on the rheological and structural properties of unentangled polymeric materials under shear flow. At the mesoscopic level, the polymer molecules are modeled as bead-spring chains using the finitely extensible nonlinear elastic (FENE) force law. Brownian dynamics (BD) simulations of this coarse-grained model displayed remarkable quantitative agreement with NEMD simulations of dense liquids and experiments of semi-dilute DNA solutions for system properties with a single adjustable parameter representing the relative magnitude of diffusive enhancement along the chain backbone. Furthermore, the BD simulations revealed the dependence of system response on the chain stretching at low values of Weissenberg number ( Wi) and on the rotational motion of individual chains induced by shear flow at high values of Wi, similarly to the NEMD simulation data. The continuum model matched the mesoscopic model at low shear rates, but greatly diverged at high values of Wi where the tumbling dynamics of the individual chains dominated the system response. This provides direct evidence that the onset of rotational motion under shear in these liquids is responsible for the well-known breakdown in pre-averaged constitutive equations at the continuum level of description. Furthermore, a possible explanation of the shear stress plateau at intermediate ranges of shear rate is offered for experimental data of semi-dilute solutions, wherein this phenomenon occurs with the onset of chain rotation within these fluids.

Lattice mean-field method for stationary polymer diffusion.

We present a method to study mean-field stationary diffusion (MFSD) in polymer systems. When gradients in chemical potentials vanish, our method reduces to the Scheutjens-Fleer self-consistent field (SF-SCF) method for inhomogeneous polymer systems in equilibrium. To illustrate the concept of our MFSD method, we studied stationary diffusion between two different bulk mixtures, containing, for simplicity, noninteracting homopolymers. Four alternatives for the diffusion equation are implemented. These alternatives are based on two different theories for polymer diffusion (the slow- and fast-mode theories) and on two different ways to evaluate the driving forces for diffusion, one of which is in the spirit of the SF-SCF method. The diffusion profiles are primarily determined by the diffusion theory and they are less sensitive to the evaluation of the driving forces. The numerical stationary state results are in excellent agreement with analytical results, in spite of a minor inconsistency at the system boundaries in the numerical method. Our extension of the equilibrium SF method might be useful for the study of fluxes, steady state profiles and chain conformations in membranes (e.g., during drug delivery), and for many other systems for which simulation techniques are too time consuming.