Mean-field Dynamics Research Articles

Applied-force oscillations in avalanche dynamics.

Until now most studies of discrete plasticity have focused on systems that are assumed to be driven by a monotonically increasing force; in many real systems, however, the driving force includes damped oscillations or oscillations induced by the propagation of discrete events or "slip avalanches." In both cases, these oscillations may obscure the true dynamics. Here we effectively consider both cases by investigating the effects of damped oscillations in the external driving force on avalanche dynamics. We compare model simulations of slip avalanches under mean-field dynamics with observations in slip-avalanche experiments on slowly compressed micrometer-sized Au specimens using open-loop force control. The studies show very good agreement between simulations and experiments. We find that an oscillatory external driving force changes the average avalanche shapes only for avalanches with durations close to the period of oscillation of the external force. This effect on the avalanche shapes can be addressed in experiments by choosing suitable specimen dimensions so that the mechanical resonance does not interact with the avalanche dynamics. These results are important for the interpretation of avalanche experiments with built-in oscillators, and for the prediction and analysis of avalanche dynamics in systems with resonant vibrations.

Classical and quantum chaos in a three-mode bosonic system

We study the dynamics of a three-mode bosonic system with mode-changing interactions. For large mode occupations the short-time dynamics is well described by classical mean-field equations allowing us to study chaotic dynamics in the classical system and its signatures in the corresponding quantum dynamics. By introducing a symmetry-breaking term we tune the classical dynamics from integrable to strongly chaotic which we demonstrate by calculating Poincar\'e sections and Lyapunov exponents. The corresponding quantum system features level statistics that change from Poissonian in the integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of out-of-time-ordered correlators (OTOCs), specifically the squared commutator, for initial states located in regular and chaotic regions of the classical mixed phase space and find marked differences between the two cases. The short-time behavior is well captured by semi-classical truncated Wigner simulations directly relating these features to properties of the underlying classical mean field dynamics. We discuss a possible experimental realization of this model system in a Bose-Einstein condensate of rubidium atoms which allows reversing the sign of the Hamiltonian required for measuring OTOCs experimentally.

Unstable modes and bistability in delay-coupled swarms.

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.

Mesoscopic population equations for spiking neural networks with synaptic short-term plasticity

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks–Markram STP. The approximate mean-field dynamics accounts for both finite number of synapses and correlation between the two synaptic variables of the model (utilization and available resources) and its numerical implementation is simple. Comparisons with Monte Carlo simulations of the microscopic model show that in both feedforward and recurrent networks, the mesoscopic mean-field model accurately reproduces the first- and second-order statistics of the total synaptic input into a postsynaptic neuron and accounts for stochastic switches between Up and Down states and for population spikes. The extended mesoscopic population theory of spiking neural networks with STP may be useful for a systematic reduction of detailed biophysical models of cortical microcircuits to numerically efficient and mathematically tractable mean-field models.

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler\u2013Boussinesq Equation with Transport Noise

The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler–Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied by Erbar and Maas. We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of mean-field limits of various classical models from statistical mechanics.

State-dependent mean-field formalism to model different activity states in conductance-based networks of spiking neurons.

More interest has been shown in recent years to large-scale spiking simulations of cerebral neuronal networks, coming both from the presence of high-performance computers and increasing details in experimental observations. In this context it is important to understand how population dynamics are generated by the designed parameters of the networks, which is the question addressed by mean-field theories. Despite analytic solutions for the mean-field dynamics already being proposed for current-based neurons (CUBA), a complete analytic description has not been achieved yet for more realistic neural properties, such as conductance-based (COBA) network of adaptive exponential neurons (AdEx). Here, we propose a principled approach to map a COBA on a CUBA. Such an approach provides a state-dependent approximation capable of reliably predicting the firing-rate properties of an AdEx neuron with noninstantaneous COBA integration. We also applied our theory to population dynamics, predicting the dynamical properties of the network in very different regimes, such as asynchronous irregular and synchronous irregular (slow oscillations). This result shows that a state-dependent approximation can be successfully introduced to take into account the subtle effects of COBA integration and to deal with a theory capable of correctly predicting the activity in regimes of alternating states like slow oscillations.

Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.

Understanding the nature of mean-field semiclassical light-matter dynamics: An investigation of energy transfer, electron-electron correlations, external driving, and long-time detailed balance

Semiclassical electrodynamics is an appealing approach for studying light-matter interactions, especially for realistic molecular systems. However, there is no unique semiclassical scheme. On the one hand, intermolecular interactions can be described instantaneously by static two-body interactions connecting different molecules plus a classical transverse E-field; we will call this Hamiltonian #I. On the other hand, intermolecular interactions can also be described as effects that are mediated exclusively through a classical one-body E-field without any quantum effects at all (assuming we ignore electronic exchange); we will call this Hamiltonian #II. Moreover, one can also mix these two Hamiltonians into a third, hybrid Hamiltonian, which preserves quantum electron-electron correlations for lower excitations but describes higher excitations in a mean-field way. To investigate which semiclassical scheme is most reliable for practical use, here we study the real-time dynamics of a pair of identical two-level systems (TLSs) undergoing either resonance energy transfer (RET) or collectively driven dynamics. While all approaches perform reasonably well when there is no strong external excitation, we find that no single approach is perfect for all conditions. Each method has its own distinct problems: Hamiltonian #I performs best for RET but behaves in a complicated manner for driven dynamics. Hamiltonian #II is always stable, but obviously fails for RET at short distances. One key finding is that, under externally driving, a full configuration interaction description of Hamiltonian #I strongly overestimates the long-time electronic energy, highlighting the not obvious fact that, if one plans to merge quantum molecules with classical light, a full, exact treatment of electron-electron correlations can actually lead to worse results than a simple mean-field treatment.

Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks.

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.

Mean-field dynamics for mixture condensates via Fock space methods

We consider a mean-field model to describe the dynamics of [Formula: see text] bosons of species one and [Formula: see text] bosons of species two in the limit as [Formula: see text] and [Formula: see text] go to infinity. We embed this model into Fock space and use it to describe the time evolution of coherent states which represent two-component condensates. Following this approach, we obtain a microscopic quantum description for the dynamics of such systems, determined by the Schrödinger equation. Associated to the solution to the Schrödinger equation, we have a reduced density operator for one particle in the first component of the condensate and one particle in the second component. In this paper, we estimate the difference between this operator and the projection onto the tensor product of two functions that are solutions of a system of equations of Hartree type. Our results show that this difference goes to zero as [Formula: see text] and [Formula: see text] go to infinity.

Infection promotes species coexistence: Rock–paper–scissors game with epidemic on graphs

We combine the rock–paper–scissors (RPS) game with SIS epidemic model. We study the effect of infection on population dynamics of RPS games with epidemic by using the metapopulation dynamic model on graphs. Via infection, an individual R changes to I with infection rate β. An infected individual I returns to R with recovery rate γ. All agents move by random walk between patches (subpopulations) and the RPS game with epidemic is performed in each patch. The dynamics are represented by the reaction–diffusion equations in diffusively coupled reactors. We obtain the solutions of the reaction–diffusion equations for the mean-field dynamics in a single patch analytically. The numerical solutions are obtained for metapopulation dynamics on graphs with three nodes. The analytical solutions are also obtained for metapopulation dynamics on complete graph. When the recovery rate is lower than the critical value, infected individuals survive, the dynamics exhibit stable focuses, and all species coexist. Infection promotes the coexistence of species.

Burst statistics in an early biofilm quorum sensing model: the role of spatial colony-growth heterogeneity

Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called “autoinducers”) and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a “cluster” of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get “induced” into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.

Driven Bose-Hubbard dimer under nonlocal dissipation: A bistable time crystal

We investigate the critical behavior of the open coherently-driven Bose-Hubbard dimer under nonlocal dissipation. A conserved quantity arises from the nonlocal nature of the dissipation, rendering the dimer multistable. In the weak-coupling semiclassical limit, the displayed criticality takes the form of amplitude bistability and breaking of spatial and temporal symmetries. A period-bistable time crystal is formed, consisting of Josephson-like oscillations. Mean-field dynamics and quantum trajectories complement the spectral analysis of the Liouvillian in the approach to the semiclassical limit.

Physical flow effects can dictate plankton population dynamics.

Oceanic flows do not necessarily mix planktonic species. Differences in individual organisms’ physical and hydrodynamic properties can cause changes in drift normal to the mean flow, leading to segregation between species. This physically driven heterogeneity may have important consequences at the scale of population dynamics. Here, we describe how one form of physical forcing, circulating flows with different inertia effects between phytoplankton and zooplankton, can dramatically alter excitable plankton bloom dynamics. This may impact our understanding of the initiation and development of harmful algal blooms (HABs), which have significant negative ecological and socio-economic consequences. We study this system in detail, providing spatio-temporal dynamics for particular scenarios and summarizing large-scale behaviour via spatially averaged bifurcation diagrams. The key message is that, across a large range of parameter values, fluid flow can induce plankton blooms and mean-field population dynamics that are distinct from those predicted for well-mixed systems. The implications for oceanic population dynamic studies are manifest: we argue that the formation of HABs will depend strongly on the physical and biological state of the ecosystem, and that local increases in zooplankton heterogeneity are likely to precede phytoplankton blooms.

Quantum chaos in a Bose-Hubbard dimer with modulated tunneling

In the large-$N$, classical limit, the Bose-Hubbard dimer undergoes a transition to chaos when its tunnelling rate is modulated in time. We use exact and approximate numerical simulations to determine the features of the dynamically evolving state that are correlated with the presence of chaos in the classical limit. We propose the statistical distance between initially similar number distributions as a reliable measure to distinguish regular from chaotic behaviour in the quantum dynamics. Besides being experimentally accessible, number distributions can be efficiently reconstructed numerically from binned phase-space trajectories in a truncated Wigner approximation. Although the evolving Wigner function becomes very irregular in the chaotic regions, the truncated Wigner method is nevertheless able to capture accurately the beyond mean-field dynamics.

Stochastic Resonance and Bifurcation of Order Parameter in a Coupled System of Underdamped Duffing Oscillators

The long-term mean-field dynamics of coupled underdamped Duffing oscillators driven by an external periodic signal with Gaussian noise is investigated. A Boltzmann-type [Formula: see text]-theorem is proved for the associated nonlinear Fokker–Planck equation to ensure that the system can always be relaxed to one of the stationary states as time is long enough. Based on a general framework of the linear response theory, the linear dynamical susceptibility of the system order parameter is explicitly deduced. With the spectral amplification factor as a quantifying index, calculation by the method of moments discloses that both mono-peak and double-peak resonance might appear, and that noise can greatly signify the peak of the resonance curve of the coupled underdamped system as compared with a single-element bistable system. Then, with the input signals taken from laboratory experiments, further observations show that the mean-field coupled stochastic resonance system can amplify the periodic input signal. Also, it reveals that for some driving frequencies, the optimal stochastic resonance parameter and the critical bifurcation parameter have a close relationship. Moreover, it is found that the damping coefficient can also give rise to nontrivial nonmonotonic behaviors of the resonance curve, and the resultant resonant peak attains its maximal height if the noise intensity or the coupling strength takes the critical value. The new findings reveal the role of the order parameter in a coupled system of chaotic oscillators.

Layered chaos in mean-field and quantum many-body dynamics

We investigate the dimension of the phase-space attractor of a quantum chaotic many-body ratchet in the mean-field limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes---Rabi oscillations, chaos, and self-trapping regimes---and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free noninteracting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of local density in each of the dynamical regimes has an attractor characterized by a higher fractal dimension, ${D}_{R}=2.59\ifmmode\pm\else\textpm\fi{}0.01$, ${D}_{C}=3.93\ifmmode\pm\else\textpm\fi{}0.04$, and ${D}_{S}=3.05\ifmmode\pm\else\textpm\fi{}0.05$, compared to the global measure of current, ${D}_{R}=2.07\ifmmode\pm\else\textpm\fi{}0.02$, ${D}_{C}=2.96\ifmmode\pm\else\textpm\fi{}0.05$, and ${D}_{S}=2.30\ifmmode\pm\else\textpm\fi{}0.02$. The deviation between local and global measurements of the attractor's dimension corresponds to an increase towards higher condensate depletion, which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number $N$, namely, as ${N}^{\ensuremath{\beta}}$ with ${\ensuremath{\beta}}_{R}=0.51\ifmmode\pm\else\textpm\fi{}0.004$ and ${\ensuremath{\beta}}_{C}=0.18\ifmmode\pm\else\textpm\fi{}0.004$ for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapping regimes but not in the chaotic regime, with revival times scaling linearly in particle number for Rabi dynamics. Based on the obtained results, we outline pathways for the identification and characterization of emergent phenomena in driven many-body systems. This includes the identification of many-body localization from the many-body measures of the system, the influence of entanglement on the rate of the convergence to the mean-field limit, and the establishment of a polynomial scaling of the Ehrenfest time at which the mean-field description fails to describe the dynamics of the system.

Heterogeneous populations in a network model of collective motion

Biological systems are typically heterogeneous as individuals vary in their response to the external environment and each other. Here, the effect of heterogeneity on the properties of collective motion is studied analytically in the context of a simple network models of collective motion. We consider a population that is composed of two or more sub-groups, each with different sensitivities to external noise. We find that within a mean-field model, in which particles are sampled uniformly, the dynamics in the heterogeneous case is equivalent to an effective homogeneous model with a noise term that is obtained analytically. This is different than recent simulation results with the well-known Vicsek model of collective motion, in which sub-populations with different noise interact non-additively. In particular, if one of the sub-populations is sufficiently ”cold”, it dominates the dynamics of the group as a whole. By introducing an ad-hoc bias in the sampling of neighbors, it is shown that the differences between the mean-field and Vicsek dynamics can be explained by the tendency of colder particles to cluster and partially separate from hotter ones. It is analytically shown that, provided the clustering property is sufficiently strong, the mixed system can become ordered regardless of the noisier populations.

Transport through a magnetic impurity: A slave-spin approach

We study transport across a magnetic impurity by means of a recently developed slave-spin technique that does not require any constraint. Within a conserving mean-field approximation we find a conductance that displays both the known zero-bias anomaly but also the expected peak at bias of order U. We extend the slave-spin mean-field approximation to study the out of equilibrium transient evolution of a quantum dot. We apply the method to investigate the time-evolution of a quantum dot induced by a time-dependent electrochemical potential applied to the contacts. Similarly to the time-dependent Gutzwiller approximation, the mean-field slave-spin dynamics is able to capture dissipation in the leads, so that a steady-state is reached after a characteristic relaxation time.