Mean-field Rate Equations Research Articles

Imperfect bifurcations in opinion dynamics under external fields

We investigate, through a kinetic-exchange model, the impact that an external field, like advertising and propaganda, has on opinion dynamics. We address the situations where two opposite alternatives can be selected but the possibility of indecision also exists. In this model, individuals influence each other through pairwise interactions, which can be of agreement or disagreement, and there are also external fields that can skew decision making. Two parameters are used to model the external interactions: one measures the sensitivity of the individuals to be influenced, another quantifies in which direction. We study this model in a fully connected social network scenario, by means of numerical simulations of the kinetic exchange dynamics and analytical results derived from the mean-field rate equations. We show how the external bias gives rise to imperfect bifurcations, and cusp catastrophes, allowing abrupt changes and hysteresis depending on the level of disagreement in interpersonal interactions and on the strength of the external influence.

Absorbing phase transition in the coupled dynamics of node and link states in random networks

We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states represent positive or negative relationships. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite-size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuations take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a class of finite-size topological transition associated with group splitting in the network of these final frozen configurations.

Mathematical theory of exchange-driven growth

Exchange-driven growth is a process in which pairs of clusters interact by exchanging single unit of mass at a time. The rate of exchange is given by an interaction kernel which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field rate equations of this process for the first time. We find two different classes of behavior depending on whether is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying . This result is optimal in the sense that we show for a large class of initial conditions and kernels satisfying the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for ( while existence is lost for ( In the intermediate regime we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.

Derivation of mean-field equations for stochastic particle systems

We study stochastic particle systems on a complete graph and derive effective mean-field rate equations in the limit of diverging system size, which are also known from cluster aggregation models. We establish the propagation of chaos under generic growth conditions on particle jump rates, and the limit provides a master equation for the single site dynamics of the particle system, which is a non-linear birth death chain. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary distributions. Our findings are consistent with recent results on exchange driven growth, and provide a connection between the well studied phenomena of gelation and condensation.

Dynamics of Patchy Particles in and out of Equilibrium.

We combine particle-based simulations, mean-field rate equations, and Wertheim's theory to study the dynamics of patchy particles in and out of equilibrium, at different temperatures and densities. We consider an initial random distribution of nonoverlapping three-patch particles, with no bonds, and analyze the time evolution of the breaking and bonding rates of a single bond. We find that the asymptotic (equilibrium) dynamics differs from the initial (out of equilibrium) one. These differences are expected to depend on the initial conditions, temperature, and density.

Sink strength simulations using the Monte Carlo method: Applied to spherical traps

The sink strength is an important parameter for the mean-field rate equations to simulate temporal changes in the micro-structure of materials. However, there are noteworthy discrepancies between sink strengths obtained by the Monte Carlo and analytical methods. In this study, we show the reasons for these differences. We present the equations to estimate the statistical error for sink strength calculations and show the way to determine the sink strengths for multiple traps.We develop a novel, very fast Monte Carlo method to obtain sink strengths. The results show that, in addition to the well-known sink strength dependence of the trap concentration, trap radius and the total sink strength, the sink strength also depends on the defect diffusion jump length and the total trap volume fraction. Taking these factors into account, allows us to obtain a very accurate analytic expression for the sink strength of spherical traps.

Mechanistic Picture and Kinetic Analysis of Surface-Confined Ullmann Polymerization.

Surface-confined polymerization via Ullmann coupling is a promising route to create one- and two-dimensional covalent π-conjugated structures, including the bottom-up growth of graphene nanoribbons. Understanding the mechanism of the Ullmann reaction is necessary to provide a platform for rationally controlling the formation of these materials. We use fast X-ray photoelectron spectroscopy (XPS) in kinetic measurements of epitaxial surface polymerization of 1,4-dibromobenzene on Cu(110) and devise a kinetic model based on mean field rate equations, involving a transient state. This state is observed in the energy landscapes calculated by nudged elastic band (NEB) within density functional theory (DFT), which assumes as initial and final geometries of the organometallic and polymeric structures those observed by scanning tunneling microscopy (STM). The kinetic model accounts for all the salient features observed in the experimental curves extracted from the fast-XPS measurements and enables an enhanced understanding of the polymerization process, which is found to follow a nucleation-and-growth behavior preceded by the formation of a transient state.

Synchronization and extinction in cyclic games with mixed strategies.

We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing β balls of three and four types, respectively, where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of β, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.

Spirals and coarsening patterns in the competition of many species: a complex Ginzburg–Landau approach

In order to model real ecological systems one has to consider many species that interact in complex ways. However, most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions. The few studies dealing with larger number of species and/or more complex interaction schemes are mostly restricted to numerical explorations. In this paper we determine, starting from the deterministic mean-field rate equations, for large classes of systems the space of coexistence fixed points at which biodiversity is maximal. For systems with a single coexistence fixed point we derive complex Ginzburg–Landau equations that allow to describe space-time pattern realized in two space dimensions. For selected cases we compare the theoretical predictions with the pattern observed in numerical simulations.

Evolution behavior of catalytically activated replication—decline in a coagulation process

We propose a catalytically activated replication—decline model of three species, in which two aggregates of the same species can coagulate themselves, an A aggregate of any size can replicate itself with the help of B aggregates, and the decline of A aggregate occurs under the catalysis of C aggregates. By means of mean-field rate equations, we derive the asymptotic solutions of the aggregate size distribution ak (t) of species A, which is found to depend strongly on the competition among three mechanisms: the self-coagulation of species A, the replication of species A catalyzed by species B, and the decline of species A catalyzed by species C. When the self-coagulation of species A dominates the system, the aggregate size distribution ak(t) satisfies the conventional scaling form. When the catalyzed replication process dominates the system, ak(t) takes the generalized scaling form. When the catalyzed decline process dominates the system, ak(t) approaches the modified scaling form.

Noise-induced metastability in biochemical networks

Intracellular biochemical reactions exhibit a rich dynamical phenomenology which cannot be explained within the framework of mean-field rate equations and additive noise. Here, we show that the presence of metastable states and radically different time scales are general features of a broad class of autocatalytic reaction networks, and that this fact may be exploited to gain analytical results. The latter point is demonstrated by a treatment of the paradigmatic Togashi-Kaneko reaction, which has resisted theoretical analysis for the last decade.

On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model

We aim to clarify the relationship between interacting three-species models and the two-species Lotka-Volterra (LV) model. We utilize mean-field theory and Monte Carlo simulations on two-dimensional square lattices to explore the temporal evolution characteristics of two different interacting three-species predator-prey systems: (1) a cyclic rock-paper-scissors (RPS) model with conserved total particle number but strongly asymmetric reaction rates that lets the system evolve towards one corner of configuration space; (2) a hierarchical food chain where an additional intermediate species is inserted between the predator and prey in the LV model. For model variant (1), we demonstrate that the evolutionary properties of both minority species in the steady state of this stochastic spatial three-species corner RPS model are well approximated by the LV system, with its emerging characteristic features of localized population clustering, persistent oscillatory dynamics, correlated spatio-temporal patterns, and fitness enhancement through quenched spatial disorder in the predation rates. In contrast, we could not identify any regime where the hierarchical model (2) would reduce to the two-species LV system. In the presence of pair exchange processes, the system remains essentially well-mixed, and we generally find the Monte Carlo simulation results for the spatially extended model (2) to be consistent with the predictions from the corresponding mean-field rate equations. If spreading occurs only through nearest-neighbor hopping, small population clusters emerge; yet the requirement of an intermediate species cluster obviously disrupts spatio-temporal correlations between predator and prey, and correspondingly eliminates many of the intriguing fluctuation phenomena that characterize the stochastic spatial LV system.

Modeling pulsed-laser melting of embedded semiconductor nanoparticles

Pulsed-laser melting (PLM) is commonly used to achieve a fast quench rate in both thin films and nanoparticles. A model for the size evolution during PLM of nanoparticles confined in a transparent matrix, such as those created by ion-beam synthesis, is presented. A self-consistent mean-field rate equations approach that has been used successfully to model ion beam synthesis of germanium nanoparticles in silica is extended to include the PLM process. The PLM model includes classical optical absorption, multiscale heat transport by both analytical and finite difference methods, and melting kinetics for confined nanoparticles. The treatment of nucleation and coarsening behavior developed for the ion beam synthesis model is modified to allow for a nonuniform temperature gradient and for interacting liquid and solid particles with different properties. The model allows prediction of the particle size distribution after PLM under various laser fluences, starting from any particle size distribution including as-implanted or annealed simulated samples. A route for narrowing the size distribution of embedded nanoparticles is suggested, with simulated distribution widths as low as 15% of the average size.

Kinetics of catalytically activated aggregation—fragmentation process

We propose a catalytically activated aggregation—fragmentation model of three species, in which two clusters of species A can coagulate into a larger one under the catalysis of B clusters; otherwise, one cluster of species A will fragment into two smaller clusters under the catalysis of C clusters. By means of mean-field rate equations, we derive the asymptotic solutions of the cluster-mass distributions ak(t) of species A, which is found to depend strongly on the competition between the catalyzed aggregation process and the catalyzed fragmentation process. When the catalyzed aggregation process dominates the system, the cluster-mass distribution ak(t) satisfies the conventional scaling form. When the catalyzed fragmentation process dominates the system, the scaling description of ak(t) breaks down completely and the monodisperse initial condition of species A would not be changed in the long-time limit. In the marginal case when the effects of catalyzed aggregation and catalyzed fragmentation counteract each other, ak(t) takes the modified scaling form and the system can eventually evolve to a steady state.

Impact of soft impingement on the kinetics of diffusion-controlled growth of immiscible alloys

The effect of the soft impingement mechanism on the kinetics of diffusion-controlled phase transitions of binary alloys is investigated. Soft impingement arises from the interference of the diffusional fields and implies a different morphology of the product phase when compared to that attained in the case of genuine impingement. Mean field rate equations for nucleus radius and supersaturation are coupled with the mass balance equation in terms of the geometrical parameter (time dependent) which defines the soft impingement process. The kinetics is studied as a function of the initial supersaturation and it is compared with that attained in the case of genuine impingement. Solutions of the kinetics are obtained in the model case of time dependent supersaturation which scales according to Ham’s law and in the case of simultaneous nucleation. The phase transformation of immiscible alloys in one-dimensional system is also investigated. In this case and for point islands, the exact solution of the kinetics is obtained and allows one to gain an insight into the validity of the Kolmogorov–Johnson–Mehl–Avrami model and Avrami’s exponent for one-dimensional growths.

Adlayer inhomogeneity without lateral interactions: Rationalizing correlation effects in CO oxidation at RuO2(110) with first-principles kinetic Monte Carlo

Microkinetic modeling of surface chemical reactions still relies heavily on the mean-field based rate equation approach. This approach is expected to be most accurate for systems without appreciable lateral interactions among the adsorbed chemicals, and there in particular for the uniform adlayers resulting in poisoned regimes with predominant coverage of one species. Using first-principles kinetic Monte Carlo simulations and the CO oxidation at RuO(2)(110) as a showcase, we demonstrate that even in this limit mean-field rate equations fail to predict the catalytic activity by orders of magnitude. This deficiency is traced back to the inability to account for the vacancy pair formation that is kinetically driven by the ongoing reactions.

Island size distributions in submonolayer growth: Prediction by mean field theory with coverage dependent capture numbers

We show that mean field rate equations (MFRE) for submonolayer growth with hit and stick aggregation can successfully predict the island size distributions (ISDs) in the precoalescence regime if the full dependence of capture numbers on both the island size and the coverage is taken into account. This is demonstrated by comparison of the integrated MFRE with results from extensive kinetic Monte Carlo simulations. The attempt to use various simplified expressions for the capture numbers in the integration of the MFRE turns out to be insufficient to yield a good description of the ISD.

Aggregation Processes with Catalysis-Driven Decomposition

We propose a three-species aggregation model with catalysis-driven decomposition. Based on the mean-field rate equations, we investigate the evolution behavior of the system with the size-dependent catalysis-driven decomposition rate J(i; j; k) = Jijkv and the constant aggregation rates. The results show that the cluster size distribution of the species without decomposition can always obey the conventional scaling law in the case of 0 ≤ v ≤ 1, while the kinetic evolution of the decomposed species depends crucially on the index v. Moreover, the total size of the species without decomposition can keep a nonzero value at large times, while the total size of the decomposed species decreases exponentially with time and vanishes finally.

Numerical solutions of the isotropic 3-wave kinetic equation

We show that the isotropic 3-wave kinetic equation is equivalent to the mean-field rate equations for an aggregation–fragmentation problem with an unusual fragmentation mechanism. This analogy is used to write the theory of 3-wave turbulence almost entirely in terms of a single scaling parameter. A new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending Lee’s method for solving aggregation equations. The new algorithm is validated against some analytical calculations of the Kolmogorov–Zakharov (KZ) constant for some families of model interaction coefficients. The algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature. Firstly, it is shown that for finite capacity cascades, the dissipation of energy becomes independent of the cut-off frequency as this cut-off is taken to infinity. This is an explicit indication of the presence of a dissipative anomaly. Secondly, a preliminary numerical study is presented of the so-called bottleneck effect in a wave turbulence context. It is found that the structure of the bottleneck depends non-trivially on the interaction coefficient. Finally, some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so-called mixed solutions of the kinetic equation which exhibit aspects of both KZ and equilibrium equipartition spectra.

Mean-field level analysis of epidemics in directed networks

The susceptible–infected–removed spreading model in a directed graph is studied. The mean-field level rate equations are built with the degree–degree connectivity correlation element and the (in, out)-degree distribution. And the outbreak threshold is obtained analytically—it is determined by the combination of connectivity probability and the degree distribution. Furthermore, the methods of calculating the degree–degree correlations in directed networks are presented. The numerical results of the discrete epidemic processes in networks verify our analyses.