Approximate Master Equations Research Articles

Approximate master equationsfor the spatial public goods game.

The spatial public goods game has been used to examine factors that promote cooperation. Owing to the complexity of the dynamics of this game, previous studies on this model neglected analytical approaches and relied entirely on numerical calculations using the Monte Carlo (MC) simulations. In this paper, we present the approximate master equations(AMEs) for this model. We report that the results obtained by the AMEs are mostly qualitatively consistent with those obtained by the MC simulations. Furthermore, we show that it is possible to obtain phase boundaries analytically in certain parameter regions. In the region where the noise in strategy decisions is very large, the phase boundary can be obtained analytically by considering perturbations from the steady state of the voter model. In the noiseless region, discontinuous phase transitions occur because of the characteristics of the function that represents strategy updating. Our approach is useful for clarifying the details of the mechanisms that promote cooperation and can be easily applied to other group interaction models.

Ordering dynamics and aging in the symmetrical threshold model

The so-called Granovetter–Watts model was introduced to capture a situation in which the adoption of new ideas or technologies requires a certain redundancy in the social environment of each agent to take effect. This model has become a paradigm for complex contagion. Here we investigate a symmetric version of the model: agents may be in two states that can spread equally through the system via complex contagion. We find three possible phases: a mixed one (dynamically active disordered state), an ordered one, and a heterogeneous frozen phase. These phases exist for several configurations of the contact network. Then, we consider the effect of introducing aging as a non-Markovian mechanism in the model, where agents become increasingly resistant to change their state the longer they remain in it. We show that when aging is present, the mixed phase is replaced, for sparse networks, by a new phase with different dynamical properties. This new phase is characterized by an initial disordering stage followed by a slow ordering process toward a fully ordered absorbing state. In the ordered phase, aging modifies the dynamical properties. For random contact networks, we develop a theoretical description based on an approximate master equation that describes with good accuracy the results of numerical simulations for the model with and without aging.

High-accuracy approximation of evolutionary pairwise games on complex networks

Previous studies have shown that the topological properties of a complex network, such as heterogeneity and average degree, affect the evolutionary game dynamics on it. However, traditional numerical simulations are usually time-consuming and demand a lot of computational resources. In this paper, we propose the method of dynamical approximate master equations (DAMEs) to accurately predict the evolutionary outcomes on complex networks. We demonstrate that the accuracy of DAMEs supersedes previous standard pairwise approximation methods, and DAMEs require far fewer computational resources than traditional numerical simulations. We apply the DAMES to investigate prisoner’s dilemma and snowdrift game on regular and scale-free networks, demonstrating its effectiveness. Overall, our method facilitates the investigation of evolutionary dynamics on a broad range of complex networks, and provides new insights into the puzzle of cooperation.

Nonadiabatically driven open quantum systems under out-of-equilibrium conditions: Effect of electron-phonon interaction

In this paper we explore the effects of nonadiabatic external driving on the dynamics of an electronic system coupled to two electronic leads and to a phonon mode, with and without damping. In the limit of slow driving, we establish nonadiabatic corrections to thermodynamic and transport quantities. In particular, we study the first-order correction to the work done by the driving, the charge current, and the vibrational excitation using a perturbative expansion. We then compare the results to the numerically exact hierarchical equations of motion (HEOM) approach. Furthermore, the HEOM analysis spans both the weak and strong system-bath coupling regime and the slow- and fast-driving limits. We show that the electronic friction and the nonadiabatic corrections to the charge current provide a clear indicator for the Franck-Condon effect and for nonresonant tunneling processes. We also discuss the validity of the approximate quantum master equation approach and the benefits of using HEOM to study nonadiabatically driven open quantum systems out of equilibrium.

Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs

Contagion phenomena are often the results of multibody interactions—such as superspreading events or social reinforcement—describable as hypergraphs. We develop an approximate master equation framework to study contagions on hypergraphs with a heterogeneous structure in terms of group size (hyperedge cardinality) and of node membership (hyperdegree). By mapping multibody interactions to nonlinear infection rates, we demonstrate the influence of large groups in two ways. First, we characterize the phase transition, which can be continuous or discontinuous with a bistable regime. Our analytical expressions for the critical and tricritical points highlight the influence of the first three moments of the membership distribution. We also show that heterogeneous group sizes and nonlinear contagion promote a mesoscopic localization regime where contagion is sustained by the largest groups, thereby inhibiting bistability. Second, we formulate an optimal seeding problem for hypergraph contagion and compare two strategies: allocating seeds according to node or group properties. We find that, when the contagion is sufficiently nonlinear, groups are more effective seeds than individual hubs.

Origin of Chirality Induced Spin Selectivity in Photoinduced Electron Transfer.

Here we propose a mechanism by which spin-polarization can be generated dynamically in chiral molecular systems undergoing photoinduced electron transfer. The proposed mechanism explains how spin-polarization emerges in systems where charge transport is dominated by incoherent hopping, mediated by spin-orbit and electronic exchange couplings through an intermediate charge transfer state. We derive a simple expression for the spin-polarization that predicts a nonmonotonic temperature dependence, consistent with recent experiments, and a maximum spin-polarization that is independent of the magnitude of the spin-orbit coupling. We validate this theory using approximate quantum master equations and the numerically exact hierarchical equations of motion. The proposed mechanism of chirality induced spin selectivity should apply to many chiral systems, and the ideas presented here have implications for the study of spin transport at temperatures relevant to biology and provide simple principles for the molecular control of spins in fluctuating environments.

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks.

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

Binary-state dynamics on complex networks: Stochastic pair approximation and beyond

Theoretical approaches to binary-state models on complex networks are generally restricted to infinite size systems, where a set of non-linear deterministic equations is assumed to characterize its dynamics and stationary properties. We develop in this work the stochastic formalism of the different compartmental approaches, these are: approximate master equation (AME), pair approximation (PA) and heterogeneous mean field (HMF), in descending order of accuracy. Using different system-size expansions of a general master equation, we are able to obtain approximate solutions of the fluctuations and finite-size corrections of the global state. On the one hand, far from criticality, the deviations from the deterministic solution are well captured by a Gaussian distribution whose properties we derive, including its correlation matrix and corrections to the average values. On the other hand, close to a critical point there are non-Gaussian statistical features that can be described by the finite-size scaling functions of the models. We show how to obtain the scaling functions departing only from the theory of the different approximations. We apply the techniques for a wide variety of binary-state models in different contexts, such as epidemic, opinion and ferromagnetic models.

Correlation-Picture Approach to Open-Quantum-System Dynamics

We introduce a new dynamical picture, referred to as correlation picture,' which connects a correlated state to its uncorrelated counterpart. Using this picture allows us to derive an exact dynamical equation for a general open-system dynamics with system--environment correlations included. This exact dynamics is in the form of a Lindblad-like equation even in the presence of initial system-environment correlations. For explicit calculations, we also develop a weak-correlation expansion formalism that allows us to perform systematic perturbative approximations. This expansion provides approximate master equations which can feature advantages over existing weak-coupling techniques. As a special case, we derive a Markovian master equation, which is different from existing approaches. We compare our equations with corresponding standard weak-coupling equations by two examples, where our correlation picture formalism is more accurate, or at least as accurate as weak-coupling equations.

Non-Markovian Evolution of Multi-level System Interacting with Several Reservoirs. Exact and Approximate

An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequities between the exact and approximate rates.

Social clustering in epidemic spread on coevolving networks.

Even though transitivity is a central structural feature of social networks, its influence on epidemic spread on coevolving networks has remained relatively unexplored. Here we introduce and study an adaptive susceptible-infected-susceptible (SIS) epidemic model wherein the infection and network coevolve with nontrivial probability to close triangles during edge rewiring, leading to substantial reinforcement of network transitivity. This model provides an opportunity to study the role of transitivity in altering the SIS dynamics on a coevolving network. Using numerical simulations and approximate master equations (AMEs), we identify and examine a rich set of dynamical features in the model. In many cases, AMEs including transitivity reinforcement provide accurate predictions of stationary-state disease prevalence and network degree distributions. Furthermore, for some parameter settings, the AMEs accurately trace the temporal evolution of the system. We show that higher transitivity reinforcement in the model leads to lower levels of infective individuals in the population, when closing a triangle is the dominant rewiring mechanism. These methods and results may be useful in developing ideas and modeling strategies for controlling SIS-type epidemics.

Absence of Coulomb Blockade in the Anderson Impurity Model at the Symmetric Point

In this work, we investigate the characteristics of the electric current in the so-called symmetric Anderson impurity model. We study the nonequilibrium model using two complementary approximate methods, the perturbative quantum master equation approach to the reduced density matrix, and a self-consistent equation of motion approach to the nonequilibrium Green's function. We find that at a particular symmetry point, an interacting Anderson impurity model recovers the same steady-state current as an equivalent non-interacting model, akin a two-band resonant level model. We show this in the Coulomb blockade regime for both high and low temperatures, where either the approximate master equation approach and the Green's function method provide accurate results for the current. We conclude that the steady-state current in the symmetric Anderson model at this regime does not encode characteristics of a many-body interacting system.

Majority-vote dynamics on multiplex networks with two layers

Majority-vote model is a much-studied model for social opinion dynamics of two competing opinions. With the recent appreciation that our social network comprises a variety of different ‘layers’ forming a multiplex network, a natural question arises on how such multiplex interactions affect the social opinion dynamics and consensus formation. Here, the majority-vote processes will be studied on multiplex networks with two layers to understand the effect of multiplexity on opinion dynamics. We will discuss how global consensus is reached by different types of voters: AND- and OR-rule voters on multiplex-network and voters on single-network system. The AND-model reaches the largest consensus below the critical noise parameter qc. It needs, however, much longer time to reach consensus than other models. In the vicinity of the transition point, the consensus collapses abruptly. The OR-model attains smaller level of consensus than the AND-rule but reaches the consensus more quickly. Its consensus transition is continuous. The numerical simulation results are supported qualitatively by analytical calculations based on the approximate master equation.

Dynamics of Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level open quantum system with finite number $$N$$ of quantum eigenstates interacting with arbitrary external deterministic fields and dissipative environment simultaneously. Unlike conventional master equations derived by the projection operator method for the quantum systems of this kind so far, these equations are homogeneous in the reduced statistical operator while, at the same time, taking into account initial correlations between the multi-level system and the environment. Consistent representation of the multi-level system evolution in terms of the $$SU(N)$$ operator algebra facilitates efficient numerical analysis of these equations derived for various physically realistic quantum models.

Reconciliation of quantum local master equations with thermodynamics

The study of open quantum systems often relies on approximate master equations derived under the assumptions of weak coupling to the environment. However when the system is made of several interacting subsystems such a derivation is in many cases very hard. An alternative method, employed especially in the modeling of transport in mesoscopic systems, consists in using local master equations (LMEs) containing Lindblad operators acting locally only on the corresponding subsystem. It has been shown that this approach however generates inconsistencies with the laws of thermodynamics. In this paper we demonstrate that using a microscopic model of LMEs based on repeated collisions all thermodynamic inconsistencies can be resolved by correctly taking into account the breaking of global detailed balance related to the work cost of maintaining the collisions. We provide examples based on a chain of quantum harmonic oscillators whose ends are connected to thermal reservoirs at different temperatures. We prove that this system behaves precisely as a quantum heat engine or refrigerator, with properties that are fully consistent with basic thermodynamics.

Sudden spreading of infections in an epidemic model with a finite seed fraction

We study a simple case of the susceptible-weakened-infected-removed model in regular random graphs in a situation where an epidemic starts from a finite fraction of initially infected nodes (seeds). Previous studies have shown that, assuming a single seed, this model exhibits a kind of discontinuous transition at a certain value of infection rate. Performing Monte Carlo simulations and evaluating approximate master equations, we find that the present model has two critical infection rates for the case with a finite seed fraction. At the first critical rate the system shows a percolation transition of clusters composed of removed nodes, and at the second critical rate, which is larger than the first one, a giant cluster suddenly grows and the order parameter jumps even though it has been already rising. Numerical evaluation of the master equations shows that such sudden epidemic spreading does occur if the degree of the underlying network is large and the seed fraction is small.

Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.

Lumping of degree-based mean-field and pair-approximation equations for multistate contact processes.

Contact processes form a large and highly interesting class of dynamic processes on networks, including epidemic and information-spreading networks. While devising stochastic models of such processes is relatively easy, analyzing them is very challenging from a computational point of view, particularly for large networks appearing in real applications. One strategy to reduce the complexity of their analysis is to rely on approximations, often in terms of a set of differential equations capturing the evolution of a random node, distinguishing nodes with different topological contexts (i.e., different degrees of different neighborhoods), such as degree-based mean-field (DBMF), approximate-master-equation (AME), or pair-approximation (PA) approaches. The number of differential equations so obtained is typically proportional to the maximum degree k_{max} of the network, which is much smaller than the size of the master equation of the underlying stochastic model, yet numerically solving these equations can still be problematic for large k_{max}. In this paper, we consider AME and PA, extended to cope with multiple local states, and we provide an aggregation procedure that clusters together nodes having similar degrees, treating those in the same cluster as indistinguishable, thus reducing the number of equations while preserving an accurate description of global observables of interest. We also provide an automatic way to build such equations and to identify a small number of degree clusters that give accurate results. The method is tested on several case studies, where it shows a high level of compression and a reduction of computational time of several orders of magnitude for large networks, with minimal loss in accuracy.

Algebraic aspects of the driven dynamics in the density operator and correlation functions calculation for multi-level open quantum systems

Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level quantum system with finite number N of quantum eigenstates interacting with arbitrary external classical fields and dissipative environment simultaneously. It was shown that the structure of these equations can be simplified significantly if the free Hamiltonian driven dynamics of an arbitrary quantum multi-level system under the influence of the external driving fields as well as its Markovian and non-Markovian evolution, stipulated by the interaction with the environment, are described in terms of the SU(N) algebra representation. As a consequence, efficient numerical methods can be developed and employed to analyze these master equations for real problems in various fields of theoretical and applied physics. It was also shown that literally the same master equations hold not only for the reduced density operator but also for arbitrary nonequilibrium multi-time correlation functions as well under the only assumption that the system and the environment are uncorrelated at some initial moment of time. A calculational scheme was proposed to account for these lost correlations in a regular perturbative way, thus providing additional computable terms to the correspondent master equations for the correlation functions.

Quantum dynamics intervened by repeated nonselective measurements

We derive the theory of open quantum system dynamics intervened by a series of nonselective measurements. We analyze the cases of time-independent and time-dependent Hamiltonian dynamics between the measurements and find the approximate master equation in the stroboscopic limit. We also consider a situation, in which the measurement basis changes in time, and illustrate it by nonselective measurements in the basis of diabatic states of the Landau–Zener model.