Fluorescence correlation spectroscopy (FCS) is commonly used to estimate diffusion and reaction rates. In FCS the fluorescence coming from a small volume is recorded and the autocorrelation function (ACF) of the fluorescence fluctuations is computed. Scaling out the fluctuations due to the emission process, this ACF can be related to the ACF of the fluctuations in the number of observed fluorescent molecules. In this paper the ACF of the molecule number fluctuations is studied theoretically, with no approximations, for a reaction-diffusion system in which the fluorescence changes with binding and unbinding. Theoretical ACFs are usually derived assuming that fluctuations in the number of molecules of one species are instantaneously uncorrelated to those of the others and obey Poisson statistics. Under these assumptions, the ACF derived in this paper is characterized only by the diffusive timescale of the fluorescent species and its total weight is the inverse of the mean number of observed fluorescent molecules. The theory is then scrutinized in view of previous experimental results which, for a similar system, gave a different total weight and correct estimates of other diffusive timescales. The total weight mismatch is corrected by assuming that the variance of the number of fluorescent molecules depends on the variance of the particle numbers of the other species, as in the variance decomposition formula. Including the finite acquisition time in its computation, it is shown that the ACF depends on various timescales of the system and that its total weight coincides with the one obtained with the variance decomposition formula. This calculation implies that diffusion coefficients of nonobservable species can be estimated with FCS experiments performed in reaction-diffusion systems. Ways to proceed in future experiments are also discussed.

We studied the dynamics of solvated electrons in the early stage of plasma generation in water induced with an intense femtosecond laser pulse. According to the decay kinetics of solvated electrons, a fast recombination process of solvated electrons (geminate recombination) occurred with a more prolonged lifetime (500 ps to 1 ns) than that observed in previous pulse photolysis studies (10–100 ps). This unusually longer lifetime is attributed to additional production of solvated electrons due to abundant free electrons generated with the laser-induced plasma, implying significant influence of free electrons on the dynamics of solvated electrons.

We numerically investigate the nonequilibrium critical dynamics in three-dimensional anisotropic antiferromagnets in the presence of an external magnetic field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. The nonconserved components of the staggered magnetization order parameter couple dynamically to the conserved component of the magnetization density along the direction of the external field. Employing a hybrid computational algorithm that combines reversible spin precession with relaxational Monte Carlo updates, we study the aging scaling dynamics for the model C critical line, identifying the critical initial slip, autocorrelation, and aging exponents for both the order parameter and the conserved field, thus also verifying the dynamic critical exponent. We further probe the model F critical line by investigating the system size dependence of the characteristic spin wave frequencies near criticality and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point.

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a “light” cone $-v_{0}t<x<v_{0}t$. In particular we study this density close to its maximum in the vicinity of the light cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations. However, this law blows up in the vicinity of the spreading horizon, which is nonphysical in the sense that any finite-time observation will never diverge. We claim that one can find two laws for the spatial density: The first one is the mentioned Lamperti-arcsine law describing the central part of the distribution, and the second is an infinite density illustrating the dynamics for $x\simeq v_{0}t$. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short “time” using a rate formalism.

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. An approach to obtain the exact nonequilibrium steady state of such systems and the mean first passage time to reach the origin is presented. Furthermore, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting, such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement.

Mapping network flows provides insight into the organization of networks, but even though many real networks are bipartite, no method for mapping flows takes advantage of the bipartite structure. What do we miss by discarding this information and how can we use it to understand the structure of bipartite networks better? The map equation models network flows with a random walk and exploits the information-theoretic duality between compression and finding regularities to detect communities in networks. However, it does not use the fact that random walks in bipartite networks alternate between node types, information worth 1 bit. To make some or all of this information available to the map equation, we developed a coding scheme that remembers node types at different rates. We explored the community landscape of bipartite real-world networks from no node-type information to full node-type information and found that using node types at a higher rate generally leads to deeper community hierarchies and a higher resolution. The corresponding compression of network flows exceeds the amount of extra information provided. Consequently, taking advantage of the bipartite structure increases the resolution and reveals more network regularities.

Interconnected ensembles of biological entities are perhaps some of the most complex systems that modern science has encountered so far. In particular, scientists have concentrated on understanding how the complexity of the interacting structure between different neurons, proteins, or species influences the functioning of their respective systems. It is well established that many biological networks are constructed in a highly hierarchical way with two main properties: short average paths that join two apparently distant nodes (neuronal, species, or protein patches) and a high proportion of nodes in modular aggregations. Although several hypotheses have been proposed so far, still little is known about the relation of the modules with the dynamical activity in such biological systems. Here we show that network modularity is a key ingredient for the formation of self-organizing patterns of functional activity, independently of the topological peculiarities of the structure of the modules. In particular, we propose a self-organizing mechanism which explains the formation of macroscopic spatial patterns, which are homogeneous within modules. This may explain how spontaneous order in biological networks follows their modular structural organization. We test our results on real-world networks to confirm the important role of modularity in creating macroscale patterns.

We investigate spreading and recovery of disease in a square lattice, and, in particular, emphasize the role of the initial distribution of infected patches in the network on the progression of an endemic and initiation of a recovery process, if any, due to migration of both the susceptible and infected hosts. The disease starts in the lattice with three possible initial distribution patterns of infected and infection-free sites, viz., infected core patches (ICP), infected peripheral patches (IPP), and randomly distributed infected patches (RDIP). Our results show that infection spreads monotonically in the lattice with increasing migration without showing any sign of recovery in the ICP case. In the IPP case, it follows a similar monotonic progression with increasing migration; however, a self-organized healing process starts for higher migration, leading the lattice to full recovery at a critical rate of migration. Encouragingly, for the initial RDIP arrangement, chances of recovery are much higher with a lower rate of critical migration. An eigenvalue-based semianalytical study is made to determine the critical migration rate for realizing a stable infection-free lattice. The initial fraction of infected patches and the force of infection play significant roles in the self-organized recovery. They follow an exponential law, for the RDIP case, that governs the recovery process. For the frustrating case of ICP arrangement, we propose a random rewiring of links in the lattice allowing long-distance migratory paths that effectively initiate a recovery process. Global prevalence of infection thereby declines and progressively improves with the rewiring probability that follows a power law with the critical migration and leads to the birth of emergent infection-free networks.

We study the bimodal Edwards-Anderson spin glass comparing established methods, namely the multicanonical method, the $1/k$ ensemble, and parallel tempering, to an approach where the ensemble is modified by simulating power-law-shaped histograms in energy instead of flat histograms as in the standard multicanonical case. We show that by this modification a significant speed-up in terms of mean round-trip times can be achieved for all lattice sizes taken into consideration.

In recent years, researchers have realized the difficulties of fitting power-law distributions properly. These difficulties are higher in Zipfian systems, due to the discreteness of the variables and to the existence of two representations for these systems, i.e., two versions depending on the random variable to fit: rank or size. The discreteness implies that a power law in one of the representations is not a power law in the other, and vice versa. We generate synthetic power laws in both representations and apply a state-of-the-art fitting method to each of the two random variables. The method (based on maximum likelihood plus a goodness-of-fit test) does not fit the whole distribution but the tail, understood as the part of a distribution above a cutoff that separates non-power-law behavior from power-law behavior. We find that, no matter which random variable is power-law distributed, using the rank as the random variable is problematic for fitting, in general (although it may work in some limit cases). One of the difficulties comes from recovering the “hidden” true ranks from the empirical ranks. On the contrary, the representation in terms of the distribution of sizes allows one to recover the true exponent (with some small bias when the underlying size distribution is a power law only asymptotically).

Intervals between discrete events representing human activities, as well as other types of events, often obey heavy-tailed distributions, and their impacts on collective dynamics on networks such as contagion processes have been intensively studied. The literature supports that such heavy-tailed distributions are present for interevent times associated with both individual nodes and individual edges in networks. However, the simultaneous presence of heavy-tailed distributions of interevent times for nodes and edges is a nontrivial phenomenon, and its origin has been elusive. In the present study, we propose a generative model and its variants to explain this phenomenon. We assume that each node independently transits between a high-activity and low-activity state according to a continuous-time two-state Markov process and that, for the main model, events on an edge occur at a high rate if and only if both end nodes of the edge are in the high-activity state. In other words, two nodes interact frequently only when both nodes prefer to interact with others. The model produces distributions of interevent times for both individual nodes and edges that resemble heavy-tailed distributions across some scales. It also produces positive correlation in consecutive interevent times, which is another stylized observation for empirical data of human activity. We expect that our modeling framework provides a useful benchmark for investigating dynamics on temporal networks driven by non-Poissonian event sequences.

The constituents of a complex system exchange information to function properly. Their signaling dynamics often leads to the appearance of emergent phenomena, such as phase transitions and collective behaviors. While information exchange has been widely modeled by means of distinct spreading processes—such as continuous-time diffusion, random walks, synchronization and consensus—on top of complex networks, a unified and physically grounded framework to study information dynamics and gain insights about the macroscopic effects of microscopic interactions is still eluding us. In this paper, we present this framework in terms of a statistical field theory of information dynamics, unifying a range of dynamical processes governing the evolution of information on top of static or time-varying structures. We show that information operators form a meaningful statistical ensemble and their superposition defines a density matrix that can be used for the analysis of complex dynamics. As a direct application, we show that the von Neumann entropy of the ensemble can be a measure of the functional diversity of complex systems, defined in terms of the functional differentiation of higher-order interactions among their components. Our results suggest that modularity and hierarchy, two key features of empirical complex systems—from the human brain to social and urban networks—play a key role to guarantee functional diversity and, consequently, are favored.

Timing and its variability are crucial for behavior. Consequently, neural circuits that take part in the control of timing and in the measurement of temporal intervals have been the subject of much research. Here we provide an analytical and computational account of the temporal variability in what is perhaps the most basic model of a timing circuit—the synfire chain. First we study the statistical structure of trial-to-trial timing variability in a reduced but analytically tractable model: a chain of single integrate-and-fire neurons. We show that this circuit's variability is well described by a generative model consisting of local, global, and jitter components. We relate each of these components to distinct neural mechanisms in the model. Next we establish in simulations that these results carry over to a noisy homogeneous synfire chain. Finally, motivated by the fact that a synfire chain is thought to underlie the circuit that takes part in the control and timing of the zebra finch song, we present simulations of a biologically realistic synfire chain model of the zebra finch timekeeping circuit. We find the structure of trial-to-trial timing variability to be consistent with our previous findings and to agree with experimental observations of the song's temporal variability. Our study therefore provides a possible neuronal account of behavioral variability in zebra finches.

Polymer networks are pervasive in biological organisms and engineering materials. Topological defects such as cyclic loops and dangling chains are ubiquitous in polymer networks. While fracture is a dominant mechanism for mechanical failures of polymer networks, existing models for fracture of polymer networks neglect the presence of topological defects. Here, we report a defect-network fracture model that accounts for the impact of various types of topological defects on fracture of polymer networks. We show that the fracture energy of polymer networks should account for the energy from multiple layers of polymer chains adjacent to the crack. We further show that the presence of topological defects tends to toughen a polymer network by increasing the effective chain length, yet to weaken the polymer network by introducing inactive polymer chains. Such competing effects can either increase or decrease the overall intrinsic fracture energy of the polymer network, depending on the types and densities of topological defects. Our model provides theoretical explanations for the experimental data on the intrinsic fracture energy of polymer networks with various types and densities of topological defects.

We study a two-dimensional ferromagnetic Ising model in which spins are updated using modified versions of the Metropolis and Glauber algorithms. These update rules do not obey the detailed balance condition. The steady-state behavior of the model is studied using molecular field theory and Monte Carlo simulations. This model is found to exhibit a nonequilibrium phase transition from a “paramagnetic” state with zero magnetization to a “ferromagnetic” state with nonzero magnetization as the variable that plays the role of temperature in the spin updates is decreased. From detailed Monte Carlo simulations using the modified Metropolis algorithm, we demonstrate explicitly the nonequilibrium nature of the transition and show that it cannot be described as an equilibrium transition with an effective temperature different from the temperature used in the spin updates. The critical exponents that characterize the singular behavior near this continuous phase transition are calculated from finite size scaling of specific heat, magnetization, susceptibility, and correlation length. We find that the values of these exponents are the same (within error bars) as those of the equilibrium Ising model in two dimensions.

For a noisy spin system, we derive a nonlocal stochastic version of the overdamped Landau-Lipshitz equation designed to respect the underlying Hamiltonian structure and sample the canonical or Gibbs distribution while being driven by spatially correlated (colored) noise that regularizes the dynamics, making this Stochastic partial differential equation mathematically well-posed. We begin from a microscopic discrete-time model motivated by the Metropolis-Hastings algorithm for a finite number of spins with periodic boundary conditions whose values are distributed on the unit sphere. We thus propose a future state of the system by adding to each spin colored noise projected onto the sphere, and then accept this proposed state with probability given by the ratio of the canonical distribution at the proposed and current states. For uncorrelated (white) noise this process is guaranteed to sample the canonical distribution. We demonstrate that for colored noise, the method used to project the noise onto the sphere and conserve the magnitude of the spins impacts the equilibrium distribution of the system, as coloring projected noise is not equivalent to projecting colored noise. In a specific scenario we show this break in symmetry vanishes with vanishing proposal size; the resulting continuous-time system of Stochastic differential equations samples the canonical distribution and preserves the magnitude of the spins while being driven by colored noise. Taking the continuum limit of infinitely many spins we arrive at the aforementioned version of the overdamped Landau-Lipshitz equation. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.

We propose a classification of bifurcations of Vlasov equations, based on the strength of the resonance between the unstable mode and the continuous spectrum on the imaginary axis. We then identify and characterize a new type of generic bifurcation where this resonance is weak, but the unstable mode couples with a stable mode and a Casimir invariant of the system to form a size-3 Jordan block. We derive a three-dimensional reduced noncanonical Hamiltonian system describing this bifurcation. Comparison of the reduced dynamics with direct numerical simulations on a test case gives excellent agreement. We finally discuss the relevance of this bifurcation to specific physical situations.

Many real-world social networks constantly change their global properties over time, such as the number of edges, size, and density. While temporal and local properties of social networks have been extensively studied, the origin of their dynamical nature is not yet well understood. Networks may grow or shrink if (a) the total population of nodes changes and/or (b) the chance of two nodes being connected varies over time. Here, we develop a method that allows us to classify the source of time-varying nature of temporal networks. In doing so, we first show empirical evidence that real-world dynamical systems could be categorized into two classes, the difference of which is characterized by the way the number of edges grows with the number of active nodes, i.e., densification scaling. We develop a dynamic hidden-variable model to formally characterize the two dynamical classes. The model is fitted to the empirical data to identify whether the origin of scaling comes from a changing population in the system or shifts in the connecting probabilities.

Many biological functions require dynamics to be necessarily driven out of equilibrium. In contrast, in various contexts, a nonequilibrium dynamics at fast timescales can be described by an effective equilibrium dynamics at a slower timescale. In this work, we study two different aspects: (i) the energy-efficiency tradeoff for a specific nonequilibrium linear dynamics of two variables with feedback and (ii) the cost of effective parameters in a coarse-grained theory as given by the “hidden” dissipation and entropy production rate in the effective equilibrium limit of the dynamics. To meaningfully discuss the tradeoff between energy consumption and the efficiency of the desired function, a one-to-one mapping between function(s) and energy input is required. The function considered in this work is the variance of one of the variables. We get a one-to-one mapping by considering the minimum variance obtained for a fixed entropy production rate and vice versa. We find that this minimum achievable variance is a monotonically decreasing function of the given entropy production rate. When there is a timescale separation, in the effective equilibrium limit, the cost of the effective potential and temperature is the associated “hidden” entropy production rate.

We investigate a granular gas in a shaken quasi-two-dimensional box in molecular dynamics computer simulations. After a sudden change (quench) of the shaking amplitude, transient density correlations are observed orders of magnitude beyond the steady-state correlation length scale. Propagation of the correlations is ballistic, in contrast to recently investigated quenches of Brownian particles that show diffusive propagation [Rohwer et al., Phys. Rev. Lett. 118, 015702 (2017), Rohwer et al., Phys. Rev. E 97, 032125 (2018)]. At sufficiently strong cooling of the fluid the effect is overlaid by clustering instability of the homogeneous cooling state with different scaling behavior. We are able to identify different quench regimes. In each regime correlations exhibit remarkably universal position dependence. In simulations performed with side walls we find confinement effects for temperature and pressure in steady-state simulations and an additional transient wall pressure contribution when changing the shaking amplitude. The transient contribution is ascribed to enhanced relaxation of the fluid in the presence of walls. From incompatible scaling behavior we conclude that the observed effects with and without side walls constitute distinct phenomena.