To fully characterize the information that two source variables carry about a third target variable, one must decompose the total information into redundant, unique, and synergistic components, i.e., obtain a partial information decomposition (PID). However, Shannon's theory of information does not provide formulas to fully determine these quantities. Several recent studies have begun addressing this. Some possible definitions for PID quantities have been proposed and some analyses have been carried out on systems composed of discrete variables. Here we present an in-depth analysis of PIDs on Gaussian systems, both static and dynamical. We show that, for a broad class of Gaussian systems, previously proposed PID formulas imply that (i) redundancy reduces to the minimum information provided by either source variable and hence is independent of correlation between sources, and (ii) synergy is the extra information contributed by the weaker source when the stronger source is known and can either increase or decrease with correlation between sources. We find that Gaussian systems frequently exhibit net synergy, i.e., the information carried jointly by both sources is greater than the sum of information carried by each source individually. Drawing from several explicit examples, we discuss the implications of these findings for measures of information transfer and information-based measures of complexity, both generally and within a neuroscience setting. Importantly, by providing independent formulas for synergy and redundancy applicable to continuous time-series data, we provide an approach to characterizing and quantifying information sharing amongst complex system variables.

The largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, the appearance of half-integer and other subharmonic steps between the large harmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, however, the half-integer steps are larger than harmonic ones, and Farey construction is only present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents has also shown that regardless of the substrate potential or deformation, there was no chaos in the system.

We studied the behavior of a dumbbell bouncing upon a sinusoidally vibrating plate. By introducing chiral asymmetry to the geometry of the dumbbell, we observed a cascade of bifurcations with an increase in the vibration amplitude: spinning, orbital, and rolling. In contrast, for an achiral dumbbell, bifurcation is generated by a change from random motion to vectorial inchworm motion. A simple model particle was considered in a numerical simulation that reproduced the essential aspects of the experimental observation. The mode bifurcation from directional motion to random motion is interpreted analytically by a simple mechanical discussion.

Clustering of inertial particles is analyzed in chaotic compressible flows. A simplified dynamical model for the motion of inertial particles in a compressible flow has been derived. Clustering enhancement has been observed for intermediate Stokes times and characterized in terms of the number of particles with negative finite-time Lyapunov exponents and the Lyapunov dimension of the model attractor. Cluster formation has been observed to depend on the nature of the flow; vortical or shear. The motion of heavy and light particles is analyzed in terms of the compressibility and correlation length of the density field.

Employing a numerical simulation, the normal modes are investigated for finite, one-dimensional horizontal dust chains in complex plasma. Mode couplings induced by the ion flow within the sheath are identified in the mode spectra and the coupling rules are determined. Two types of resonance-induced instabilities are observed, one bidirectional and one unidirectional. Bidirectional instability is found to cause melting of the chain with the melting proceeding via a two-step process which obeys the Lindemann criterion. The relationship between the normal mode spectra observed in finite systems and the wave dispersion relations seen in larger systems was also examined using a dust chain model. For this case, the dispersion relation was obtained through multiplication of the mode spectra matrix by a transition matrix. The resulting dispersion relations exhibit both the general features observed in larger crystals as well as several characteristics unique to finite systems, such as discontinuities and strong energy-density fluctuations.

We develop a theoretical framework for analyzing ecological models with a multidimensional niche space. Our approach relies on the fact that ecological niches are described by sequences of symbols, which allows us to include multiple phenotypic traits. Ecological drivers, such as competitive exclusion, are modeled by introducing the Hamming distance between two sequences. We show that a suitable transform diagonalizes the community interaction matrix of these models, making it possible to predict the conditions for niche differentiation and, close to the instability onset, the asymptotically long time population distributions of niches. We exemplify our method using the Lotka-Volterra equations with an exponential competition kernel.

We study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface. For Euclidean domains in which Brownian motion can be split into independent lateral and transverse displacements, we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior. This analysis is applicable to slab domains, general cylindrical domains, and a half-space. We investigate the spreading effect due to multiple reflections on the surface, and the underlying role of finite reactivity. We discuss further extensions and applications of analytical results to describe Laplacian transfer phenomena such as permeation through semipermeable membranes, secondary current distribution on partially blocking electrodes, and surface relaxation in nuclear magnetic resonance.

In a previous publication [R. Wittmann, M. Marechal, and K. Mecke, Europhys. Lett. 109, 26003 (2015)], we introduced fundamental mixed measure theory (FMMT) for mixtures of anisotropic hard bodies, which shows that earlier results with an empirical parameter are inaccurate. Now we provide a deeper insight into the background of this theory in integral geometry. We study the Frank elastic coefficients in the nematic phase of the hard spherocylinder fluid. The framework of FMMT provides us with the required direct correlation function without additional input of an equation of state. A series representation of the mixed measure gives rise to closed analytical formulas for the elastic constants that only depend on the density, order parameters, and the particle geometry, pointing out a significant advantage of our geometry-based approach compared to other density functionals. Our elastic coefficients are in good agreement with computer simulations and increase with the density and the nematic order parameter. We confirm earlier mean-field predictions in the limits of low orientational order and infinitely long rods.

The solution of the many-body statistical mechanical theory of elasticity formulated by James and Guth in the 1940s [H. M. James, J. Chem. Phys. 15, 651 (1947)] is presented. The remarkable aspect of the solution is that it gives an elastic free energy that is essentially equivalent to that developed by Flory over a period of several decades.

We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first-passage time can depend nonmonotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first-passage time is maximal for nonvanishing noise. We provide an explanation for this effect by analytically transforming the original model into a first-passage-time problem for Brownian motion. This transformation also allows for a perturbative calculation of the first-passage-time histograms. In turn this provides quantitative insights into the mechanisms that lead to the nonmonotonic behavior of the mean first-passage time. The perturbation expansion is in excellent agreement with direct numerical simulations. The approach developed here can be applied to any deterministic subthreshold dynamics and any Gauss-Markov processes for the firing threshold. This opens up the possibility to incorporate biophysically detailed components into the subthreshold dynamics, rendering our approach a powerful framework that sits between traditional integrate-and-fire models and complex mechanistic descriptions of neural dynamics.

The spin-echo approach is extended to include biocomplexes for which the interaction with dynamical noise, produced by the protein environment, is strong. Significant restoration of the free induction decay signal due to homogeneous (decoherence) and inhomogeneous (dephasing) broadening is demonstrated analytically and numerically for both an individual dimer of interacting chlorophylls and for an ensemble of dimers. Our approach does not require the use of small interaction constants between the electron states and the protein fluctuations. It is based on an exact and closed system of ordinary differential equations that can be easily solved for a wide range of parameters that are relevant for bioapplications.

In many real-world complex systems, the time evolution of the network's structure and the dynamic state of its nodes are closely entangled. Here we study opinion formation and imitation on an adaptive complex network which is dependent on the individual dynamic state of each node and vice versa to model the coevolution of renewable resources with the dynamics of harvesting agents on a social network. The adaptive voter model is coupled to a set of identical logistic growth models and we mainly find that, in such systems, the rate of interactions between nodes as well as the adaptive rewiring probability are crucial parameters for controlling the sustainability of the system's equilibrium state. We derive a macroscopic description of the system in terms of ordinary differential equations which provides a general framework to model and quantify the influence of single node dynamics on the macroscopic state of the network. The thus obtained framework is applicable to many fields of study, such as epidemic spreading, opinion formation, or socioecological modeling.

The effects of amplitude-dependent coupling on oscillator death (OD) are investigated for two repulsively coupled Lorenz oscillators. Based on numerical simulations, it is shown that as constraint strengths on the amplitude-dependent coupling change, an oscillatory state may undergo a transition to an OD state. The parameter regimes of the OD domain are theoretically determined, which coincide well with the numerical results. An electronic circuit is set up to exhibit the transition process to the OD state with an amplitude-dependent coupling. These findings may have practical importance on chaos control and oscillation depression.

We investigate the geometry of the edge of chaos for a nine-dimensional sinusoidal shear flow model and show how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically compute the scaling of the minimum perturbation required to drive the laminar attracting state into the turbulent region. We find this minimum perturbation to scale with the Reynolds number as ${\mathrm{Re}}^{-2}$.

The stimulus-dynamic response is an important topic in physics. In this work, we study the dynamics in the reaction-diffusion system subjected to a weak signal and a spatially periodic force. We find that the response of the system to the weak signal is enhanced largely by the spatially periodic force, which is termed spatially periodic-force-induced resonance. In particular, the response becomes stronger when the spatial frequency is chosen such that the system synchronizes with spatially periodic force. This combinative behavior, i.e., the spatially periodic-force-induced resonance and the spatial-synchronization-enhanced resonance, is of great interest and may shed light on our understanding of the dynamics of nonlinear systems subjected to spatially periodic force in responding to a weak signal.

We investigate finite-size effects at first-order quantum transitions. For this purpose we consider the one-dimensional $q$-state quantum Potts chain, in particular with $q=10$, which undergoes a first-order transition, separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground state. In agreement with the general theory, around the transition the low-energy properties show finite-size scaling with respect to appropriate scaling variables. Their size dependence is particularly sensitive to boundary conditions, which is a specific feature of first-order quantum transitions. Finally, we also discuss the finite-size behavior of the $q$-state Potts model $\left(q\ge 2\right)$ at the first-order transitions driven by a parallel magnetic field, occurring in the ferromagnetic phase.

We study the convex hull of the set of points visited by a two-dimensional random walker of $T$ discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter $L$ and area $A$. While the mean perimeter $\langle L\rangle$ and the mean area $\langle A\rangle$ have been studied before, analytically and numerically, and exact results are known for large $T$ (Brownian motion limit), little is known about the full distributions $P\left(A\right)$ and $P\left(L\right)$. In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities $P\left(A\right)$ and $P\left(L\right)$ are as small as ${10}^{-300}$. We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large $T$, a universal scaling behavior (independent of the details of the jump distributions) as a function of $A/T$ and $L/\sqrt{T}$, respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

Networks can exhibit an abrupt transition in the form of a spontaneous self-organization of a sizable fraction of the population into a giant component of connected members. This behavior has been demonstrated in random graphs under suppressive rules that passively or actively attempt to delay the formation of the giant cluster. We show that suppressive rules are not a necessary condition for a sharp transition at the percolation threshold. Rather, a finite system with aggressive tendency to form a giant cluster may exhibit an instability at the percolation threshold that is relieved through an abrupt and discontinuous transition to the stable branch. We develop the theory for a class of equilibrated networks that produce this behavior and find that the discontinuous jump is especially pronounced in small networks but disappears when the size of the system is infinite.

Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified $q$-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.

The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and it vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed.