Polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and nontrivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allows to take into account the specific features of polynomial chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, and finally a chemical oscillator.

Quintic derivative nonlinear Schrödinger equations arise in various physical contexts, notably in the study of hydrodynamic wave packets and media with negative refractive index. A procedure to isolate propagating wave patterns in such nonlinear Schrödinger equations is proposed which is based on two integrals of motion. As an illustration of the method, a “gray” solitary pulse, a “dark” localized mode with nonzero minimum in intensity on a continuous-wave background is identified.

The formal structure of glass singularities in the mode-coupling theory (MCT) of supercooled liquids dynamics is closely related to that appearing in the analysis of heterogeneous bootstrap percolation on Bethe lattices, random graphs, and complex networks. Starting from this observation one can build up microscopic on-lattice realizations of schematic MCT based on cooperative facilitated spin mixtures. I discuss a microscopic implementation of the ${\mathsf{F}}_{13}$ schematic model including multiple glassy states and the glass-glass transition. Results suggest that our approach is flexible enough to bridge alternative theoretical descriptions of glassy matter based on the notions of quenched disorder and dynamic facilitation.

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

We introduce a family of rock-paper-scissors-type models with $Z_N$ symmetry ($N$ is the number of species), and we show that it has a very rich structure with many completely different phases. We study realizations that lead to the formation of domains, where individuals of one or more species coexist, separated by interfaces whose (average) dynamics is curvature driven. This type of behavior, which might be relevant for the development of biological complexity, leads to an interface network evolution and pattern formation similar to the ones of several other nonlinear systems in condensed matter and cosmology.

Two-phase flow through heterogeneous media leads to scale-free distributions of irregularly shaped pockets of one fluid trapped within the other. Although reactions within these fluids are often modeled at the homogeneous continuum scale, there exists no current framework for upscaling from the pore scale that accounts for the complex and scale-free geometry of the bubbles. In this paper, we apply a linear-kinetics reaction-diffusion model to characterize the steady-state chemical environment inside the irregular pockets. Using a combination of theory and invasion-percolation simulations, we derive scaling laws describing the distribution of diffusion times within bubbles. We show that chemical concentrations within the bubbles are determined by the Laplace transform of the entire distribution of diffusion times from each location. This serves as a means to compute average concentrations of reactant within a bubble of unique geometry and size. Furthermore, the overall system size imposes upper bounds on the distribution of bubble sizes, thereby imposing a system-size dependence on the statistics and average concentrations. These conclusions have profound implications for continuum models of porous reactive flow, where kinetic and equilibrium parameters are often chosen from laboratory measurements made at centimeter scales.

We study the motion of a ferromagnetic helical nanostructure under the action of a rotating magnetic field. A variety of dynamical configurations were observed that depended strongly on the direction of magnetization and the geometrical parameters, which were also confirmed by a theoretical model, based on the dynamics of a rigid body under Stokes flow. Although motion at low Reynolds numbers is typically deterministic, under certain experimental conditions the nanostructures showed a surprising bistable behavior, such that the dynamics switched randomly between two configurations, possibly induced by thermal fluctuations. The experimental observations and the theoretical results presented in this paper are general enough to be applicable to any system of ellipsoidal symmetry under external force or torque.

A variety of chemical and biological nonlinear excitable media, including heart tissue, exhibit vortices (spiral waves) that can anchor to nonexcitable obstacles. Such anchored vortices can be terminated by the application of high-frequency wave trains, as shown previously in isotropic excitable media. In this study, we examined the basic dependencies of the conduction velocities of planar waves and waves around curved obstacles as a function of anisotropy through numerical simulations of excitable media that mimic the fiber orientation in a real heart. We also investigated the unpinning of anchored spiral waves by high-frequency wave trains in an anisotropic excitable medium. Unlike the findings regarding the termination of spiral waves in isotropic excitable systems, we found a nonmonotonic relationship between the maximum unpinning period and the obstacle radius depending on the fiber orientation, where the formation of unwanted secondary pinned vortices or chaotic waves is seen over a wide range of parameters.

Cross sections (CSs) for the interaction of low-energy electrons (LEE) with condensed macromolecules are essential parameters for accurate modeling of radiation-induced molecular decomposition and chemical synthesis. Electron irradiation of dry nanometer-scale macromolecular solid films has often been employed to measure CSs and other quantitative parameters for LEE interactions. Since such films have thicknesses comparable with electron thermalization distances, energy deposition varies throughout the film. Moreover, charge accumulation occurring inside the films shields a proportion of the macromolecules from electron irradiation. Such effects complicate the quantitative comparison of the CSs obtained in films of different thicknesses and limit the applicability of such measurements. Here, we develop a simple mathematical model, termed the molecular survival model, that employs a CS for a particular damage process together with an attenuation length related to the total CS, to investigate how a measured CS might be expected to vary with experimental conditions. As a case study, we measure the absolute CS for the formation of DNA strand breaks (SBs) by electron irradiation at 10 and 100 eV of lyophilized plasmid DNA films with thicknesses between 10 and 30 nm. The measurements are shown to depend strongly on the thickness and charging condition of the nanometer-scale films. Such behaviors are in accord with the model and support its validity. Via this analysis, the CS obtained for SB damage is nearly independent of film thickness and charging effects. In principle, this model can be adapted to provide absolute CSs for electron-induced damage or reactions occurring in other molecular solids across a wider range of experimental conditions.

The translocation of a bond fluctuation polymer through an interacting nanopore is studied using dynamic Monte Carlo simulation. A driving force $F$ is applied only for monomers inside the pore. The influence of polymer-pore interaction on the scaling relation $\tau \sim N^{\alpha }$ is studied for both unbiased and biased translocations, with $\tau$ the translocation time and $N$ the polymer length. Results show that the exponent $\alpha$ is dependent on the polymer-pore interaction. For a noninteracting pore, we find $\alpha =2.48$ for unbiased translocation and $\alpha =1.35$ for strong biased translocation; for strong attraction, we find $\alpha =2.35$ for unbiased translocation and $\alpha =1.22$ for strong biased translocation. The unbiased translocation corresponds to the low-$NF$ regime whereas the strong biased translocation corresponds to the high-$NF$ regime.

A numerical analysis of mechanical frictionless contact between rough self-affine elastic manifolds was carried out. It is shown that the lower cutoff wave number in surface spectra is a key parameter controlling the representativity of the numerical model. Using this notion we demonstrate that for representative surfaces the evolution of the real contact area with load is universal and independent of the Hurst roughness exponent. By introducing a universal law containing three constants, we extend the study of this evolution beyond the limit of infinitesimal area fractions.

We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, to characterize, and to simulate networks with a broad range of properties. Any network can be embedded on a surface with sufficiently high genus and therefore the study of topologically embedded graphs is non-restrictive. We show that the local properties of the network are affected by the surface genus which determines the average degree, which influences the degree distribution, and which controls the clustering coefficient. The global properties of the graph are also strongly affected by the surface genus which is constraining the degree of interwovenness, changing the scaling properties of the network from large-world kind (small genus) to small- and ultrasmall-world kind (large genus). Two elementary moves allow the exploration of all networks embeddable on a given surface and naturally introduce a tool to develop a statistical mechanics description for these networks. Within such a framework, we study the properties of topologically embedded graphs which dynamically tend to lower their energy towards a ground state with a given reference degree distribution. We show that the cooling dynamics between high and low “temperatures” is strongly affected by the surface genus with the manifestation of a glass-like transition occurring when the distance from the reference distribution is low. We prove, with examples, that topologically embedded graphs can be built in a way to contain arbitrary complex networks as subgraphs. This method opens a new avenue to build geometrically embedded networks on hyperbolic manifolds.

We study the dynamics of the naming game as an opinion formation model on time-varying social networks. This agent-based model captures the essential features of the agreement dynamics by means of a memory-based negotiation process. Our study focuses on the impact of time-varying properties of the social network of the agents on the naming game dynamics. In particular, we perform a computational exploration of this model using simulations on top of real networks. We investigate the outcomes of the dynamics on two different types of time-varying data: (1) the networks vary on a day-to-day basis and (2) the networks vary within very short intervals of time (20 sec). In the first case, we find that networks with strong community structure hinder the system from reaching global agreement; the evolution of the naming game in these networks maintains clusters of coexisting opinions indefinitely leading to metastability. In the second case, we investigate the evolution of the naming game in perfect synchronization with the time evolution of the underlying social network shedding new light on the traditional emergent properties of the game that differ largely from what has been reported in the existing literature.

We have developed a model for studying the axial propagation of elliptically polarized electromagnetic waves in a spatially random helical media. We start by writing Maxwell equations for a structurally chiral medium whose dielectric permittivities, polar, and helical angles contain both a stochastic contribution and a deterministic one. We write the electromagnetic equations into a Marcuvitz-Schwigner representation to transform them afterward in a simpler expression by using the Oseen transformation. We exhibit that in the Oseen frame the Marcuvitz-Schwigner equations turns out to be a linear vector stochastic system of differential equations with multiplicative noise. Applying to the resulting equation a formalism for treating stochastic differential equations, we find the governing equations for the first moments of the electromagnetic field amplitudes for a general autocorrelation function for the system diffractive indexes, and calculate their corresponding band structure for a particular spectral noise density. We have shown that the average resulting electromagnetic fields exhibit a decaying exponential dependence which stems from by dissipation and the presence of qualitative modifications in the band structure including a considerable widening of the band gap and the existence of new local maxima for the modes without a band gap.

We present a model of bidirectional vesicular transport between two intracellular organelles, which takes into account intermediate stages of transport that occur between vesicular budding from one organelle and subsequent fusion with the other organelle. These are incorporated into the model by associating with each organelle a donor pool of newly budded vesicles and an acceptor pool of transported vesicles ready for fusion. By constructing a system of differential equations that keeps track of the distribution of vesicles and protein concentrations within the various pools and along cytoskeletal tracks, we show how a stable steady state can emerge that consists of organelles that maintain different protein concentrations in spite of the continuous exchange of materials. In particular, exploiting the fact that the surface area of individual vesicles is much smaller than the surface area of organelles, we use an adiabatic approximation to eliminate the vesicular variables. This results in a major simplification of the dynamics and provides a systematic procedure for deriving phenomenological models of cooperative transport.

Pathogen-host relationships are the result of an ongoing coevolutionary race where the immune system of the host attempts to eliminate the pathogen, while the successful pathogen mutates to become invisible for the host's immune system. We here propose a minimal pathogen-host evolution model that takes into account cross immunization and allows for evolution of a spatially heterogeneous immune status of a population of hosts. With only the mutation rate as a determining parameter, the model allows us to produce an evolutionary tree of diseases which is highly branched, but hardly ever splits into separate long-lived trunks. Side branches remain short lived and seldom diverge to the extent of losing all cross immunizations.

The role of diffraction is investigated for two-dimensional area-preserving maps with sharply or almost sharply divided phase space, in relation to the issue of dynamical tunneling. The diffraction effect is known to appear in general when the system contains indifferentiable or discontinuous points. We find that it controls the quantum transition between regular and chaotic regions in mixed phase space in the case where the border between these regions is set to be sharp. However, its manifestation is rather subtle: it would be possible to identify the diffraction effect under suitable coordinates if the support of the wave function contains indifferentiable or discontinuous points, whereas it is mixed with the tunneling effect and the whole process becomes hybrid if the support does not contain the sources of diffraction. We make detailed analyses, including the semiclassical treatment of edge contributions of the one-step propagator, to clarify the nature of diffraction in mixed phase space. Our result implies that chaos does not play any roles in the regular-to-chaotic transition process when the phase space is sharply divided.

The decay of the plasma grating formed at the intersection of two femtosecond filaments is measured in several molecular and atomic gases. The grating evolution is ruled by ambipolar diffusion in atomic gases and by a combination of ambipolar diffusion and collision-assisted free electron recombination in molecular gases. Electron diffusion and recombination coefficients are extracted for Ne, Ar, Kr, Xe, N${}_2$, O${}_2$, CO${}_2$, and air at 1 bar.

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder–network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.

We extend the definition of nonadiabatic entropy production given for Markovian systems by Esposito and Van den Broeck [Phys. Rev. Lett. 104, 090601 (2010)], to arbitrary non-Markov ergodic dynamics. We also introduce a notion of stability characterizing non-Markovianity. For stable non-Markovian systems, the nonadiabatic entropy production satisfies an integral fluctuation theorem, leading to the second law of thermodynamics for transitions between nonequilibrium steady states. This quantity can also be written as a sum of products of generalized fluxes and forces, thus being suitable for thermodynamics. On the other hand, the generalized fluctuation-dissipation relation also holds, clarifying that the conditions for it to be satisfied are ergodicity and stability instead of Markovianity. We show that in spite of being counterintuitive, the stability criterion introduced in this work may be violated in non-Markovian systems even if they are ergodic, leading to a violation of the fluctuation theorem and the generalized fluctuation-dissipation relation. Stability represents then a necessary condition for the above properties to hold and explains why the generalized fluctuation-dissipation relation has remained elusive in the study of non-Markov systems exhibiting nonequilibrium steady states.