We study the occurrence of vibrational resonance as well as the underlying mechanism in excitable systems. The single vibration resonance and vibration bi-resonance are observed when tuning the amplitude and frequency of high-frequency force simultaneously. Furthermore, by virtue of the phase diagram of low-frequency-signal-free FitzHugh-Nagumo model, it is found that each maxima of response measure is located exactly at the transition boundary of phase patterns. Therefore, it is the transition between different phase-locking modes that induces vibrational resonance in the excitable systems. Finally, this mechanism is verified in the Hodgkin-Huxley neural model. Our results provide insights into the transmission of weak signals in nonlinear systems, which are valuable in engineering for potential applications.

We present experimental results on the study of spatiotemporally localized surface wave events on deep water that can be modeled using the Peregrine breather solution of the nonlinear Schrödinger equation. These are often considered as prototypes of oceanic rogue waves that can focus wave energy into a single wave packet. For small steepness values of the carrier gravity waves the Peregrine breathers are relatively wide, thus providing an excellent agreement between the theory and experimental results. For larger steepnesses the focusing leads to temporally and spatially shorter events. Nevertheless, agreement between measurements and the Peregrine breather theory remains reasonably good, with discrepancies of modulation gradients and spatiotemporal symmetries being tolerable. Lifetimes and travel distances of the spatiotemporally localized wave events determined from the experiment are in good agreement with the theory.

This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width $H_b$ for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, $A_c$, has a minimum for a beam width $H_b\approx 0.32$. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [$A_x=\text{(length/height)}=10$]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter $A$ on the stability threshold is investigated for various values of the beam width $H_b$ and different Prandtl numbers $\text{Pr}$. It is shown that, for not too small values of the Prandtl number ($\text{Pr}>{\text{Pr}}_l$), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter $A$. The critical curves display two behaviors. For small or moderate values of $A$, the critical Rayleigh number ${\text{Ra}}_c$ increases with $A$ up to a maximum. Then, when $A$ is further increased, ${\text{Ra}}_c$ undergoes a decrease and eventually goes to 0 at $A=A_c$, i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value ${\text{Pr}}_l$ (which depends on $H_b$), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.

The fundamental assumptions of the adiabatic theory do not apply in the presence of sharp field gradients or in the presence of well-developed magnetohydrodynamic turbulence. For this reason, in such conditions the magnetic moment $\mu$ is no longer expected to be constant. This can influence particle acceleration and have considerable implications in many astrophysical problems. Starting with the resonant interaction between ions and a single parallel propagating electromagnetic wave, we derive expressions for the magnetic moment trapping width $\Delta \mu$ (defined as the half peak-to-peak difference in the particle magnetic moments) and the bounce frequency ${\omega }_b$. We perform test-particle simulations to investigate magnetic moment behavior when resonance overlapping occurs and during the interaction of a ring-beam particle distribution with a broadband slab spectrum. We find that the changes of magnetic moment and changes of pitch angle are related when the level of magnetic fluctuations is low, $\delta B/B_0=({10}^{-3},{10}^{-2})$, where $B_0$ is the constant and uniform background magnetic field. Stochasticity arises for intermediate fluctuation values and its effect on pitch angle is the isotropization of the distribution function $f\left(\alpha \right)$. This is a transient regime during which magnetic moment distribution $f\left(\mu \right)$ exhibits a characteristic one-sided long tail and starts to be influenced by the onset of spatial parallel diffusion, i.e., the variance $\langle {\left(\Delta z\right)}^2\rangle$ grows linearly in time as in normal diffusion. With strong fluctuations $f\left(\alpha \right)$ becomes completely isotropic, spatial diffusion sets in, and the $f\left(\mu \right)$ behavior is closely related to the sampling of the varying magnetic field associated with that spatial diffusion.

We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest $O$(10) bound-state energies and wave functions to machine precision. The method extends also to Hamiltonians that are neither Hermitian nor $\mathcal{PT}$ symmetric and thus allows one to investigate whether or not the spectra in such cases are still real. Furthermore, the approach is especially useful for problems in which a position-dependent mass is adopted, for example in effective-mass models in solid-state physics or in the approximate treatment of coupled nuclear motion in molecular physics or quantum chemistry. The performance of the algorithm is demonstrated by considering the inversion motion of different isotopes of ammonia molecules within a position-dependent mass model and some other examples of one- and two-dimensional Hamiltonians that allow for the comparison to analytical or numerical results in the literature.

Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.

We present a method that enables the direct simulation of Donnan equilibria. The method is based on a grand-canonical Monte Carlo scheme that properly accounts for the unequal partitioning of small ions on the two sides of a semipermeable membrane, and can be used to determine the Donnan electrochemical potential, osmotic pressure, and other system properties. Positive and negative ions are considered separately in the grand-canonical moves. This violates instantaneous charge neutrality, which is usually considered a prerequisite for simulations using the Ewald sum to compute the long-range charge-charge interactions. In this work, we show that if the system is neutral only in an average sense, it is still possible to get reliable results in grand-canonical simulations of electrolytes performed with Ewald summation of electrostatic interactions. We compare our Donnan method with a theory that accounts for differential partitioning of the salt, and find excellent agreement for the electrochemical potential, the osmotic pressure, and the salt concentrations on the two sides. We also compare our method with experimental results for a system of charged colloids confined by a semipermeable membrane and to a constant-$NVT$ simulation method, which does not account for salt partitioning. Our results for the Donnan potential are much closer to the experimental results than the constant-$NVT$ method, highlighting the important effect of salt partitioning on the Donnan potential.

Techniques are proposed for solving integral equations of the first kind with an input known not precisely. The requirement that the solution sought for includes a given number of maxima and minima is imposed. It is shown that when the deviation of the approximate input from the true one is sufficiently small and some additional conditions are fulfilled the method leads to an approximate solution that is necessarily close to the true solution. No regularization is required in the present approach. Requirements on features of the solution at integration limits are also imposed. The problem is treated with the help of an ansatz proposed for the derivative of the solution. The ansatz is the most general one compatible with the above mentioned requirements. The techniques are tested with exactly solvable examples. Inversions of the Lorentz, Stieltjes, and Laplace integral transforms are performed, and very satisfactory results are obtained. The method is useful, in particular, for the calculation of quantum-mechanical reaction amplitudes and inclusive spectra of perturbation-induced reactions in the framework of the integral transform approach.

A general approach for investigating transport phenomena in porous media is presented. This approach has the capacity to represent various kinds of irregularity in porous media without the need for excessive detail or computational effort. The overall method combines a generalized effective medium approximation (EMA) with a macroscopic continuum model in order to derive a transport equation with explicit analytical expressions for the transport coefficients. The proposed form of the EMA is an anisotropic and heterogeneous extension of Kirkpatrick's EMA [Rev. Mod. Phys. 45, 574 (1973)] which allows the overall model to account for microscopic alterations in connectivity (with the locations of the pores and the orientation and length of the throat) as well as macroscopic variations in transport properties. A comparison to numerical results for randomly generated networks with different properties is given, indicating the potential for this methodology to handle cases that would pose significant difficulties to many other analytical models.

In this paper we introduce a conditional imitation rule into an evolutionary game, in which the imitation probabilities of individuals are determined by a function of payoff difference and two crucial parameters $\mu$ and $\sigma$. The parameter $\mu$ characterizes the most adequate goal for individuals and the parameter $\sigma$ characterizes the tolerance of individuals. By using the pair approximation method and numerical simulations, we find an anomalous cooperation enhancement in which the cooperation level shows a nonmonotonic variation with the increase of temptation. The parameter $\mu$ affects the regime of the payoff parameter which supports the anomalous cooperation enhancement, whereas the parameter $\sigma$ plays a decisive role on the appearance of the nonmonotonic variation of the cooperation level. Furthermore, to give explicit implications for the parameters $\mu$ and $\sigma$ we present an alterative form of the conditional imitation rule based on the benefit and the cost incurred to individuals during strategy updates. In this way, we also provide a phenomenological interpretation for the nonmonotonic behavior of cooperation with the increase of temptation. The results give a clue that a higher cooperation level could be obtained under adverse environments for cooperation by applying the conditional imitation rule, which is possible to be manipulated in real life. More generally, the results in this work might point out an efficient way to maintain cooperation in the risky environments to cooperators.

We analyze the chemical and jump surface diffusion coefficients, $D_c$ and $D_J$, near a first-order phase transition at which two phases coexist and the surface coverage, $\theta$, jumps between single-phase values ${\theta }_{-}^{*}$ and ${\theta }_{+}^{*}$. Contrary to other studies, we consider temperatures that are sufficiently subcritical. Using the local equilibrium approximation, we obtain approximate analytical formulas for the dependences of $D_c$ and $D_J$ on the coverage and system size, $N$, near such a transition. In the two-phase regime, when $\theta$ ranges between ${\theta }_{-}^{*}$ and ${\theta }_{+}^{*}$, the diffusion coefficients behave as the sums of two hyperbolas, $D_c\approx A_{-}/N|\theta -{\theta }_{-}^{*}|+A_{+}/N|\theta -{\theta }_{+}^{*}|$ and $D_J\approx A_{-}|\theta -{\theta }_{+}^{*}|/\theta +A_{+}|\theta -{\theta }_{-}^{*}|/\theta$. This behavior rapidly changes as the system goes from the two-phase regime to either of the single-phase regimes (when $\theta$ goes below ${\theta }_{-}^{*}$ or above ${\theta }_{+}^{*}$). The crossover behavior of $D_c\left(\theta \right)$ and $D_J\left(\theta \right)$ between the two-phase and single-phase regimes is described by rather complex formulas involving the Lambert function. We consider a lattice-gas model on a triangular lattice to illustrate these general results, applying them to four specific examples of transitions exhibited by the model.

Nonspecific binding of regulatory proteins to DNA can be an important mechanism for target search and storage. This seems to be the case for the lambda repressor protein (CI), which maintains lysogeny after infection of E. coli. CI binds specifically at two distant regions along the viral genome and induces the formation of a repressive DNA loop. However, single-molecule imaging as well as thermodynamic and kinetic measurements of CI-mediated looping show that CI also binds to DNA nonspecifically and that this mode of binding may play an important role in maintaining lysogeny. This paper presents a robust phenomenological approach using a recently developed method based on the partition function, which allows calculation of the number of proteins bound nonspecific to DNA from measurements of the DNA extension as a function of applied force. This approach was used to analyze several cycles of extension and relaxation of λ DNA performed at several CI concentrations to measure the dissociation constant for nonspecific binding of CI ($\sim 100$ nM), and to obtain a measurement of the induced DNA compaction ($\sim 10\%$) by CI.

A wide array of biological processes occur at rates that vary significantly with force. Instantaneous molecular forces fluctuate due to thermal noise and active processes, leading to concomitant fluctuations in biomolecular rate constants. We demonstrate that such fluctuations have a dramatic effect on the transition kinetics of force-dependent processes. As an illustrative, biologically relevant example, we model the pausing of eukaryotic RNA polymerase as it transcribes nucleosomal DNA. Incorporating force fluctuations in the model yields qualitatively different predictions for the pausing time scales when compared to behavior under the average force alone. We use our model to illustrate the broad range of behaviors that can arise in biomolecular processes that are susceptible to force fluctuations. The fluctuation time scale, which varies significantly for in vivo biomolecular processes, yields very different results for overall rates and dramatically alters the force regime of relevance to the transition. Our results emphasize the importance of transient high-force behavior for determining kinetics in the fluctuating environment of a living cell.

Using network models consisting of gap-junction-coupled Wang-Buszaki neurons, we demonstrate that it is possible to obtain not only synchronous activity between neurons but also a variety of constant phase shifts between 0 and $\pi$. We call these phase shifts intermediate stable phase-locked states. These phase shifts can produce a large variety of wavelike activity patterns in one-dimensional chains and two-dimensional arrays of neurons, which can be studied by reducing the system of equations to a phase model. The $2\pi$ periodic coupling functions of these models are characterized by prominent higher order terms in their Fourier expansion, which can be varied by changing model parameters. We study how the relative contribution of the odd and even terms affects what solutions are possible, the basin of attraction of those solutions, and their stability. These models may be applicable to the spinal central pattern generators of the dogfish and also to the developing neocortex of the neonatal rat.

Synchronized oscillations of large numbers of central neurons are believed to be important for a wide variety of cognitive functions, including long-term memory recall and spatial navigation. It is therefore plausible that evolution has optimized the biophysical properties of central neurons in some way for synchronized oscillations to occur. Here, we use computational models to investigate the relationships between the presumably genetically determined parameters of stellate cells in layer II of the entorhinal cortex and the ability of coupled populations of these cells to synchronize their intrinsic oscillations: in particular, we calculate the time it takes circuits of two or three cells with initially randomly distributed phases to synchronize their oscillations to within one action potential width, and the metabolic energy they consume in doing so. For recurrent circuit topologies, we find that parameters giving low intrinsic firing frequencies close to those actually observed are strongly advantageous for both synchronization time and metabolic energy consumption.

In a recent paper, Valdès-Parada and Alvarez-Ramirez [Phys. Rev. E 84, 031201 (2011)] used the technique of volume averaging to derive a “frequency-dependent” dispersion tensor, ${\mathsf{\text{D}}}_{\gamma }^{*}$, the goal of which is to describe solute transport in porous media undergoing periodic processes. We describe two issues related to this dispersion tensor. First, we demonstrate that the definition of ${\mathsf{\text{D}}}_{\gamma }^{*}$ is erroneous and derive a corrected version, ${\mathsf{\text{D}}}_{\gamma }^{*c}$. With this modification, the approach of Valdès-Parada and Alvarez-Ramirez becomes strictly equivalent to the one devised by Moyne [Adv. Water Res. 20, 63 (1997)]. Second, we show that the term “frequency-dependent dispersion” is misleading because ${\mathsf{\text{D}}}_{\gamma }^{*}$ and ${\mathsf{\text{D}}}_{\gamma }^{*c}$ do not depend on the process operating frequency, $\chi$. The study carried out by Valdès-Parada and Alvarez-Ramirez represents a spectral analysis of the relaxation of ${\mathsf{\text{D}}}_{\gamma }^{*}$ towards its steady-state, independent of any periodic operation or excitation.

In a recent paper [Valdés-Parada and Alvarez-Ramirez, Phys. Rev. E 84, 031201 (2011)], we studied the passive transport of a solute in a rigid and homogeneous porous medium in the domain of frequency and obtained good agreement between upscaled and pore-scale results for several values of the frequency. This work was commented on by Davit and Quintard, who wrote about the closure problem in the Laplace domain and provided the correct definition of the dispersion coefficient. However, the obtention of the inverse Laplace transform is not always an easy (or even possible) task to perform. For this reason, in this Reply we show that the same result reached by Davit and Quintard can be obtained using integral equation formulations based on Green's functions without the need of the Laplace transform. In addition, we identify three time stages for the transport process and we point out that the derivations provided by Davit and Quintard are only valid once a pre-asymptotic stage has been reached in which there is a separation of characteristic length scales that allows treating volume averaged quantities as constants in space but not in time.

In recent years the prisoner's dilemma has become a paradigm for the study of the emergence of cooperation in spatially structured populations. Such a structure is usually assumed to be given by a graph. In general, the success of cooperative strategies is associated with the possibility of forming globular clusters, which in turn depends on a feature of the network that is measured by its clustering coefficient. In this work we study the dependence of the success of cooperation on this coefficient for regular networks. Additionally, for both stochastic and deterministic dynamics we show that there is a strong dependence on the initial composition of the population. This hints at the existence of several different mechanisms that could promote or hinder cluster expansion. We have studied in detail some of these mechanisms by concentrating on completely ordered networks (large clustering coefficient) or completely random networks (vanishing clustering coefficient).

We consider a wave-front model for the spread of neolithic culture across Europe, and use Bayesian inference techniques to provide estimates for the parameters within this model, as constrained by radiocarbon data from southern and western Europe. Our wave-front model allows for both an isotropic background spread (incorporating the effects of local geography) and a localized anisotropic spread associated with major waterways. We introduce an innovative numerical scheme to track the wave front, and use Gaussian process emulators to further increase the efficiency of our model, thereby making Markov chain Monte Carlo methods practical. We allow for uncertainty in the fit of our model, and discuss the inferred distribution of the parameter specifying this uncertainty, along with the distributions of the parameters of our wave-front model. We subsequently use predictive distributions, taking account of parameter uncertainty, to identify radiocarbon sites which do not agree well with our model. These sites may warrant further archaeological study or motivate refinements to the model.