In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): $U+2V\stackrel{\lambda }{\to }3V$ and $V\stackrel{\mu }{\to }P$, $U\stackrel{\nu }{\to }Q$, with a constant feed rate for $U$. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model, which now involves two composite fields and also intrinsic noise terms. One of the composites is ${\psi }_1={\varphi }_v^2$, where ${\langle {\varphi }_v\rangle }_{\eta }=v$ is the concentration of the species $V$ and the averaging is over the internal noise ${\eta }_{u,v,{\psi }_1}$. The second composite field is the product of three fields $\chi =\lambda {\varphi }_u{\varphi }_v^2$ and requires a noise source to ensure probability conservation. A third composite ${\psi }_2={\varphi }_u{\varphi }_v$ can also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero-energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields ${\varphi }_u,{\varphi }_v,\chi$ that induce higher-order processes such as $n\to n$ scattering for $n>3$. The amplitude of the noise acting on ${\varphi }_v$ is itself stochastic in nature.

Resonance-assisted tunneling is investigated within the framework of one-dimensional integrable systems. We present a systematic recipe, based on Hamiltonian normal forms, to construct one-dimensional integrable models that exhibit resonance island chain structures with accurately controlled sizes and positions of the islands. Using complex classical trajectories that evolve along suitably defined paths in the complex time domain, we construct a semiclassical theory of the resonance-assisted tunneling process. This semiclassical approach yields a compact analytical expression for tunnelling-induced level splittings which is found to be in very good agreement with the exact splittings obtained through numerical diagonalization.

A continuation method developed from a three-dimensional spectral finite element code is used to study natural convection in a tilted rectangular cavity. The cavity has its length equal to two times the side of its square cross section and it contains a fluid with a Prandtl number $\mathrm{Pr}=1$. A detailed bifurcation diagram is first obtained in the case without inclination in order to get the sequence of the different branches of solutions and determine the stable solutions. The focus is then put on the stable solutions in the inclined cavity, when the tilt occurs around its longest axis. The subtle changes induced by the tilt on the convective system are clarified. Three different stable solutions are obtained: the longitudinal roll $L-$ solution (with the same sense of rotation as the inclination angle), which develops smoothly from zero Rayleigh number on the leading branch; the longitudinal roll $L+$ solution (with a sense of rotation opposite to the inclination angle), which is on a disconnected branch and is stabilized beyond a secondary bifurcation point; the oblique roll $O\pm$ solutions (corresponding to transverse roll solutions perturbed by the longitudinal flow induced by the tilt), which quickly appear beyond saddle-node points on new disconnected branches. The domain of existence of these stable solutions is eventually obtained and described in the Rayleigh number-inclination parameter space. Finally, the Nusselt number is determined as a function of the inclination at a constant Rayleigh number for the different stable solutions. The Nusselt number is maximum at an inclination of $49.{55}^{\circ }$ for the leading longitudinal roll $L-$ solution.

According to dielectric formalism, the energy loss of the heavy ion depends on its velocity and its charge density. Also, it depends on the target through its dielectric function; here the random phase approximation is used because it correctly describes fully ionized plasmas at any degeneracy. On the other hand, the Brandt-Kitagawa (BK) model is employed to depict the projectile charge space distribution, and the stripping criterion of Kreussler et al. is used to determine its mean charge state $\langle Q\rangle$. This latter criterion implies that the mean charge state depends on the electron density and temperature of the plasma. Also, the initial charge state of the heavy ion is crucial for calculating $\langle Q\rangle$ inside the plasma. Comparing our models and estimations with experimental data, a very good agreement is found. It is noticed that the energy loss in plasmas is higher than that in the same cold gas cases, confirming the well-known enhanced plasma stopping (EPS). In this case, EPS is only due to the increase in projectile effective charge $Q_{\mathrm{eff}}$, which is obtained as the ratio between the energy loss of each heavy ion and that of the proton in the same plasma conditions. The ratio between the effective charges in plasmas and in cold gases is higher than 1, but it is not as high as thought in the past. Finally, another significant issue is that the calculated effective charge in plasmas $Q_{\mathrm{eff}}$ is greater than the mean charge state $\langle Q\rangle$, which is due to the incorporation of the BK charge distribution. When estimations are performed without this distribution, they do not fit well with experimental data.

We present a simple calculation of the Lorentz transformation of the spectral distribution of blackbody radiation at temperature $T$. Here we emphasize that $T$ is the temperature in the blackbody rest frame and does not change. We thus avoid the confused and confusing question of how temperature transforms. We show by explicit calculation that at zero temperature the spectral distribution is invariant. At finite temperature we find the well-known result familiar in discussions of the 2.7 K cosmic radiation.

Delay-coupled networks are investigated with nonidentical delay times and the effects of such heterogeneity on the emergent dynamics of complex systems are characterized. A simple decomposition method is presented that decouples the dynamics of the network into node-size modal equations in the vicinity of equilibria. The resulting independent components contain distributed delays that map the spatiotemporal complexity of the system to the time domain. We demonstrate that this approach can be used to reveal physical phenomena in heterogenous vehicular traffic when vehicles are linked via vehicle-to-vehicle communication.

Dynamical quorum sensing, the population based phenomenon, is believed to occur when the elements of a system interact via dynamic coupling. In the present work, we demonstrate an alternate scenario, involving static coupling, that could also lead to quorum sensing behavior. These static and dynamic coupling terms have already been employed by Konishi [Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2781 (2007)]. In our context, the coupling is defined as static or dynamic, on the basis of the relative time scales at which the surrounding dynamics and the elements' dynamics evolve. According to this, if the variation in the surrounding dynamics happens on a much larger (fast) time scale than that at which the elements' dynamics are varying (such as seconds and $\mu \text{s}$), then the coupling is considered to be static, otherwise it is considered to be dynamic. A series of experiments have been performed starting from a system of three Chua's circuits to a system of 20 Chua's circuits to study two types of quorum transitions: the emergence and the extinction of global oscillations (period-1). The numerics involving up to 100 Chua's circuits validate the experimental observations.

Up to now, there have been a great number of studies that demonstrate the effect of spatial topology on the promotion of cooperation dynamics (namely, the so-called “spatial reciprocity”). However, most researchers probably attribute it to the positive assortment of strategies supported by spatial arrangement. In this paper, we analyze the time course of cooperation evolution under different evolution rules. Interestingly, a typical evolution process can be divided into two evident periods: the enduring (END) period and the expanding (EXP) period where the former features that cooperators try to endure defectors’ invasion and the latter shows that perfect $C$ clusters fast expand their area. We find that the final cooperation level relies on two key factors: the formation of the perfect $C$ cluster at the end of the END period and the expanding fashion of the perfect $C$ cluster during the EXP period. For deterministic rule, the smooth expansion of $C$ cluster boundaries enables cooperators to reach a dominant state, whereas, the rough boundaries for stochastic rule cannot provide a sufficient beneficial environment for the evolution of cooperation. Moreover, we show that expansion of the perfect $C$ cluster is closely related to the cluster coefficient of interaction topology. To some extent, we present a viable method for understanding the spatial reciprocity mechanism in nature and hope that it will inspire further studies to resolve social dilemmas.

Using a method for stochastic data analysis borrowed from statistical physics, we analyze synthetic data from a Markov chain model that reproduces measurements of wind speed and power production in a wind park in Portugal. We show that our analysis retrieves indeed the power performance curve, which yields the relationship between wind speed and power production, and we discuss how this procedure can be extended for extracting unknown functional relationships between pairs of physical variables in general. We also show how specific features, such as the rated speed of the wind turbine or the descriptive wind speed statistics, can be related to the equations describing the evolution of power production and wind speed at single wind turbines.

Nonthermal Brownian motion is investigated experimentally by injecting a particle into soft-mode turbulence (SMT), in the electroconvection of a nematic liquid crystal. It is clarified that the particle motion can be classified into two phases: fast motion, where particles move with the local convective flow, and slow motion, where they are carried by global slow pattern dynamics. We propose a simplified model to clarify the mechanism of the short-time and asymptotic behavior of diffusion. In our model, the correlation time is estimated as a function of a control parameter $\varepsilon$. The scaling of the SMT pattern correlation time, ${\tau }_d\sim {\varepsilon }^{-1}$, is estimated from the particle dynamics, which is consistent with a previous report observed from the Eulerian viewpoint. The origin of the non-Gaussian distribution of the displacement in the short-time regime is also discussed and an analytical curve is introduced that quantitatively agrees with the experimental data. Our results clearly illustrate the characteristics of diffusive motion in SMT, which are considerably different from the conventional Brownian motion.

The structure of colloids with competing interactions which are confined in a harmonic external trap potential is analyzed numerically by energy minimization in two spatial dimensions. A wealth of different cluster structures is found to be stable including clusters with a fringed outer rim (reminiscent to an ornamental border), clusters perforated with voids, as well as clusters with a crystalline core and a disordered rim. All cluster structures occur in a two-dimensional parameter space. The structural ordering can therefore be efficiently tuned by changing few parameters only providing access to a controlled fabrication of colloidal clusters.

This paper presents a two-dimensional numerical investigation of deformation and motion of erythrocytes in stenotic microvessels using the immersed boundary-fictitious domain method. The erythrocytes were modeled as biconcave-shaped closed membranes filled with cytoplasm. We studied the biophysical characteristics of human erythrocytes traversing constricted microchannels with the narrowest cross-sectional diameter as small as 3 $\mu$m. The effects of essential parameters, namely, stenosis severity, shape of the erythrocytes, and erythrocyte membrane stiffness, were simulated and analyzed in this study. Moreover, simulations were performed to discuss conditions associated with the shape transitions of the cells along with the relative effects of radial position and initial orientation of erythrocytes, membrane stiffness, and plasma environments. The simulation results were compared with existing experiment findings whenever possible, and the physical insights obtained were discussed. The proposed model successfully simulated rheological behaviors of erythrocytes in microscale flow and thus is applicable to a large class of problems involving fluid flow with complex geometry and fluid-cell interactions. Our study would be helpful for further understanding of pathology of malaria and some other blood disorders.

We examine electron transfer between two quantum states in the presence of a dissipative environment represented as a set of independent harmonic oscillators. For this simple model, the Marcus transfer rates can be derived from the equations of motion for electronic operators and we show that these rates are associated to an explicit expression for the environment correlation time. We demonstrate that as a manifestation of the Goldilocks principle, the optimal transfer is governed by a single parameter which is equal to just the inverse square root of 2.

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincaré recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.

Reactivity boundaries that divide the origin and destination of trajectories are of crucial importance to reveal the mechanism of reactions, which was recently found to exist robustly even at high energies for index 1 saddles [Phys. Rev. Lett. 105, 048304 (2010)]. Here we revisit the concept of the reactivity boundary and propose a more general definition that can involve a single reaction associated with a bottleneck composed of higher-index saddles and/or several saddle points with different indices, where the normal form theory, based on expansion around a single stationary point, does not work. We numerically demonstrate the reactivity boundary by using a reduced model system of the ${{\mathrm{H}}_5}^{+}$ cation where the proton exchange reaction takes place through a bottleneck composed of two index 2 saddle points and two index 1 saddle points. The cross section of the reactivity boundary in the reactant region of the phase space reveals which initial conditions are effective in making the reaction happen and thus sheds light on the reaction mechanism.

We report a theoretical investigation of the dust lattice (DL) mode in two-dimensional Yukawa crystals in the presence of asymmetric ion flow and an external magnetic field perpendicular to the crystal plane. Two mutually perpendicular modes are found to be coupled due to Lorentz force. Interaction among the dust grains along the vertical direction of ion flow is strongly affected due to the formation of an ion wake. This causes anisotropy in interaction strength along two mutually perpendicular directions. Both hybrid modes are studied as characteristics of different ion flow speeds and magnetic field strengths. The study shows a fluctuation in DL mode frequency driven by the strength of the particle-wake interaction. The effect of ion flow on polarization of the hybrid wave amplitudes is discussed in detail. Results show a possible mechanism of anomalous phase transition in dusty plasma.

Track structure Monte Carlo simulations are frequently applied in micro- and nanodosimetry to calculate the radiation transport in detail. The use of a well-validated set of cross section data in such simulation codes ensures accurate calculations of transport parameters, such as ionization yields. These cross section data are, however, scarce and often discrepant when measured by different groups. This work surveys literature data on ionization and charge-transfer cross sections of nitrogen, methane, and propane for electrons, protons, and helium particles, focusing on the energy range between 100 keV and 20 MeV. Based on the evaluated data, different models for the parametrization of the cross section data are implemented in the code ptra, developed for simulating proton and alpha particle transport in an ion-counting nanodosimeter. The suitability of the cross section data is investigated by comparing the calculated mean ionization cluster size and energy loss with experimental results in either nitrogen or propane. For protons, generally good agreement between measured and simulated data is found when the Rudd model is used in ptra. For alpha particles, however, a considerable influence of different parametrizations of cross sections for ionization and charge transfer is observed. The ptra code using the charge-transfer data is, nevertheless, successfully benchmarked by the experimental data for the calculation of nanodosimetric quantities, but remaining discrepancies still have to be further investigated (up to 13% lower energy loss and 19% lower mean ionization cluster size than in the experiment). A continuation of this work should investigate data for the energy loss per interaction as well as differential cross section data of nitrogen and propane. Interpolation models for ionization and charge-transfer data are proposed. The Barkas model, frequently used for a determination of the effective charge in the ionization cross section, significantly underestimates both the energy loss (by up to 19%) and the mean ionization cluster size (up to 65%) for alpha particles. It is, therefore, not recommended for particle-track simulations.

We studied the influence of fluorescent polystyrene beads on both entangled and cross-linked actin networks. Thermal bead fluctuations were observed via video particle tracking and analyzed with one-point microrheology. Illumination of fluorescent beads with their appropriate excitation wavelength leads to a drastic softening of actin gels. Other wavelengths and bright field microscopy do not increase thermal bead fluctuations. This effect cannot be significantly reduced by adding common oxygen scavengers. We conclude that the usage of fluorescent beads impairs results when studying the microrheology of actin networks.

We show, using differential dynamic microscopy, that the diffusivity of nonmotile cells in a three-dimensional (3D) population of motile E. coli is enhanced by an amount proportional to the active cell flux. While nonmotile mutants without flagella and mutants with paralyzed flagella have quite different thermal diffusivities and therefore hydrodynamic radii, their diffusivities are enhanced to the same extent by swimmers in the regime of cell densities explored here. Integrating the advective motion of nonswimmers caused by swimmers with finite persistence-length trajectories predicts our observations to within 2%, indicating that fluid entrainment is not relevant for diffusion enhancement in 3D.