On Complex Dynamics and the Schrödinger Equation

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

Because their conformational entropy is a nonlinear function of size, polymer molecules can only solidify under conditions far from equilibrium with the formation of dissipative structures. A dissipative structure is nonhomogeneous, it evolves with time from an initially space‐dependent fluctuation, and has a characteristic size that depends upon the initial perturbation and the conditions that apply during its entire history. The glass transition is envisioned as the formation of the ultimate dissipative structure where all of the latent heat of crystallization is dissipated within the structure. The mechanism of glass formation can be described as the spinodal decomposition of a homogeneous liquid into two interpenetrating networks of high‐ and low‐energy solids with a characteristic distance and a gradient of density and/or composition extending through both high‐ and low‐energy regions.

This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schr\"odinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.

Transition to spatiotemporal chaos via stationary branching shocks and holes

We previously showed that the adult ocellated lizard skin colour pattern is effectively generated by a stochastic cellular automaton (CA) of skin scales. We additionally suggested that the canonical continuous 2D reaction-diffusion (RD) process of colour pattern development is transformed into this discrete CA by reduced diffusion coefficients at the borders of scales (justified by the corresponding thinning of the skin). Here, we use RD numerical simulations in 3D on realistic lizard skin geometries and demonstrate that skin thickness variation on its own is sufficient to cause scale-by-scale coloration and CA dynamics during RD patterning. In addition, we show that this phenomenon is robust to RD model variation. Finally, using dimensionality-reduction approaches on large networks of skin scales, we show that animal growth affects the scale-colour flipping dynamics by causing a substantial decrease of the relative length scale of the labyrinthine colour pattern of the lizard skin.

In this paper, complex Ginzburg-Landau (CGL) equations governed by p-Laplacian are studied. We discuss the global existence of solutions for the initial-boundary value problem of the equation in general domains. The global solvability of the initialboundary value problem for the case when p = 2 is already examined by several authors provided that parameters appearing in CGL equations satisfy a suitable condition. Our approach to CGL equations is based on the theory of parabolic equations with nonmonotone perturbations. By using this method together with some approximate procedure and a diagonal argument, the global solvability is shown without assuming any growth conditions on the nonlinear terms.

A general, the simplest model of a spatial dissipative structure arising in an excitable medium is constructed, containing at least two components interacting with each other with their own mobility. One of these components (active) uses the other component as food. It is shown that such a model leads to a stationary stable spatial distribution of the components in the form of Liesegang bands. As specific examples of the formation of spatial dissipative structures, structures arising in plankton consisting of phytoplankton and zooplankton and in the soil containing the bacterial population and the nutrient substrate are considered. Bifurcation diagrams are constructed in the parameter space, characteristic for each of the considered excitable media, which determine the conditions for the formation of dissipative structures in these media. The existence in the plankton of a strange attractor of a previously unknown shape in four-dimensional phase space has been discovered.

Particle-based Brownian dynamics simulations offer the opportunity to not only simulate diffusion of particles but also the reactions between them. They therefore provide an opportunity to integrate varied biological data into spatially explicit models of biological processes, such as signal transduction or mitosis. However, particle based reaction-diffusion methods often are hampered by the relatively small time step needed for accurate description of the reaction-diffusion framework. Such small time steps often prevent simulation times that are relevant for biological processes. It is therefore of great importance to develop reaction-diffusion methods that tolerate larger time steps while maintaining relatively high accuracy. Here, we provide an algorithm, which detects potential particle collisions prior to a BD-based particle displacement and at the same time rigorously obeys the detailed balance rule of equilibrium reactions. We can show that for reaction-diffusion processes of particles mimicking proteins, the method can increase the typical BD time step by an order of magnitude while maintaining similar accuracy in the reaction diffusion modelling.

This paper is the second of a two-stage exposition, in which we study the nonequilibrium dynamics of the complex Ginzburg-Landau (CGL) equation. We use spiral defects to characterize the system evolution and morphologies. In the first paper of this exposition [S.K. Das, S. Puri, and M.C. Cross, Phys. Rev E 64, 046206 (2001)], we presented analytical results for the correlation function of a single spiral defect, and its short-distance singular behavior. We had also examined the utility of the Gaussian auxiliary field ansatz for characterizing multispiral morphologies. In this paper, we present results from an extensive numerical study of nonequilibrium dynamics in the CGL equation with dimensionality d=2,3. We discuss the behavior of domain growth laws; real-space correlation functions; and momentum-space structure factors. We also compare numerical results for the correlation functions and structure factors with analytical results presented in our first paper.

The structurization of holographic planes in holograms recorded in CaF 2 crystal with color centers was found. The structurization is apparent in the formation of spiral bundles, which pierce the holographic planes. It testifies to self-organization of color centers in these planes. This process is believed to be linked with colloidal centers, 2D metal islets in the crystal lattice, whose formation and decay during hologram recording at temperatures of 150–190 °C may be considered as a dynamic phase transition that facilitates the generation of stable spatially inhomogeneous (dissipative) structures in the form of bundles. The bundles arise during hologram recording process and remain frozen on cooling of the crystal after the process is finished.

The paper aims to examine the suitability of A. S. Balankin brittle fracture criterion in predicting the of large-scale coal and gas outbursts in underground coal mines under compression load. Formation of dissipative structures as cracks and clusters of local microcracking occurs under the influence of the rock pressure. The formation of microdefects in dissipative structures occurs mainly on local anisotropic inclusions. Mechanism of pre-outburst spalling of gas-saturated coal is analyzed. Destruction (primary outburst) occurs when the brittle fracture conditions are met. And also in the case of such a ratio of “coal seam thickness – the size of micro defects aggregations”, which provides reduction in coal strength to a minimum. Further destruction occurs by known scenario.

This paper investigates new optical soliton solutions to the complex Ginzburg-Landau equation in the presence of white noise, a fundamental model in nonlinear optics that describes soliton dynamics. The study focuses on nine distinct forms of self-phase modulation structures, each exhibiting unique nonlinear characteristics and dispersion properties. To derive the soliton solutions, the generalized -expansion approach is employed, which is known for its effectiveness in handling nonlinear differential equations and extracting exact solutions systematically. Through this analytical framework, a variety of soliton profiles are retrieved, demonstrating the influence of nonlinear dispersion and gain-loss terms on soliton propagation. A key observation from the analysis is that the presence of white noise primarily affects the phase component of the solitons, while their amplitude remains intact. This result suggests that the robustness of the soliton amplitude against stochastic perturbations could have significant implications for practical optical communication systems and laser pulse propagation, where maintaining stable intensity profiles is crucial. The obtained results provide valuable insights into the interplay between noise and nonlinear wave dynamics, offering potential applications in fiber-optic communication, mode-locked lasers, and other areas of photonics where controlled soliton evolution is essential.•The paper investigates optical soliton solutions of the complex Ginzburg-Landau equation with white noise, focusing on nine forms of self-phase modulation with varied nonlinear and dispersive properties.•Using the generalized -expansion method, the study derives exact soliton profiles, revealing how nonlinear dispersion and gain-loss mechanisms shape soliton dynamics.•A key finding shows that white noise affects soliton phase without disturbing amplitude, highlighting amplitude robustness with implications for optical communications and laser systems.

Dynamics of defects in the vector complex Ginzburg–Landau equation

Image smoothing and super-resolution are realized by using a reaction diffusion model which is a typical partial differential equation. The new method is based on the theory of self-organization. In the Turing instability at the bistable region diffusion process in space leads to the availability of image smoothing and decides the smoothing effects. After comparing with average filter and median filter it is found that the effects of image smoothing by the reaction diffusion model are better than that by other filters. Super-resolution can also be achieved by the reaction diffusion model in a suitable region of control parameters.

The popular reaction–diffusion model for the negative bias temperature instability is discussed from the viewpoint of stochastic chemical kinetics. We present a microscopic formulation of the reaction–diffusion model based on the reaction–diffusion master equation and solve it using the stochastic simulation algorithm. The calculations are compared to the macroscopic version as well as established experimental data. The degradation predicted by the microscopic reaction–diffusion model strongly deviates from the macroscopic version and the experimentally observed behavior. Those deviations are explained as necessary consequences of the physical processes involved. The presented results show the impact of the unphysical assumptions in the reaction–diffusion model. Further, we generally question the suitability of the mathematical framework of reaction rate equations for a reactive-diffusive system at the given particle densities.