Knud Zabrocki

Influence of layer defects on the damping in ferroelectric thin films

A Greens function technique for a modified Ising model in a transverse field is applied, which allows to calculate the damping of the elementary excitations and the phase transition temperature of ferroelectric thin films with structural defects. Based on an analytical expression for the damping function, we analyze its dependence on temperature, film thickness and interaction strength numerically. The results demonstrate that defect layers in ferroelectric thin films, layers with impurities or vacancies as well as layers with dislocations are able to induce a strong increase of the damping due to different exchange interactions within the defect layers. The results are in good agreement with experimental data for thin ferroelectric films with different thickness.

Delay-controlled reactions

Abstract When the entities undergoing a chemical reaction are not available simultaneously, the classical rate equation in the reaction-limited regime, should be extended by including non-Markovian memory effects. We consider the two cases of an external feedback, realized by fixed functions and an internal feedback originated in a self-organized manner by the relevant concentration itself. Whereas in the first case the fixed points are not changed, although the dynamical process is altered, the second case offers a complete new behavior, characterized by the existence of a time persistent solution. As an example we consider a single-species pair annihilation A + A →∅ process combined with a spontaneous creation of particles ∅→ A .

Memory Driven Pattern Formation

The diffusion equation is extended by including spatial-temporal memory in such a manner that the conservation of the concentration is maintained. The additional memory term gives rise to the formation of non-trivial stationary solutions. The steady state pattern in an infinite domain is driven by a competition between conventional particle current and a feedback current. We give a general criteria for the existence of a non-trivial stationary state. The applicability of the model is tested in case of a strongly localized, time independent memory kernel. The resulting evolution equation is exactly solvable in arbitrary dimensions and the analytical solutions are compared with numerical simulations. When the memory term offers a spatially decaying behavior, we find also the exact stationary solution in form of a screened potential.

Spatiotemporal memory in a diffusion–reaction system

We consider a reaction–diffusion process with retardation. The particles, initially immersed in traps, remain inactive until another particle is annihilated spontaneously with a rate λ at a certain point . In that case the traps within a sphere of radius R(t) = vtα around will be activated and a particle is released with a rate μ. Due to the competition between both reactions the system evolves three different time regimes. While in the initial time interval the diffusive process dominates the behaviour of the system, there appears a transient regime. The system shows a travelling wave solution which tends to a non-trivial stationary solution for v → 0. In that regime one observes a very slow decay of the concentration. In the final long time regime a crossover to an exponentially decaying process is observed. In case of λ = μ the concentration is a conserved quantity, whereas for μ > λ the total particle number tends to zero after a finite time. The mean square displacement offers an anomalous diffusive behaviour where the dynamic exponent is determined by the exponent α. In one dimension the model can be solved exactly. The situation could be applied for the development of a bacterial colony, gene pool or chemical signalling.

Relationship between a non-Markovian process and Fokker-Planck equation

Abstract We demonstrate the equivalence of a non-Markovian evolution equation with a linear memory-coupling and a Fokker–Planck equation (FPE). In case the feedback term offers a direct and permanent coupling of the current probability density to an initial distribution, the corresponding FPE offers a non-trivial drift term depending itself on the diffusion parameter. As the consequence the deterministic part of the underlying Langevin equation is likewise determined by the noise strength of the stochastic part. This memory induced stochastic behavior is discussed for different, but representative initial distributions. The analytical calculations are supported by numerical results.

Memory-controlled diffusion

Memory effects require for their incorporation into random-walk models an extension of the conventional equations. The linear Fokker-Planck equation for the probability density p(r,t) is generalized by including nonlinear and nonlocal spatial-temporal memory effects. The realization of the memory kernel is restricted due the conservation of the basic quantity p. A general criteria is given for the existence of stationary solutions. In case the memory kernel depends on p polynomially, transport may be prevented. Owing to the delay effects a finite amount of particles remains localized and the further transport is terminated. For diffusion with nonlinear memory effects we find an exact solution in the long-time limit. Although the mean square displacement exhibits diffusive behavior, higher order cumulants offer differences to diffusion and they depend on the memory strength.

RETARDATION EFFECTS IN A FINITE DIFFUSION-REACTION SYSTEM

The reaction-diffusion process is generalized by including spatiotemporal delay effects. As a first example, we study the influence of a constant production term which is switched off after a finite time. In a second case, all diffusion-reaction processes within a distance R(t) = κtα around a certain spatial point are assumed to contribute to the instantaneous dynamics of the system. There occurs a competition between reaction-diffusion and the accumulation process which leads to a non-trivial stationary state. The evolving concentration profiles are calculated analytically for both a ballistic behavior with α = 1 and a diffusion-like transport with α = 1/2. Because the spatiotemporal delay breaks the reflection symmetry, the profiles reveal an anisotropic behavior. The exact solution in one dimension is supported by numerical simulations.

Relationship Between Non-Markovian- and Drift-Fokker-Planck Equation

Based on the stochastic description of transport phenomena the relationship between a non-Markovian evolution equation and the Fokker-Planck equation with drift is investigated. Memory is included by direct coupling between initial and current values of probability density. We present the result for three different initial distributions.

Feedback Coupling and Chemical Reactions

When the entities undergoing a chemical reaction are not available simultaneously, the classical rate equation in the reaction‐limited regime, should be extended by including non‐Markovian memory effects. We consider the two cases of an external feedback, realized by fixed functions and an internal feedback originated in a self‐organized manner by the relevant concentration itself. Whereas in the first case the fixed points are not changed, although the dynamical process is altered, the second case offers a complete new behavior, characterized by the existence of a time persistent solution. As an example we consider a single‐species pair annihilation A + A → O process combined with a spontaneous creation of particles O → A.

Memory in diffusive systems

The classical rate equations for the concentration p(x,t) or the probability density in the diffusion-limited regime are extended by including non-Markovian terms. We present analytical and numerical results for a whole class of evolution models with conserved p, where the underlying equations are of convolution type with temporally and spatially varying memory kernels. Based on our recent studies in the reaction-limited case with memory, we study now the influence of time and spatial couplings. Due to the balance between the conventional diffusive current and the additional force, originated by the feedback, the system exhibits a non-trivial stationary solution which depends on both the initial distribution and the memory strength. For a non-linear memory kernel of KPZ-type we get an asymptotic exact solution. Although the mean square displacement offers ultimately diffusion, the distribution function is determined by the memory strength, too. Differences to diffusion are observed in higher order cumulants. For an arbitrary memory kernel we find a criteria which enables us to get a non-trivial stationary solution.