Deterministic Diffusion Research Articles

Understanding deterministic diffusion by correlated random walks

Low-dimensional periodic arrays of scatterers with a moving point particle are ideal models for studying deterministic diffusion. For such systems the diffusion coefficient is typically an irregular function under variation of a control parameter. Here we propose a systematic scheme of how to approximate deterministic diffusion coefficients of this kind in terms of correlated random walks. We apply this approach to two simple examples which are a one-dimensional map on the line and the periodic Lorentz gas. Starting from suitable Green-Kubo formulas we evaluate hierarchies of approximations for their parameter-dependent diffusion coefficients. These approximations converge exactly yielding a straightforward interpretation of the structure of these irregular diffusion coeficients in terms of dynamical correlations.

Suppression and enhancement of diffusion in disordered dynamical systems.

The impact of quenched disorder on deterministic diffusion in chaotic dynamical systems is studied. As a simple example, we consider piecewise linear maps on the line. In computer simulations we find a complex scenario of multiple suppression and enhancement of normal diffusion, under variation of the perturbation strength. These results are explained by a theoretical approximation, showing that the oscillations emerge as a direct consequence of the unperturbed diffusion coefficient, which is known to be a fractal function of a control parameter.

Deterministic diffusion generated by a chaotic map with intrinsic bias

We investigate diffusive motion that results from random walks generated by the deterministic Kim–Kong map. The orbital density is not symmetric in this map and random walks generated by the map have an intrinsic bias. An external bias parameter is adjusted to compensate this intrinsic bias so that the average step length of walkers vanishes. The mean square displacement of diffusing walkers is computed for parameter values in the chaotic window and is found to scale linearly with time for typical parameter values, including the reverse bifurcation point and the end of the chaotic window. For parameter values near the period three cycle, where the orbit alternates intermittently between regular and irregular behaviours, normal diffusion occurs after the elapse of an initial transient time.

Deterministic diffusion in one-dimensional maps—calculation of diffusion constants by harmonic inversion of periodic orbit sums

A method is proposed for the calculation of diffusion constants for one-dimensional maps exhibiting deterministic diffusion. The procedure is based on harmonic inversion and uses a known relation between the diffusion constant and the periodic orbits of a map. The method is tested on an example map for which results calculated by different other techniques are available for comparison.

Radiolysis of Aqueous Solutions with Pulsed Ion Beams. 4. Product Yields for Proton Beams in Solutions of Thiocyanate and Methyl Viologen/Formate

The yields of radicals from water decomposition produced in the radiolysis of two types of aqueous solutions, thiocyanate and methyl viologen (MV2+), were determined using proton pulses of 5.2 MeV energy. Aerated thiocyanate solutions in the concentration range of 0.001−0.75 M gave yields of (SCN)2•-, formed from scavenging OH radicals, that were lower than those for high-energy electrons and higher than those for 4He ions of 21 MeV energy. The (SCN)2•- yield increased with increasing thiocyanate concentration, but the decay of thiocyanate radicals through intratrack reactions appears to be substantial in proton radiolysis. Methyl viologen radical cations (MV•+) formed by scavenging eaq-, H atoms, and OH radicals were measured in deaerated 0.5 mM MV2+ solutions containing formate. The MV•+ yields agreed with the results of steady-state proton beam radiolysis, which confirms earlier results that this system is a suitable chemical dosimeter for ion beam pulse radiolysis. The yields of MV•+ in deaerated MV2+ solutions containing formate and formate/tertiary butanol were used to determine the yields of OH radicals and the sum of the eaq- and H atom yields. Both sets of yields for proton beams were lower than the corresponding ones for high-energy electrons and higher than those for 21 MeV 4He ions. The predicted hydroxyl radical yields for proton beams increase with increasing scavenging capacity and approach the value found for high-energy electron radiolysis. The sum of the eaq- and H atom yields is about 1.8 molecules/100 eV and nearly independent of the scavenging capacity for OH radical. Intratrack reactions were simulated using a deterministic diffusion kinetic model, and the results qualitatively predict the observed yields in the thiocyanate and the MV2+ solutions.

Effects of a periodic perturbation on a discrete-time model of coupled oscillators

We study a model consisting of a unidimensional lattice of coupled circle maps. The coupling constant changes periodically and connects nearest-neighbour sites. The trajectory in phase space oscillates between a repeller and a sink whose stability is exchanged periodically. The system shows deterministic diffusion and spatio-temporal chaos. Within each period of modulation a succession of spatial structures is found whose extent of spatial correlation changes drastically. We study the evolution of the system in terms of its order parameters and other indicators.

A Vortex/Impulse Method for Immersed Boundary Motion in High Reynolds Number Flows

A Lagrangian method that combines vortices and impulse elements (vortex dipoles) is introduced. The applications addressed are flows induced by the motion of thin flexible boundaries immersed in a two-dimensional incompressible fluid. The impulse elements are attached to the boundaries and are used to account for the forces affecting the motion. The vortices occupy a region surrounding the boundaries and are used to account for the viscous effects via a deterministic diffusion method. The convergence of the method is demonstrated numerically. The method is then used to track the motion of an undulating filament, simulating the swimming of an organism in a slightly viscous fluid.

Chaotic transport and current reversal in deterministic ratchets

We address the problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type. We take into account the inertial term in order to understand the role of the chaotic dynamics in the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a periodic regime. Close to this bifurcation, we observed trajectories revealing intermittent chaos and anomalous deterministic diffusion.

Invariant measures for the Musiela equation with deterministic diffusion term

In this article the forward rates equation of the Musiela model is analysed. The equation is studied in the Sobolev spaces \(H^1_\gamma({\Bbb R}^+)\) and \(H^1({\Bbb R}^+)\). Explicit mild solutions and equivalent conditions for the existence and uniqueness of invariant measures are presented.

Relationships among coefficients in deterministic and stochastic transient diffusion.

Systems are studied in which transport is possible due to large extensions with open boundaries in certain directions, but the particles responsible for transport can disappear from it by leaving it in other directions, by chemical reaction or by adsorption. The connection of the total escape rate, the rate of the disappearance, and the diffusion coefficient is investigated. It leads to the observation that the diffusion coefficient defined by <x(2)> is in general different from the one present in the effective Fokker-Planck equation. The result makes it possible to generalize the Gaspard-Nicolis formula [Phys. Rev. Lett. 65, 1693 (1990)] to this transient case in deterministic systems.

Simple deterministic dynamical systems with fractal diffusion coefficients.

We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional array of scatterers with moving point particles. The particles move from one scatterer to the next according to a piecewise linear, expanding, deterministic map on unit intervals. The microscopic chaotic scattering process of the map can be changed by a control parameter. The macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation. Our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter. We provide qualitative and quantitative arguments to explain features of this fractal structure.

Heat Conductivity and Dynamical Instability

We present a series of numerical and analytical computations on heat conduction for a strongly chaotic system\char22{}the Lorentz gas. Heat conduction is characterized by nontrivial features: While the heat conductivity is well defined in the thermodynamic limit, a linear gradient appears only for quite small temperature differences. The key dynamical feature inducing such a behavior is recognized as deterministic diffusion (along transport direction) which is usually associated to full hyperbolicity.

When noise decreases deterministic diffusion

Dynamical noise, acting homogeneously in each time step, can enhance the stability of an unstable fixed point. However, if dynamical noise is added locally in state space, additional clear enhancement can be achieved, if this restriction is chosen properly. A systematic analysis of the influence of local and global dynamical noise on the residence time of an unstable state is presented, and optimal parameters for the stabilizing mechanisms are discussed. As a consequence, it is demonstrated that local dynamical noise can yield increased localization in deterministic diffusion models.

Radiolysis of aqueous solutions with pulsed helium ion beams—2. Yield of SO4− formed by scavenging hydrated electron as a function of S2O2−8 concentration

Yields of SO4− in the radiolysis of aqueous (NH4)2S2O8 solutions with pulsed 24 MeV 4He2+ ion beams were measured in the concentration range from 1×10−4 to 1 mol dm−3. The yield increased with increasing the concentration, and it was smaller than that in electron beam pulse radiolysis in the concentration range measured. The results were compared with the reported Ge−aq derived from various scavengers. Though the yields agree well for low LET radiations, the result of the present research for 4He ion beams is smaller than the yield determined by a product analysis method. This suggests that once formed SO4− disappears by the intra-track reactions and that this effect is more important in ion beam radiolysis because of more dense production of the initial species. Simulation by the deterministic diffusion kinetic model supports it qualitatively.

Theory of slow dynamics in highly charged colloidal suspensions

A systematic theory for the dynamics of charge-stabilized colloidal suspensions of interacting Brownian particles with both Coulomb and hydrodynamic interactions is presented. A nonlinear deterministic diffusion equation for the average local volume fraction of macroions $\ensuremath{\Phi}(x,t)$ and a linear stochastic diffusion equation for the nonequilibrium density fluctuations around \ensuremath{\Phi}, both of which contain an anomalous self-diffusion coefficient ${D}_{S}\ensuremath{\sim}[1\ensuremath{-}\ensuremath{\Phi}(x,t)/{\ensuremath{\varphi}}_{g}{]}^{\ensuremath{\gamma}},$ are derived, where $\ensuremath{\gamma}=1$ here. The glass transition volume fraction ${\ensuremath{\varphi}}_{g}$ is found to be small as ${\ensuremath{\varphi}}_{g}\ensuremath{\sim}{(Zql}_{B}{/a)}^{\ensuremath{-}3}$ for highly charged colloidal suspensions with $Z\ensuremath{\gg}q,$ where Ze is the charge of the macroions, $\ensuremath{-}\mathrm{qe}$ is the charge of counterions, ${l}_{B}$ is the Bjerrum length, and a is the radius of the macroions. The dynamic anomaly of ${D}_{S}(\ensuremath{\Phi}),$ which results from the correlations among macroions and counterions, due to long-range Coulomb interactions, is shown to cause slow dynamical behavior near ${\ensuremath{\varphi}}_{g}.$ This situation is exactly the same as that of the hard sphere suspensions previously discussed by the present author, where $\ensuremath{\gamma}=2.$

On the long-time behaviour of ensembles in a model of deterministic diffusion

We describe a large model system, based on the baker transformation, where deterministic diffusion occurs. The model is similar to one recently considered by Gaspard, and by Hasegawa and Driebe. We point out the close relationship between this system and a simple random walk, and analyse the evolution with time of ensembles in the system using the resolvent-based version of the `subdynamics' formalism developed by Prigogine and his collaborators. We obtain an exact and rigorous description of the long-time behaviour of ensembles, including the irreversible approach to equilibrium, for the case where the system has finite size. We also consider the `thermodynamic' limit where the size of the system becomes infinite, and derive a description of the long-time behaviour in this case, where correlations decay non-exponentially with time.

Slow dynamics of nonequilibrium density fluctuations in concentrated hard-sphere suspensions

The coupled diffusion equations recently proposed for concentrated hard-sphere suspensions of interacting Brownian particles, the nonlinear deterministic diffusion equation with the self-diffusion coefficient DS(Φ(x, t)) for the average local volume fraction Φ(x, t), and the linear stochastic diffusion equation with DS(Φ(x, t)) for the density fluctuations δn(x, t) are numerically solved under a spatially inhomogeneous, nonequilibrium initial state. Thus, in a supercooled region where Φβ ≤ Φ < Φg, the slow evolution of the cluster-like glassy domains with Φ(x, t) ≥ Φg and the slow relaxation of the nonequilibrium density fluctuations are shown to be caused by the dynamic singularity of the self-diffusion coefficient, DS(Φ(x, t)) ∼ (1−Φ(x, t)/Φg)2, where Φ is a particle volume fraction, Φg = (4/3)3/(7 ln 3 − 8 ln 2 + 2) is the colloidal glass transition volume fraction, and Φβ is the crossover volume fraction.

Modelling of linear energy transfer effects on track core processes in the radiolysis of water up to 300°C

Deterministic diffusion–kinetic modelling of the radiolysis of water by low linear energy transfer (LET) radiation in terms of our recently formulated extended spur diffusion model (J. Phys. Chem., 1995, 99, 11464) has been applied to high LET radiation for 20 eV nm-1<LET<70 eV nm-1 up to 300°C for a cylindrical track. The calculations show that an increase in LET accelerates the decay of the radicals e-aq and OH, and the formation of the molecular products H2 and H2O2 and increases the extents of the main track reactions. This results in changes in the chemical yields (g-values), i.e., a decrease in g(e-aq) and g(OH) and an increase in g(H2) and g(H2O2), with increasing LET over the whole temperature range. For a given high LET an increase in temperature results in a marked increase in g(OH) and g(H2), a small increase in g(e-aq) and a slight rise and fall in g(H2O2). These changes are largely accounted for by the different temperature dependences of the various radical–radical reactions. The temperature dependences of these g-values decrease with increasing LET. Reasonable agreement between modelling and experiment allows estimates to be made of the g-values for fast neutron radiolysis of water.

Mixing in reaction dynamics under enhanced diffusion conditions

We first review the Lévy-flight approach for the description of enhanced diffusion and then study the effects of mixing on the transient A + B → 0 reaction at stoichiometric conditions in d = 1−3 dimensions under enhanced diffusion conditions. The particles are assumed to follow Lévy walks, characterized by an index γ, which result in enhanced diffusion for 1 < γ < 2. Particle density decays and particle-particle correlation functions are calculated using deterministic reaction diffusion equations and computer simulations of this multi-particle problem. We concentrate on the segregation phenomenon and demonstrate that, depending on γ, segregation may disappear in d = 3 dimensions under enhanced diffusion conditions. Hence, Lévy walks lead to an efficient mixing of reacting particles.

Diffusive Hydrodynamic Limits for Systems of Interacting Diffusions with Finite Range Random Interaction

We consider a system of interacting diffusive particles with finite range random interaction. The variables can be interpreted as charges at sites indexed by a periodic multidimensional lattice. The equilibrium states of the system are canonical Gibbs measures with finite range random interaction. Under the diffusive scaling of lattice spacing and time, we derive a deterministic nonlinear diffusion equation for the time evolution of the macroscopic charge density. This limit is almost sure with respect to the random environment.