Hopping Dynamics Research Articles

Dynamics of electron hopping in redox polymer electrodes using kinetic Ising model

The analysis of diffusion-migration equations pertaining to electron hopping and physical displacement in redox polymer electrodes is carried out using kinetic Ising model formalism. It is shown that, by the appropriate choice of transition probabilities obeying detailed balancing conditions, a hierarchy of transport equations can be derived. Earlier transport equations due to Nernst-Planck and Saveant are derived as special cases. The dependence of apparent diffusion coefficient on number density of redox centres, polymer morphology etc. are pointed out. Several new insights concerning the microscopic basis underlying the hitherto known phenomenological equations are demonstrated.

Coherent and diffusive features of polaron transport in exciton–phonon systems: A Mori approach

Employing Mori-formalism we calculate the polaron transport of an exciton–phonon system with linear site-occupational coupling. The main feature of our approach is a consistent treatment of the memory function, which is of 2nd order in an effective “smallness parameter” appearing after a Lang–Firsov transformation has been made. The memory function is divided into a quasicoherent part, which can be interpreted as a band type motion, decreasing with increasing temperature, an incoherent part describing usual hopping dynamics and two crossterms. In contrast to previous similar approaches our results do not display any divergence problem. It is found that the two cross terms yield a small transport contribution of coherent nature, at least within the accuracy of the present approximation. Due to this term, although higher order effects may result in a damping, the transport will display coherent features for a long time.

Self-organized criticality in 1D stochastic traffic flow model

Results of computer simulations of a 1D particle hopping model of traffic flow are presented. The model is characterized by parallel update and fully asymmetric stochastic hopping dynamics which allows unbounded series of jumps to empty neighbour sites on the right. The considered case of open boundary conditions can be used to model a “bottleneck” situation in traffic. Evidence for self-organized criticality is found in two aspects: the presence of long-range spatial correlations manifested in the shape of density profiles, and long-time temporal correlations showing up in the low-frequency behaviour of the spectral density of the total particle number and flow. A plausible conjecture is to interpret the observed qualitative changes in these features, as a function of the injection rate and the hopping probability, in terms of a nonequilibrium phase transition between a low-density phase and a maximal current phase. This conjecture is supported by the phase diagram obtained in mean-field approximation.

Dynamics of phason hopping in AlFeCu and AlMnPd quasicrystals

We report the present state of knowledge about phason hopping in quasicrystals (QC) and discuss the results of quasielastic neutron scattering experiments in the systems AlFeCu and AlMnPd.

On the dynamics of Frenkel defect formation and ionic hopping in AgCl, AgBr and β-AgI

Complete conductivity spectra extending from dc to mid-IR frequencies have been taken of crystalline AgCl, AgBr and β-AgI, at various temperatures. The spectra contain two new dynamic features: (i) a thermally activated Debye-type relaxation which is explained by the frequent hopping of silver ions from their regular lattice sites to adjacent interstitial sites and back again and (ii) contributions to the conductivity below and around 1 THz which are attributed to high-amplitude vibrational motions, mostly of the silver ions. We also report results of a neutron scattering study on AgBr. These contain thermally activated quasielastic components caused by the localized cation back-and-forth hopping and fast correlated movements of neighboring ions.

Multichromophoric Cyclodextrins. 3. Investigation of Dynamics of Energy Hopping by Frequency-Domain Fluorometry

A β-cyclodextrin labeled with seven naphthoyloxy chromophores was studied by steady-state and time-resolved fluorescence spectroscopy in order to get information on the dynamics of energy hopping between chromophores. The steady-state fluorescence anisotropy was recorded as a function of excitation wavelength in a mixture of methanol and ethanol at 110 K (rigid glass). The fluorescence anisotropy decay was obtained under the same conditions by the multifrequency phase-modulation technique upon excitation at 290 nm. The data were analyzed and interpreted on the basis of a theoretical model involving a unique rate constant for energy hopping between nearest neighbors. In particular, this model predicts a long-time leveling-off of the emission anisotropy at 1/7th of the fundamental anisotropy, which is confirmed by both steady-state and time-resolved data and thus indicates that there is no preferred mutual orientation between the chromophores. As regards the rate of energy hopping, an average value of 2 × 109 s-1 can be deduced from the comparison between the theoretical and experimental decays. This value is shown to be consistent with a dipole−dipole mechanism of energy transfer.

Dynamics of Carrier Hopping in Exponential Band Tails of Quasi‐One‐Dimensional Systems in Electric Field

AbstractThe dynamics of carrier hopping in band tails of a quasi‐one‐dimensional (QOD) system is studied in the presence of temperature and electic field. A method is used in which it is assumed that localized states are distributed randomly in both space and energy coordinates and the hopping of carriers occurs in both coordinates. The exponential form of the density of states for band tails is considered. The hopping rate and the time dependent demarcation energy are calculated in the activated and nonactivated relaxation regimes. The hopping rate varies with the electric field as (1  β2)−1/γ and exp [(1  β2)] for the AR and the NAR regions, respectively. Here β and γ are directly proportional to the electric field and the temperature, respectively. Numerical calculations are performed for the hopping rate and the demarcation energy as a function of β and γ for both regions.

Similarity in multilegged locomotion: Bouncing like a monopode

Despite impressive variation in leg number, length, position and type of skeleton, similarities of legged, pedestrian locomotion exist in energetics, gait, stride frequency and ground-reaction force. Analysis of data available in the literature showed that a bouncing, spring-mass, monopode model can approximate the energetics and dynamics of trotting, running, and hopping in animals as diverse as cockroaches, quail and kangaroos. From an animal's mechanical-energy fluctuation and ground-reaction force, we calculated the compression of a virtual monopode's leg and its stiffness. Comparison of dimensionless parameters revealed that locomotor dynamics depend on gait and leg number and not on body mass. Relative stiffness per leg was similar for all animals and appears to be a very conservative quantity in the design of legged locomotor systems. Differences in the general dynamics of gait are based largely on the number of legs acting simultaneously to determine the total stiffness of the system. Four- and six-legged trotters had a greater whole body stiffness than two-legged runners operating their systems at about the same relative speed. The greater whole body stiffness in trotters resulted in a smaller compression of the virtual leg and a higher natural frequency and stride frequency.

Periodic Motions of a Hopping Robot With Vertical and Forward Motion

This article analyzes the global dynamical behavior of sim plified hopping robot models that are analogous to Raibert's experimental machines. We first review a one-dimensional ver tical hopping model that captures both the vertical hopping dynamics and nonlinear control algorithm. Second, we present a more complicated two-dimensional model that includes both forward and vertical hopping dynamics and a foot placement algorithm. These systems are analyzed using a Poincare return map. In this approach, issues of stability and global dynamical behavior are reduced to the study of the fixed points of this map. For the one-dimensional model, a closed-form return map is obtained. For the two-dimensional model, we derive an exact return map based on the first integrals of motion. Because this map can only be constructed numerically, we also derive an analytical approximation to the return map based on perturba tion methods. The approximate return map is shown to closely predict the behavior of the exact map for small forward run ning velocities. In addition, the approximate return map can be used to quantitatively explore the coupling of vertical and lat eral dynamics and to determine the effect of the foot placement algorithm on dynamical behavior The bifurcation diagrams, which capture variations in dy namical behavior with respect to the variations in system and control parameters, are also constructed. The bifurcation dia grams exhibit a period-doubling cascade. In other words, for certain system parameter values, Raibert's control algorithm can lead to an anomalous nonuniform, but stable, hopping be havior. Using the vertical model results as a guide, we interpret the interesting dynamical behavior of this system.

Dynamics of electron hopping in assemblies of redox centers. Percolation and diffusion

The interdependence of electron hopping between redox centers and the physical motion of redox centers is investigated systematically. When physical motion is either nonexistent or much slower than electron hopping, charge propagation is fundamentally a percolation process. In the opposite extreme, rapid molecular motion thoroughly rearranges the molecular distribution between successive electron hops, thereby leading to mean-field behavior. Monte-Carlo simulations are employed to study the transition between static percolation and mean-field behaviors as a function of the relative rates of electron hopping and physical motion

Long-range spatial correlations for anisotropic zero-range processes

The spatial correlations are investigated for a homogeneous system of indistinguishable particles undergoing stochastic anisotropic hopping dynamics on the d-dimensional lattice, d>or=2. The interaction is zero range, i.e. the rate at which particles leave a given site only depends on the occupation number at that site. A series expansion around the independent particle system is given for the equal time correlations and is shown to converge for small times t. The formal t to infinity limiting expansion is analysed termwise from which a quadrupole type decay ( approximately r-d) is derived for the stationary two-points function ( eta (0) eta (r))-( eta (0))2. This phenomenon of self-organized criticality is a direct consequence of the anisotropy causing the system to violate the condition of detailed balance, combined with the conservation law forcing a diffusive decay ( approximately t-d2/) of the temporal correlations.

Theory of hydrogen hopping dynamics including hydrogen-lattice correlations

A theory is formulated to incorporate effects of correlations in the hydrogen-lattice vibrations into the description of the intersite hopping dynamics of hydrogen (and similar light interstitials) in metals. A suitable model is proposed, the theoretical treatment is sketched and the resulting hopping rate expression is presented. It is discussed that in “normal” cases the obtained hopping rate formula can be cast into the form for an uncorrelated system but with suitably renormalized parameters. The theory is shown to reproduce well the hopping rate (diffusion coefficient) for hydrogen and deuterium in tantalum and copper and for μ+ in copper.

Theory of hydrogen hopping dynamics including hydrogen-lattice correlations

Theory of hydrogen hopping dynamics including hydrogen-lattice correlations

First-passage-time distribution in a random random walk.

A detailed study of the statistical properties of first-passage times in a one-dimensional random environment is presented. It is found that the spectrum of relaxation times of the master equation describing the hopping dynamics in such a system possesses a gap, which separates the leading relaxation time from the next one by an amount that is exponentially large in the square root of the size of the system. The leading relaxation time for a given (typical) system equals \ensuremath{\tau}, the mean first-passage time for that realization. One consequence of the existence of this gap is that for times that are larger than a fraction of the mean first-passage time \ensuremath{\tau}, the probability of the first-passage time to have a value t is approximated, to a high degree of accuracy, by (1/\ensuremath{\tau})exp(-t/\ensuremath{\tau}). The appearance of the above-mentioned gap is, surprisingly, related to the existence of a dominant single effective trap (or ``potential well''), the next leading trap being of no importance for large systems. A detailed study of the properties of the mean first-passage time and its fluctuations among the members of the ensemble of realizations is presented (the probability distribution of \ensuremath{\tau} is shown to possess anomalous long-time behavior). The realization-averaged mean first-passage time is shown to be dominated by rare realizations and is contrasted with typical values of that quantity. The mean first-passage-time distribution is shown to be related to a new characterization of random walks that we coin the extent. The latter quantity is investigated in detail.

Nonequilibriurn discontinuous phase transitions in a fast ionic conductor model: Coexistence and spinodal lines

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field along a principal axis are studied in the case of off-critical densities. We describe the corresponding nonequilibrium first-order phase transitions, evaluate coexistence and spinodal lines, and make some comparisons with experimental observations on fast ionic conductors.

Nonequilibrium phase transitions in stochastic lattice systems: Influence of the hopping rates

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field ⇀E along a principal axis are studied in the case of different ratiosγ between the jump rates in the field direction and perpendicular to it using different transition probabilities. We investigate the dependence of the non-equilibrium steady-state properties on the transition mechanism. We find self-similarity with respect to (T, γ) and a coexistence curve critical exponent which, for smallγ, seems independent ofγ. There is some evidence that this exponent might be halfway between the equilibrium mean field and Onsager's values. A crossover toward mean-field behavior for largeγ seems also identified.

Phase Transitions in Lattice-Gas Models Far from Equilibrium

A lattice-gas model with particle-conserving hopping dynamics on a periodic lattice is exposed to a strong external field along one of the principal axes. The resulting stationary state is determined exactly in the limit of infinite ratio of jump rates in and perpendicular to the field direction. In this state the gas-liquid phase transition is of mean-field type. For not too strong coupling in the direction orthogonal to the field, several other phase transitions occur. All of these have mean-field character as well.

Electron hopping and magnetic field dependent spin dynamics of radical ions in solution

The role of hopping on the geminate recombination of radical ions (N,N-dimethylaniline cation and anthracene anion) in acetonitrile is studied via the nanosecond time-resolved magnetic field effect on the triplet yield and the influence of donor concentration thereon. Increasing donor concentration leads to lifetime broadening of the magnetic field dependence of the triplet yield. Responsible for this effect is the perturbation of the coherent spin motion caused by hopping of the electron spin between donor sites of different nuclear spin configuration. Comparison of experimental results with calculations based on the semiclassical theory of spin motion yields an estimate of the hopping rates. Deuteration of both radicals influences the halfwidth of the magnetic field effect: at long probing times and low donor concentrations the halfwidth measured for protonated radical ions exceeds the one for the deuterated species: at short delay times and large donor concentrations, i.e. high hopping rates, this isotopic effect is reversed.