K. P. N. Murthy

Molecular dynamics simulations of oxygen ion diffusion in yttria-stabilized zirconia

Oxygen ion diffusion in yttria-stabilized zirconia (YSZ) is studied employing molecular dynamics simulation. Oxygen ions migrate mainly by nearest neighbour hopping amongst the tetrahedral lattice sites of zirconium ions. A linear relation between the mean square displacement and time is found, after the oxygen ions have moved over distances much larger than the characteristic distances of the underlying crystal structure. In this diffusive region, the bulk oxygen tracer diffusion coefficient is 2.21×10−6cm2s−1 at 1759K and 3.53×10−6cm2s−1 at 2057K. The ionic conductivity, calculated from the bulk oxygen tracer diffusion coefficient, matches well with the experimental values. For all the ion pairs in YSZ, we have calculated the radial distribution function. We find that the peak height is smaller at higher temperature, due to the volume expansion of the YSZ crystal.

Can coarse-graining introduce long-range correlations in a symbolic sequence?

We present an exactly solvable mean-field–like theory of correlated ternary sequences which are actually systems with two independent parameters. Depending on the values of these parameters, the variance on the average number of any given symbol shows a linear or a superlinear dependence on the length of the sequence. We have shown that the available phase space of the system is made up of a diffusive region surrounded by a superdiffusive region. Motivated by the fact that the diffusive portion of the phase space is larger than that for the binary, we have studied the mapping between these two. We have identified the region of the ternary phase space, particularly the diffusive part, that gets mapped into the superdiffusive regime of the binary. This exact mapping implies that long-range correlations found in a lower-dimensional representative sequence may not, in general, correspond to the correlation properties of the original system.

Diffusion controlled multiplicative process: typical versus average behaviour

We investigate the dynamics of the number of particles diffusing in a multiplicative medium. We show that the typical behaviour of the growth process is different from the average. We develop a new formalism to study the average growth process and extend it to the calculation of higher moments and finally of the probability distribution. We show that the fluctuations of the growth process increase exponentially with time. We describe the interesting features of the distribution.

Statistical methods in nonlinear dynamics

Sensitivity to initial conditions in nonlinear dynamical systems leads to exponential divergence of trajectories that are initially arbitrarily close, and hence to unpredictability. Statistical methods have been found to be helpful in extracting useful information about such systems. In this paper, we review briefly some statistical methods employed in the study of deterministic and stochastic dynamical systems. These include power spectral analysis and aliasing, extreme value statistics and order statistics, recurrence time statistics, the characterization of intermittency in the Sinai disorder problem, random walk analysis of diffusion in the chaotic pendulum, and long-range correlations in stochastic sequences of symbols.

Intermittency in random maps

The escape probability for a random walk on a one-dimensional lattice is discussed in terms of random maps. The global dynamics is found to be intermittent: the laminar regions with long sojourn near the upper fixed point are broken by irregular bursts. Intermittency emerges even if the Sinai condition is not satisfied. It is shown that, if P is the probability of choosing the upper map, the average laminar length diverges as (1−P)−α.

First passage time on a multifurcating hierarchical structure

Asymptotic behaviour of the moments of the first passage time (FPT) on a one-dimensional lattice holding a multifurcating hierarchy of teeth is studied. There is a transition from ordinary to anomalous diffusion when the parameter controlling the relative sizes of the teeth, is varied with respect to the furcating number of the hierarchy. The scaling behaviour of the moments of FPT with the linear dimensions of the lattice segment indicates that in the anomalcus phase the probability density of the FPT is multifractal.

Persistence and life time distribution in coarsening phenomena

We investigate the lifetime distribution P(τ,t) in one and two dimensional coarsening processes modelled by Ising–Glauber dynamics at zero temperature. The lifetime τ is defined as the time that elapses between two successive flips in the time interval (0,t) or between the last flip and the observation time t. We calculate P(τ,t) averaged over all the spins in the system and over several initial disorder configurations. We find that asymptotically the lifetime distribution obeys a scaling ansatz: P(τ,t)=t−1φ(ξ), where ξ=τ/t. The scaling function φ(ξ) is singular at ξ=0 and 1, mainly due to slow dynamics and persistence. An independent lifetime model where the lifetimes are sampled from a distribution with power law tail is presented, which predicts analytically the qualitative features of the scaling function. The need for going beyond the independent lifetime models for predicting the scaling function for the Ising–Glauber systems is indicated.

Fractal measures of first passage time of a simple random walk

We consider random walks, starting at the site i = 1, on a one-dimensional lattice segment with an absorbing boundary at i = 0 and a reflecting boundary at i = L. We find that the typical value of first passage time (FPT) is independent of system size L, while the mean value diverges linearly with L. The qth moment of the FPT diverges with system size as L2q−1, for q >12. For a finite but large L, the FPT distribution has an 1/t tail cut off by an exponential of the form exp(-t/L2). However, if L is set equal to infinity, the distribution has an algebraic tail given by t-12. We find that the generalised dimensions D(q) have a nontrivial dependence on q. This shows that the FPT distribution is a multifractal. We also calculate the singularity spectrum f(α).