V. Balakrishnan

Einstein's Miraculous Year

With each passing year, the young Albert Einstein’s achievements in physics in the year 1905 seem to be ever more miraculous. We describe why the centenary of this remarkable year is worthy of celebration.

Recurrence time statistics in chaotic dynamics.: I. Discrete time maps

The dynamics of transitions between the cells of a finite-phase-space partition in a variety of systems giving rise to chaotic behavior is analyzed, with special emphasis on the statistics of recurrence times. In the case of one-dimensional piecewise Markow maps the recurrence problem is cast into a-renewal process. In the presence of intermittency, transitions between cells define a non-Markovian, non-renewal process reflected in the presence of power-law probability distributions and of divergent variances and mean values.

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.

Squeezing and higher-order squeezing of photon-added coherent states propagating in a Kerr-like medium

We investigate non-classical effects such as fractional revivals, squeezing and higher-order squeezing of photon-added coherent states propagating through a Kerr-like medium.The Wigner functions corresponding to these states at the instants of fractional revivals are obtained, and the extent of non-classicality quantified.We investigate the squeezing and higher-order squeezing properties of photon-added coherent states propagating through a Kerr-like medium, particularly close to instants of revivals and fractional revivals of the state. The Wigner functions at these instants are obtained, and the extent of non-classicality quantified.

Transport properties on a random comb

We study the random walk of a particle in a random comb structure, both in the presence of a biasing field and an the field-free case. We show that the mean-field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk (CTRW) on the backbone of the comb, with a definite waiting time density. We find an exact expression for this central quantity. The Green function for the CTRW is then obtained. Its first and second moments determine the drift and diffusion at all times. We show that the drift velocity v vanishes asymptotically for power-law and stretched-exponential distributions of branch lengths on the comb, whatever be the biasing field strength. For an exponential branch-length distribution, v is a nonmonotonic function of the bias, increasing initially to a maximum and then decreasing to zero at a critical value. In the field-free case, anomalous diffusion occurs for a range of power-law distributions of the branch length. The corresponding exponent for the mean square displacement is obtained, as is the asymptotic form of the positional probability distribution for the random walk. We show that normal diffusion occurs whenever the mean branch length is finite, and present a simple formula for the effective diffusion constant; these results are extended to regular (nonrandom) combs as well. The physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time (on a finite chain) that controls the effective waiting time density of the CTRW.

On the connection between biased dichotomous diffusion and the one-dimensional Dirac equation

The master equation for dichotomous diffusion (DD) (the integral of a random telegraph process) is the well-known telegraphers equation, which is converted to the Klein–Gordon equation by a simple transformation. After a brief recapitulation of the solution and of the analogy between DD and the Dirac equation in one spatial dimension, we consider velocity-biased DD. The corresponding master equation and its solution are presented. It is shown that these may be interpreted physically in terms of a Lorentz transformation to a frame moving with a boost velocity equal to the mean drift velocity of the diffusing particle. The modifications that arise in the connection with the Dirac equation are also exhibited. The correspondence between the rest mass of the Dirac particle and the frequency of direction reversal in the DD is shown to be modified precisely by the time dilatation correction to the latter quantity.

Wave packet dynamics of photon-added coherent states

We show in the framework of a tractable model that revivals and fractional revivals of wave packets afford clear signatures of the extent of departure from coherence and from Poisson statistics of the matter wave field in a Bose-Einstein condensate, or of a suitably chosen initial state of the radiation field propagating in a Kerr-like medium.

RECURRENCE TIME STATISTICS IN CHAOTIC DYNAMICS: MULTIPLE RECURRENCES IN INTERMITTENT CHAOS

An exact expression is derived for the distribution of successive recurrences in a finite-cell partition of a general family of one-dimensional maps exhibiting intermittent chaos due to a tangency at a marginally stable fixed point, characterized by an index α. It is shown that multiple recurrences are not statistically independent events in the strict sense of the term. The asymptotic factorization of the joint distribution into a product of first recurrence time distributions is also analyzed.

Reflection principles for biased random walks and application to escape time distributions

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such “frozen” processes.