Quantum randomness in the Sky

In this article, we study quantum randomness of stochastic cosmological particle production scenario using quantum corrected higher order Fokker Planck equation. Using the one to one correspondence between particle production in presence of scatterers and electron transport in conduction wire with impurities we compute the quantum corrections of Fokker Planck Equation at different orders. Finally, we estimate Gaussian and non-Gaussian statistical moments to verify our result derived to explain stochastic particle production probability distribution profile.Comment: 6 pages, 4 figures, Accepted for publication in European Physical Journal

Quantum Out-of-Equilibrium Cosmology

In this work, our prime focus is to study the one to one correspondence between the conduction phenomena in electrical wires with impurity and the scattering events responsible for particle production during stochastic inflation and reheating implemented under a closed quantum mechanical system in early universe cosmology. In this connection, we also present a derivation of fourth order corrected version of the Fokker Planck equation and its analytical solution for studying the dynamical features of the particle creation events in the stochastic inflation and reheating stage of the universe. It is explicitly shown from our computation that quantum corrected Fokker Planck equation describe the particle creation phenomena better for Dirac delta type of scatterer. In this connection, we additionally discuss It$\hat{o}$, Stratonovich prescription and the explicit role of finite temperature effective potential for solving the probability distribution profile. Furthermore, we extend our discussion to describe the quantum description of randomness involved in the dynamics. We also present a computation to derive the expression for the measure of the stochastic non-linearity arising in the stochastic inflation and reheating epoch of the universe, often described by Lyapunov Exponent. Apart from that, we quantify the quantum chaos arising in a closed system by a more strong measure, commonly known as Spectral Form Factor using the principles of Random Matrix Theory (RMT). Additionally, we discuss the role of out of time order correlation (OTOC) function to describe quantum chaos in the present non-equilibrium field theoretic setup. Finally, for completeness, we also provide a bound on the measure of quantum chaos arising due to the presence of stochastic non-linear dynamical interactions into the closed quantum system of the early universe in a completely model-independent way.Comment: 177 pages, 58 figures, 4 tables, Accepted for publication in European Physical Journal

Spectral Form Factor for Time-dependent Matrix model

The quantum chaos is related to a Gaussian random matrix model, which shows a dip-ramp-plateau behavior in the spectral form factor for the large size $N$. The spectral form factor of time dependent Gaussian random matrix model shows also dip-ramp-plateau behavior with a rounding behavior instead of a kink near Heisenberg time. This model is converted to two matrix model, made of $M_1$ and $M_2$. The numerical evaluation for finite $N$ and analytic expression in the large $N$ are compared for the spectral form factor.Comment: 38 pages,16 figure

Immobilization of tannery industrial sludge in ceramic membrane preparation and hydrophobic surface modification for application in atrazine remediation from water

Chromium laden waste produced from tannery industry was immobilized in ceramic matrix for fabrication of the tubular single channel microfiltration membranes by extrusion. The presence of chromia resulted in substitutional solid solution formation with alumina and catalyzed mullite phase growth, hence increasing the mechanical and chemical stability of the membranes. The structural, morphological and water permeation characteristics of the membranes were studied to analyze their formation mechanism and effect of different parameters, viz. the sintering temperature, amount of waste added, presence of organics and extent of chromium immobilization. The surface of the macroporous membrane was hydrophobically modified, by polydimethylsiloxane (PDMS), producing contact angle of 141 degrees. The process efficiency of the hydrophobic membrane was assessed in terms of the removal of atrazine, a contaminant of emerging concern, following the principle of hydrophobic interaction. Effect of different operating parameters affecting atrazine removal, viz. transmembrane pressure, cross flow velocity and filtration time was studied in cross flow filtration mode. High atrazine removal of >95% was obtained along with the maintenance of high flux during the filtration operation. The prepared cost-effective microfiltration membranes can thus be further modified for efficient water treatment applications

Sparse random matrices and Gaussian ensembles with varying randomness

Abstract We study a system of N qubits with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Our motivation is to understand the system at large N. In practice most of our calculations are carried out using exact diagonalisation techniques (up to N = 24). Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N. We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N, and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile