Quantum persistence: A random-walk scenario

Abstract

In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power-law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum walks are discussed.

Quantum walks: The first detected passage time problem

“…The quantum renewal equation (17) and the Z transforms Eqs. (36,37) give a rather general relation between the amplitude of first detection φ n and the wave function of the system free of measurement |ψ f . In that sense the problem of first detection reduces to the solution of the Schrödinger equation, e.g.…”

“…The recent advance of single particle quantum tracking and measurement, for systems where coherence is maintained for relatively long times, is clearly a reason to be optimistic with respect to possible first detection measurements. Such measurements could test our predictions as well as those of a variety of other theoretical approaches [14,23,[31][32][33][34][35][36][37][38], some of which are compared with our results towards the end of this manuscript.…”

Kinetic Exchange Models in Economics and Sociology

“…But for CCM, the direction reversal probability f is greater than 1/2 for all λ < 1 and f → 1/2 for λ → 1. Through further analysis of time correlation and other relevant quantities it was shown that direction reversal is preferred in these cases [51]. In the equivalent picture of the walk in the abstract space for gains and losses, it is similar to the fact that here individuals has a tendency to make a gain immediately after a loss and vice versa.…”

“…It was also shown in [51] that the "antipersistence effect" is maximum for no saving and decreases with saving. This is perhaps in tune with the human feeling of security associated with the saving factor.…”

“…In [51], the exact nature of the walk associated with CC and CCM model was explored and it was shown through the effective bias p associated with the walks, distribution of walk lengths at a stretch etc., that CCM is not a simple BRW and CC is not a simple RW.…”