Markovian Memory Embedded in Two-State Natural Processes

TL;DR

This paper explores how Markovian memory influences binary processes, affecting state clustering, dispersion, and long-term behavior, with applications to animal behavior and quantum state dynamics.

Contribution

It introduces a mathematical framework for analyzing Markovian memory effects in natural binary processes, demonstrating its applicability to diverse systems.

Findings

Markovian memory causes clustering or dispersion of states.

Balanced transition probabilities can mimic random process evolution.

Model successfully estimates transition probabilities in quantum Zeno effect.

Abstract

Markovian memory embedded in a binary system is shaping its evolution on the basis of its current state and introduces either clustering or dispersion of binary states. The consequence is directly observed in the lengthening or shortening of the runs of the same binary state and also in the way the proportion of a state within a sequence of state measurements scatters about its true average, which is quantifiable through the Markovian self-transition probabilities. It is shown that the Markovian memory can even imitate the evolution of a random process, regarding the long-term behavior of the frequencies of its binary states. This situation occurs when the associated binary state self-transition probabilities are balanced. To exemplify the behavior of Markovian memory, two natural processes are selected. The first example is studying the preferences of nonhuman troglodytes regarding handedness. The Markovian model in this case assesses the extent of influence two contiguous individuals may have on each other. The other example studies the hindering of the quantum state transitions that rapid state measurements introduce, known as the Quantum Zeno effect (QZE). Based on the current methodology, simulations of the experimentally observed clustering of states allowed for the estimation of the two self-transition probabilities in this quantum system. Through these, one can appreciate how the particular hindering of the evolution of a quantum state may have originated. The aim of this work is to illustrate as merits of the current mathematical approach, its wide range applicability and its potential to provide a variety of information regarding the dynamics of the studied process.