Universality of Long-Range Correlations in Expansion-Randomization Systems

We study the stochastic dynamics of sequences evolving by single site mutations, segmental duplications, deletions, and random insertions. These processes are relevant for the evolution of genomic DNA. They define a universality class of non-equilibrium 1D expansion-randomization systems with generic stationary long-range correlations in a regime of growing sequence length. We obtain explicitly the two-point correlation function of the sequence composition and the distribution function of the composition bias in sequences of finite length. The characteristic exponent $\chi$ of these quantities is determined by the ratio of two effective rates, which are explicitly calculated for several specific sequence evolution dynamics of the universality class. Depending on the value of $\chi$, we find two different scaling regimes, which are distinguished by the detectability of the initial composition bias. All analytic results are accurately verified by numerical simulations. We also discuss the non-stationary build-up and decay of correlations, as well as more complex evolutionary scenarios, where the rates of the processes vary in time. Our findings provide a possible example for the emergence of universality in molecular biology.Comment: 23 pages, 15 figure

Concepts of quantum non-Markovianity: a hierarchy

Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.Comment: Clarifications and references added; discussion of the related classical hierarchy significantly improved. To appear in Physics Report