Covariant influences for finite discrete dynamical systems

TL;DR

This paper introduces a rigorous resource-theoretic framework for understanding external influences on finite discrete dynamical systems, encompassing both deterministic and stochastic cases, and characterizes state transition possibilities.

Contribution

It applies resource theory to finite discrete dynamical systems, providing necessary and sufficient conditions for state transitions under covariant influences, a novel approach in this context.

Findings

Established a covariant influence framework for deterministic systems

Derived necessary conditions for state transitions in stochastic systems

Unified behavior laws across different finite discrete dynamical systems

Abstract

We develop a rigorous theory of external influences on finite discrete dynamical systems, going beyond the perturbation paradigm, in that the external influence need not be a small contribution. Indeed, the covariance condition can be stated as follows: if we evolve the dynamical system for $n$ time steps and then we disturb it, it is the same as first disturbing the system with the same influence and then letting the system evolve for $n$ time steps. Applying the powerful machinery of resource theories, we develop a theory of covariant influences both when there is a purely deterministic evolution and when randomness is involved. Subsequently, we provide necessary and sufficient conditions for the transition between states under deterministic covariant influences and necessary conditions in the presence of stochastic covariant influences, predicting which transitions between states are forbidden. Our approach, for the first time, employs the framework of resource theories, borrowed from quantum information theory, to the study of finite discrete dynamical systems. The laws we articulate unify the behavior of different types of finite discrete dynamical systems, and their mathematical flavor makes them rigorous and checkable.