Can anything from Noether's theorem be salvaged for discrete dynamical systems?

TL;DR

This paper investigates whether Noether's theorem, which links symmetries to conserved quantities, can be adapted for discrete dynamical systems, using the Ising model as a test case.

Contribution

It demonstrates that energy can serve as a conserved quantity in certain discrete systems with time-invariance, extending the concept of Noether's theorem.

Findings

Energy acts as a generator of dynamics in the Ising model.

Energy is conserved under time-invariant conditions in the discrete system.

The classical Noether's theorem principles can be partially applied to discrete systems.

Abstract

The dynamics of a physical system is linked to its phase-space geometry by Noether's theorem, which holds under standard hypotheses including continuity. Does an analogous theorem hold for discrete systems? As a testbed, we take the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that---and why---energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant.