# General criterion for harmonicity

## TL;DR

This paper introduces a new class of transfer matrices inspired by Markov processes, providing a criterion for perfect harmonicity in systems, exemplified by a polymer model that remains harmonic despite non-Gaussian sub-units.

## Contribution

It presents a novel algebraic approach to determine harmonicity in physical systems and constructs a polymer model that remains harmonic with non-Gaussian components.

## Key findings

- Largest eigenvalue determined by explicit algebraic equation
- Polymer with non-Gaussian sub-units remains harmonic until fully stretched
- Confirmed by Monte Carlo and Langevin simulations

## Abstract

Inspired by Kubo-Anderson Markov processes, we introduce a new class of transfer matrices whose largest eigenvalue is determined by a simple explicit algebraic equation. Applications include the free energy calculation for various equilibrium systems and a general criterion for perfect harmonicity, i.e., a free energy that is exactly quadratic in the external field. As an illustration, we construct a "perfect spring", namely a polymer with non-Gaussian, exponentially distributed sub-units which nevertheless remains harmonic until it is fully stretched. This surprising discovery is confirmed by Monte Carlo and Langevin simulations.