Extremal property of a simple cycle

TL;DR

This paper proves a sharp bound on the ratio of imaginary to real parts of eigenvalues in finite-state systems obeying first order kinetics without detailed balance, showing that a simple cycle with equal rates maximizes oscillation decay time.

Contribution

It establishes a universal inequality for eigenvalues in such systems and identifies the simple cycle with equal rates as the extremal case.

Findings

The ratio | ext{Im} \lambda| / | ext{Re} \\lambda| \\leq \\cot(\\pi/n) for all nonzero eigenvalues.

Equality is achieved by a simple irreversible cycle with equal transition rates.

This cycle exhibits the slowest decay of oscillations among systems with the same number of states.

Abstract

We study systems with finite number of states $A_i$ ($i=1,..., n$), which obey the first order kinetics (master equation) without detailed balance. For any nonzero complex eigenvalue $\lambda$ we prove the inequality $\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is sharp and it becomes an equality for an eigenvalue of a simple irreversible cycle $A_1 \to A_2 \to... \to A_n \to A_1$ with equal rate constants of all transitions. Therefore, the simple cycle with the equal rate constants has the slowest decay of the oscillations among all first order kinetic systems with the same number of states.