A unified framework for quasi-species evolution and stochastic quantization

TL;DR

This paper explores the connection between biological evolution equations and quantum mechanics, introducing a stochastic quantization framework that reveals new behaviors and steady states in particle ensembles.

Contribution

It presents a unified framework linking quasi-species evolution with stochastic quantization, including explicit solutions for harmonic oscillator potentials.

Findings

Ensemble probability distributions decompose into eigenfunctions.

Out-of-equilibrium dynamics relax to a fundamental state without interactions.

Presence of scattering leads to Bose-Einstein statistical steady states.

Abstract

We investigate the profound relation between the equations of biological evolution and quantum mechanics by writing a biologically inspired equation for the stochastic dynamics of an ensemble of particles. Interesting behavior is observed which is related to a new type of stochastic quantization. We find that the probability distribution of the ensemble of particles can be decomposed into eigenfunctions associated to a discrete spectrum of eigenvalues. In absence of interactions between the particles, the out-of-equilibrium dynamics asymptotically relaxes towards the fundamental state. This phenomenon can be related with the Fisher theorem in biology. On the contrary, in presence of scattering processes the evolution reaches a steady state in which the distribution of the ensemble of particles is characterized by a Bose-Einstein statistics. In order to show a concrete example of this stochastic quantization we have solved explicitly the case in which the potential energy has the harmonic oscillator form.