A Stationary Drake Equation Distribution as a Balance of Birth-Death Processes

TL;DR

This paper introduces a stochastic model of the Drake Equation, modeling civilization birth and collapse as Poisson and exponential processes, leading to a stationary distribution that challenges the inference of the number of civilizations.

Contribution

It presents a novel stochastic formulation of the Drake Equation, deriving a stationary distribution and analyzing implications for the Copernican Principle and the probability of being alone.

Findings

Number of civilizations follows a Poisson distribution with mean λ_C/λ_L.

Demonstrates the non-inferability of N using the Copernican Principle.

Provides a probabilistic framework for civilization dynamics in the galaxy.

Abstract

Previous critiques of the Drake Equation have highlighted its deterministic nature, implying that the number of civilizations is the same at all times. Here, I build upon earlier work and present a stochastic formulation. The birth of civilizations within the galaxy is modeled as following a uniform rate (Poisson) stochastic process, with a mean rate of $\lambda_C$. Each then experiences a constant hazard rate of collapse, which defines an exponential distribution with rate parameter $\lambda_L$. Thus, the galaxy is viewed as a frothing landscape of civilization birth and collapse. Under these assumptions, I show that N in the Drake Equation must follow another Poisson distribution, with a mean rate $(\lambda_C/\lambda_L)$. This is then used to rigorously demonstrate why the Copernican Principle does not allow one to infer N, as well evaluating the algebraic probability of being alone in the galaxy.