Physical analogies

Evolutionary theory has long benefited from analogies with statistical physics—the other field of science dealing with large, evolving populations—, as discussed in e.g. \cite{Sella_2005,Mustonen_2010,de_Vladar_2011,Smerlak_2017}. More recently, Leuthäusser \cite{Leuth_usser_1986} and others \cite{Baake_1997} highlighted a parallel between evolutionary models and certain quantum spin systems, which they could leverage to compute the quasispecies distribution \(Q\) for some special fitness landscape. The present work was inspired by the same observation: the selection-mutation operator \(A\ =\ \mu\Delta+\phi\) in the Crow-Kimura equation has the form \(A\ =\ -H\) with \(H\) a random Schrödinger operator. 
Such operators arises naturally in different physical contexts. For instance, in quantum physics, \(H\) is the energy of a system with configuration space \(X\) (electrons in two or three dimensions, spin chains for hypercube graphs);  Eq. \ref{crow_kimura} then corresponds to the Schrödinger equation in imaginary time, a central tool in quantum Monte Carlo methods \cite{Kosztin_1996}. In hydrodynamics, Eq. \ref{crow_kimura} describes the dynamics of various fields (Burgers flow, temperature, magnetic field) in external force fields \cite{Carmona_1994}. In the mathematical literature, Eq. \ref{crow_kimura} is studied under the name "parabolic Anderson model" \cite{K_nig_2016}
Although superficially different—and associated to different vocabularies—, the phenomenology of these physical systems boil down to the same mathematical fact, which is that the eigenvectors of \(H\) are "caught by disorder" \cite{Stollmann_2001}i.e. localize in small subregions of \(X\) when \(\phi\) is sufficiently rugged and \(\mu\) sufficiently small. This behavior is responsible for the Anderson localization of waves, the intermittency of turbulent Burgers flows—and the transient concentration of evolving populations at the top of fitness peaks below the Eigen error threshold. The first two columns of Table \ref{analogy} highlight the correspondence between these phenomena.
The equivalence between quantum Hamiltonians and diffusion processes discussed in this paper (the third column in Table \ref{analogy}) is a known trick used to solve the Fokker-Planck equation \cite{risken1984}, and has also been used to derive mathematical results \cite{Witten_1982}w. To my knowledge, the first application of