Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cond-mat/9601088

Diffusion on a hypercubic lattice with pinning potential: exact results for the error-catastrophe problem in biological evolution

S. Galluccio·R. Graber·Y. -C. Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the theoretical biology framework one fundamental problem is the so-called error catastrophe in Darwinian evolution models. We reexamine Eigen's fundamental equations by mapping them into a polymer depinning transition problem in a ``genotype'' space represented by a unitary hypercubic lattice. The exact solution of the model shows that error catastrophe arises as a direct consequence of the equations involved and confirms some previous qualitative results. The physically relevant consequence is that such equations are not adequate to properly describe evolution of complex life on the Earth.

Explore related subjects

cond-matq-bio

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. Galluccio, R. Graber, Y. -C. Zhang. 1996-01-20. Diffusion on a hypercubic lattice with pinning potential: exact results for the error-catastrophe problem in biological evolution. https://doi.org/10.1088/0305-4470%2F29%2F10%2F005

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2025

The $1/r^2$ Integrable system: The Universal Hamiltonian for Quantum Chaos

We summarize recent work showing that the $1/r^2$ model of interacting particles in 1-dimension is a universal Hamiltonian for quantum chaotic systems. The problem is analyzed in terms of random matrices and of the evolution of their eigenvalues under changes of parameters. The robustness of bulk space-time correlations of a many particle system to changing boundary conditions is suggested to be at the root of the universality. The explicit density-density correlation functions of the $1/r^2$ model, now available through the above mapping at two values of the coupling constant, are interpreted in the light of Bethe's {\it Ansatz}, giving a vivid picture of the fractionalization of bare particles or holes into ``quark'' like Bethe quasi-particles and holes.

cond-mat↗
arXiv2025

Super Lax Pairs and Infinite Symmetries in The $1/r^2$ System

We present an algebraic structure that provides an interesting and novel link between supersymmetry and quantum integrability. This structure underlies two classes of models that are exactly solvable in 1-dimension and belong to the $1/r^2 $ family of interactions. The algebra consists of the commutation between a ``Super- Hamiltonian'', and two other operators, in a Hilbert space that is an enlargement of the original one by introducing fermions. The commutation relations reduce to quantal Ordered Lax equations when projected to the original subspace, and to a statement about the ``Harmonic Lattice Potential'' structure of the Lax operator. These in turn lead to a highly automatic proof of the integrability of these models. In the case of the discrete $SU(n)-1/r^2$ model, the `` Super-Hamiltonian'' is again an $SU(m)-1/r^2$ model with a related $m$, providing an interesting hierarchy of models.

cond-mat↗
arXiv2025

What Does The Korringa Ratio Measure?

We present an analysis of the Korringa ratio in a dirty metal, emphasizing the case where a Stoner enhancement of the uniform susceptibilty is present. We find that the relaxation rates are significantly enhanced by disorder, and that the inverse problem of determining the bare density of states from a study of the change of the Knight shift and relaxation rates with some parameter, such as pressure, has rather constrained solutions, with the disorder playing an important role. Some preliminary applications to the case of chemical substitution in the Rb$_{3-x}$K$_x $C$_{60}$ family of superconductors is presented and some other relevant systems are mentioned.

cond-mat↗