6 research outputs found
Eigen model as a quantum spin chain: exact dynamics
We map Eigen model of biological evolution [Naturwissenschaften {\bf 58}, 465
(1971)] into a one-dimensional quantum spin model with non-Hermitean
Hamiltonian. Based on such a connection, we derive exact relaxation periods for
the Eigen model to approach static energy landscape from various initial
conditions. We also study a simple case of dynamic fitness function.Comment: 10 pages. Physical Revew E vol. 69, in press (2004
Insights into the Second Law of Thermodynamics from Anisotropic Gas-Surface Interactions
Thermodynamic implications of anisotropic gas-surface interactions in a
closed molecular flow cavity are examined. Anisotropy at the microscopic scale,
such as might be caused by reduced-dimensionality surfaces, is shown to lead to
reversibility at the macroscopic scale. The possibility of a self-sustaining
nonequilibrium stationary state induced by surface anisotropy is demonstrated
that simultaneously satisfies flux balance, conservation of momentum, and
conservation of energy. Conversely, it is also shown that the second law of
thermodynamics prohibits anisotropic gas-surface interactions in "equilibrium",
even for reduced dimensionality surfaces. This is particularly startling
because reduced dimensionality surfaces are known to exhibit a plethora of
anisotropic properties. That gas-surface interactions would be excluded from
these anisotropic properties is completely counterintuitive from a causality
perspective. These results provide intriguing insights into the second law of
thermodynamics and its relation to gas-surface interaction physics.Comment: 28 pages, 11 figure
Solution of the Crow-Kimura and Eigen models for alphabets of arbitrary size by Schwinger spin coherent states
To represent the evolution of nucleic acid and protein sequence, we express
the parallel and Eigen models for molecular evolution in terms of a functional
integral representation with an -letter alphabet, lifting the two-state,
purine/pyrimidine assumption often made in quasi-species theory. For arbitrary
and a general mutation scheme, we obtain the solution of this model in
terms of a maximum principle. Euler's theorem for homogeneous functions is used
to derive this `thermodynamic' formulation of evolution. The general result for
the parallel model reduces to known results for the purine/pyrimidine
alphabet and the nucleic acid alphabet for the Kimura 3 ST mutation
scheme. Examples are presented for the and cases. We derive the
maximum principle for the Eigen model for general . The general result for
the Eigen model reduces to a known result for . Examples are presented for
the nucleic acid and the amino acid alphabet. An error catastrophe
phase transition occurs in these models, and the order of the phase transition
changes from second to first order for smooth fitness functions when the
alphabet size is increased beyond two letters to the generic case. As examples,
we analyze the general analytic solution for sharp peak, linear, quadratic, and
quartic fitness functions.Comment: 50 pages, 8 figures, to appear in J. Stat. Phys; some typos fixe