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| 2008 49 (05): 1073-1090 ISSN: 0253-6102 CN: 11-2592/O3 |
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Emerging of Stochastic Dynamical Equalities and Steady
State Thermodynamics from Darwinian Dynamics
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| P. Ao |
Department of Mechanical Engineering and Department of Physics,
University of Washington, Seattle, WA 98195, USA
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Received
2007-11-30
Revised
Online
Accepted
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Abstract The evolutionary dynamics first conceived by Darwin and Wallace,
referring to as Darwinian dynamics in the present paper, has been
found to be universally valid in biology. The statistical mechanics and thermodynamics, while enormous
successful in physics, have been in an awkward situation of wanting a consistent dynamical understanding. Here we present from a formal point of view an exploration of the connection between thermodynamics and Darwinian dynamics and a few related topics.
We first show that the stochasticity in
Darwinian dynamics implies the existence temperature, hence the
canonical distribution of Boltzmann-Gibbs type. In term of relative
entropy the Second Law of thermodynamics is dynamically
demonstrated without detailed balance condition, and is valid regardless of size of the system. In particular, the dynamical component responsible for breaking detailed balance condition does
not contribute to the change of the relative entropy. Two types of
stochastic dynamical equalities of current interest are explicitly
discussed in the present approach: One is based on Feynman-Kac formula and another is a generalization of Einstein relation. Both are directly accessible to experimental tests. Our demonstration
indicates that Darwinian dynamics represents logically a simple
and straightforward starting point for statistical mechanics and
thermodynamics and is complementary to and consistent with conservative dynamics that dominates the physical sciences. Present exploration suggests the existence of a unified stochastic
dynamical framework both near and far from equilibrium.
Key words
non-equilibrium processes
Darwinian dynamics
generalized Einstein
relation
absence of detailed balance
stochastic dynamical equalities
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