Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics

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.

On the application of statistical physics to evolutionary biology

“…In contrast, random drift causes random fluctuations in allele frequencies, analogous to the disorganised motions of heat. Ao (2005Ao ( , 2008 has set out this analogy in detail, identifying gradual parameter changes that preserve the stationary distribution as being "reversible" in the thermodynamic sense. It is then possible to make a precise analogy with the Carnot cycle: isothermal changes, in which parameters such as the strength of selection change at constant population size, alternate with adiabatic changes, in which population size and intensive parameters change together (i.e., keeping Ns, N m etc.…”

“…Nevertheless, we can still regard the distribution of observed variables, such as the mean and variance of a quantitative trait, as an average over the space of genotype frequencies, and we still expect the population to tend towards states that can be realised in the largest number of ways. Ao (2005Ao ( , 2008 develops methods for analysing systems that do not tend towards a stationary state, with detailed balance.…”

“…Recently, Ao (2005Ao ( , 2008 and Sella and Hirsh (2005a) have drawn attention to this analogy, using a measure of entropy, S H , that was introduced by Iwasa (1988). Our paper sets their work in a wider context, and extends it to cover a broader range of models.…”

“…This latter case is difficult, because in general, there is no detailed balance in the stationary state, and it is not possible to write down the stationary density explicitly. However, Ao (2005Ao ( , 2008 has made some progress on this problem (see Discussion).…”

Nonequilibrium landscape theory of neural networks

“…To quantitatively study the global stability and functions of neural networks, we need to find out whether a Lyapunov function of monotonically going down in time exists (12,13,18,19,(26)(27)(28)(29)(30)(31)(32)(33)(34) and how it is related to our potential landscape. We expand the UðuÞ with respect to the parameter…”