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This is the fourth paper, the last one, on solution to the problem of absence of detailed balance in nonequilibrium processes. It is an approach based on another known universal dynamics: 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 enormously 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.

A tendency in biological theorizing is to formulate principles above or equal to Evolution by Variation and Selection of Darwin and Wallace. In this letter I analyze one such recent proposal which did so for the developmental ascendency. I show that though the idea of developmental ascendency is brilliant, this is in wrong order in the hierarchical structure of biological theories and can easily generate confusing. Several other examples are also briefly discussed in the note added.

The study of biological cells as a nonequilibrium, nonlinear, stochastic open chemical systems provides a paradigm for other complex, self-organizing systems with short-time deterministic and long-time evolutionary dynamics.

Darwinian dynamics is manifestly stochastic and nonconservative, but has a profound connection to conservative dynamics in physics. In the present paper the main ideas and logical steps leading to thermodynamics from Darwinian dynamics are discussed in a quantitative manner. A synthesis between nonequilibrum dynamics and conservative dynamics is outlined.

We discuss the population dynamics with selection and random diffusion, keeping the total population constant, in a fitness landscape associated with Constraint Satisfaction, a paradigm for difficult optimization problems. We obtain a phase diagram in terms of the size of the population and the diffusion rate, with a glass phase inside which the dynamics keeps searching for better configurations, and outside which deleterious `mutations' spoil the performance. The phase diagram is analogous to that of dense active matter in terms of temperature and drive.

We discuss the population dynamics with selection and random diffusion, keeping the total population constant, in a fitness landscape associated with Constraint Satisfaction, a paradigm for difficult optimization problems. We obtain a phase diagram in terms of the size of the population and the diffusion rate, with a glass phase inside which the dynamics keeps searching for better configurations, and outside which deleterious `mutations' spoil the performance. The phase diagram is analogous to that of dense active matter in terms of temperature and drive.