Relative Entropy in Biological Systems

In this paper we review various information-theoretic characterizations of the approach to equilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics of a population or probability distribution. Under suitable assumptions, the distribution will approach an equilibrium with the passage of time. Relative entropy - that is, the Kullback--Leibler divergence, or various generalizations of this - provides a quantitative measure of how far from equilibrium the system is. We explain various theorems that give conditions under which relative entropy is nonincreasing. In biochemical applications these results can be seen as versions of the Second Law of Thermodynamics, stating that free energy can never increase with the passage of time. In ecological applications, they make precise the notion that a population gains information from its environment as it approaches equilibrium.Comment: 20 page

Network Models

Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce "network models" to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to $\mathbf{Cat}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.Comment: 46 page

An Exactly Conservative Integrator for the n-Body Problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge

Descriptive oceanography during the Frontal Air‐Sea Interaction Experiment: Medium‐ to large‐scale variability

Medium‐ and large‐scale oceanographic variability in the Sargasso Sea is examined during the Frontal Air‐Sea Interaction Experiment (FASINEX), focusing primarily on processes that influence the formation of subtropical fronts. From Fall to Spring the mean meridional gradient of meridional Ekman transport in the Subtropical Convergence Zone (STCZ) enhances the meridional sea surface temperature (Ts) gradients between 26° and 32°N. In the presence of this enhanced mean gradient, baroclinic eddies with zonal wavelengths of ≈800 km and periods of ≈200 days exert the dominant influence on the formation of subtropical fronts at medium and large scales. These eddies generate westward propagating Ts anomaly features with the same dominant wavelengths and periods. They are confined between 26° and 32°N and have amplitudes that occasionally exceed ±1°C. Ts fronts tend to be found within bands ≈200 km wide that roughly follow the periphery of these anomaly features. Deformation in the horizontal eddy current field is primarily responsible for the existence of these frontal bands. The migration of the strong front originally bracketed by the FASINEX moored array was related to the westward propagation of the larger‐scale eddy/anomaly/frontal‐band pattern. The moored array was located within a warm‐anomaly feature during most of the experiment, which produced exceptionally warm conditions in the upper ocean. These anomalies are confined between 26° and 32°N, not only because the relatively large seasonal mean Tsy there allows horizontal eddy currents to force strong anomalies, but also because the baroclinic eddies with wavelengths of ≈800 km and periods of ≈200 days are confined to the STCZ. Large meridional variability exists in many properties of the eddy field, much of which can be traced to the influence of the Sargasso Sea mean current field on eddy variability