Where does divergence collide?

In vector calculus, the divergence of a vector field is a scalar operator, written as the dot of the del operator with the field, that measures the net outward flux per unit volume at a point: it is positive where the field acts like a source, negative where it acts like a sink, and zero where flow is conserved. The divergence theorem, whose development Katz traced in his 1979 The History of Stokes' Theorem, relates the total divergence inside a region to the flux crossing its boundary. The referent here is a differential operator acting on a field in space, returning a number at each point, not a comparison between two probability distributions, a split between evolving lineages, or a widening gap between national incomes.[1]

Unit

A scalar value at each point, the net outflow per unit volume of a vector field

Recovery means

Not applicable

Measured by

Computed as the sum of the partial derivatives of the field's components, or as flux per volume in the limit of a shrinking region

In information theory and statistics, a divergence is a function that measures how much one probability distribution differs from another, quantifying the information lost when the second is used to approximate the first. Kullback and Leibler's 1951 On Information and Sufficiency introduced the best known example, now called the Kullback-Leibler divergence, built from the expected log ratio of the two distributions. Crucially it is not a true distance: it is asymmetric and does not obey the triangle inequality, so swapping the two distributions changes its value. The referent is a directed measure of difference between distributions, not a source-measuring operator on a field, a split between species, or a spread in incomes.[2]

Unit

A non-negative number of nats or bits, the information lost approximating one distribution by another

Recovery means

Not applicable

Measured by

Computed as the expected logarithm of the ratio of the two distributions' densities, summed or integrated over the outcome space

In evolutionary biology, divergence is the accumulation of heritable differences between lineages after they separate from a common ancestor, so that populations, species, or their gene sequences grow steadily more unlike over time. Nei's 1972 Genetic Distance between Populations gave a widely used way to quantify this genetic divergence and to relate it to the time since two populations split. The referent is a biological process of differentiation between real lineages, measured in substitutions or in shifting allele frequencies, not an operator on a field, a gap between probability distributions, or a spread in economic output.[3]

Unit

The amount of heritable difference between two lineages, scaled by substitutions per site or a genetic-distance index

Recovery means

Not applicable

Measured by

Estimated from differences in DNA or protein sequences, or from shifts in allele frequencies, and calibrated against divergence time

In economics, divergence is the widening of gaps in income or growth between countries or regions over time, the opposite of the convergence predicted by some growth models in which poorer economies catch up to richer ones. Pritchett's 1997 Divergence, Big Time documented the large and growing spread in income per person between rich and poor countries over the long run. The referent is a pattern in aggregate economic outcomes across places, measured in the dispersion of income or growth rates, not a differential operator, a difference between distributions, or the splitting of biological lineages.[4]

Unit

The gap in income per person or growth rate between economies, tracked over time

Recovery means

Not applicable

Measured by

Assessed by the dispersion of income per capita across countries or regions, or by whether growth rates rise or fall with initial income