power_law_bin_average: Bin average of a power law over geometric bins

In mizer: Dynamic Multi-Species Size Spectrum Modelling

Bin average of a power law over geometric bins

Description

Computes the exact average of the power law w^d over each bin [w_j, w_{j+1}], i.e.

\overline{w^d}_j = \frac{1}{\Delta w_j}\int_{w_j}^{w_{j+1}} w^d\, dw.

Usage

power_law_bin_average(w, dw, d, w_max = Inf)

Arguments

w	Numeric vector of left bin edges w_j.
dw	Numeric vector of bin widths \Delta w_j (same length as w).
d	Single numeric exponent of the power law.
w_max	Optional upper cutoff. The power law is taken to be zero above w_max, with the straddling bin getting the partial average. Defaults to Inf (no cutoff), in which case the result is identical to the uncut formula.

Details

The integral has a closed form, so the result is exact (not merely second order):

\overline{w^d}_j = \frac{w_{j+1}^{d+1} - w_j^{d+1}}{(d+1)\,\Delta w_j}, \quad d \neq -1,

\overline{w^d}_j = \frac{\ln(w_{j+1}/w_j)}{\Delta w_j}, \quad d = -1.

This is used by the bin-averaged (second-order) code paths that need the average of a power-law rate over each bin, for example setExtMort() and the resource capacity and rate in setResource(). The grid does not need to be geometric; only the left bin edges w and the bin widths dw are used, with w_{j+1} = w_j + \Delta w_j.

An optional upper cutoff w_max handles a knife-edge truncation of the power law (for example the resource carrying capacity, which is \kappa w^{-\lambda} below w_pp_cutoff and zero above it). The bin straddling the cutoff then receives the partial bin-average — the power-law average over the part of the bin below w_max, divided by the full bin width — and bins entirely above the cutoff get zero.

Value

A numeric vector (same length as w) of bin averages of w^d (truncated at w_max when supplied).

mizer documentation built on Aug. 24, 2026, 9:08 a.m.