Recurrence relations recovered from Hershkowitz's Langmuir probe sketches: an exact logistic map for the double probe and discrete signatures of circuit effects

The schematic current–voltage characteristics in Hershkowitz’s chapter ‘How Langmuir probes work’ remain a standard reference for how beams, collector geometry, plasma-potential fluctuations and the probe circuit distort a Langmuir probe trace. Plotted from closed forms, they are also a noiseless test of what a curve alone reveals. The chapter’s 15 figures were split into 22 numbered panels, digitized with intensity-weighted sub-pixel centroids (repeatability 0.1–0.2 pixel) and resampled on uniform bias grids. Each panel then reduces to a recurrence identified by the quantity it conserves from one step to the next. The retarding region of a single probe is a multiplicative map with a hard ceiling; orbital-motion-limited spheres and cylinders are additive maps in the current and its square; cold ions entering a sheath are additive in the inverse squared density; and the ideal double probe is an exact Beverton–Holt map, the exact sampling of a logistic curve, reproduced to 0.2% of its swing.Neither exact map can period-double, so a cascade read from a probe trace belongs to a forward-Euler integrator with a bias step above twice the electron temperature. Mixtures over species or time have no single map and are recognized by a U-shaped per-capita rate. The recovered parameters reproduce the stated beam energies, spreads, area ratios and fluctuation amplitudes to 1–4% and expose three drafting discrepancies, including ion curves drawn at the reciprocals of their labelled Mach numbers. Circuit effects leave their own signatures: a series resistance makes the local temperature rise with current, a load line acts as a contracting relaxation map, and a low-impedance alternating-current path performs an exact convolution. In the chapter’s laboratory trace the local temperature falls from about 5 V to 1.8 V as the current rises, which excludes a series resistance. The drawings support two to five bias steps per e-fold.