Characterize the reciprocity correction to the maximal chemical-potential imbalance governing self-propulsion

Characterize the correction term ∂_ΛΔμ0*(v) in the expansion Δμ0*(v) ≈ Δμ0*(v)|_{Λ=0} + (∂_ΛΔμ0*(v)|_{Λ=0}) Λ for the maximal chemical potential imbalance Δμ0*(v) = lim_{M→∞} Δμ0(v) that determines steady-state droplet speeds in one dimension. In particular, derive its functional form and curvature with respect to v for general droplet radius R and model parameters to establish rigorous criteria for whether the onset of condensate self-propulsion at Λ>0 is subcritical or supercritical.

Background

The onset and nature of droplet self-propulsion are determined by zeros of the chemical-potential imbalance Δμ0(v). For weak reciprocity, the authors expand the maximal imbalance Δμ0*(v) in Λ; the leading term at Λ=0 (nonreciprocal limit) is known and bounded, while the Λ-dependent correction controls whether the bifurcation is supercritical or subcritical.

The authors report they could not completely map out the Λ-correction term and only obtained partial evidence (e.g., upward curvature for small R in a tested parameter set). A complete characterization would provide general, parameter-independent criteria for bifurcation type and stability boundaries.