Independence of nonlinearity from nondissipative linearity

Establish that the nonlinearity of the composite phenomenon B, defined as the sum of the dissipative phenomenon D, the nondissipative phenomenon N, and the impulsive transition term Σ_i δ[t_i, X_i] representing the boundary between dissipative and nondissipative states, is not related to the linearity of the nondissipative phenomenon N that satisfies superposition. Concretely, prove that the nonzero superposition limit arising in the composite phenomenon B due to the impulsive transition (as formalized after the derivation leading to Equation (26)) is independent of the linear superposition behavior of the standalone nondissipative phenomenon N (as characterized in Equation (21)).

Background

In Section 2, the paper introduces three phenomena: a dissipative phenomenon D, a nondissipative phenomenon N, and a composite phenomenon B that includes impulses at the boundary of the second law where the thermodynamic state transitions from dissipative to nondissipative. Using superposition arguments, both D and N independently satisfy linear superposition, while the composite phenomenon B exhibits a nonzero limit under vanishing perturbations due to the summation of delta-functions capturing instantaneous transitions, thereby indicating nonlinearity.

After establishing this nonlinearity for B, the authors explicitly posit a conjecture that the observed nonlinearity is not related to the linearity of N. Although they complete the proof that the composite phenomenon is nonlinear, they do not resolve whether this nonlinearity is independent of N’s linearity, leaving the conjecture open for rigorous verification.