Quantum–Classical Transition Analogy of the Diffusion–Mobility Relation for Hopping and Band Transport Systems: Molecules to Materials

We propose a quantum-classical transition analogy for Einstein's diffusion-mobility (D/μ) relation to reveal electron-hole dynamics in both the degenerate and nondegenerate molecular and material systems. Here, one-to-one variation between differential entropy and chemical potential (Δη/Δhs) is the proposed analogy, which unifies quantum and classical transport. The degeneracy stabilization energy on D/μ decides whether the transport is quantum or classical; accordingly, the transformation occurs in the Navamani-Shockley diode equation.