A Universal Thermodynamic Inequality: Scaling Relations Between Current, Activity, and Entropy Production

We derive a universal thermodynamic bound constraining directional transport in both discrete and continuous nonequilibrium systems. For continuous-time Markov jump processes and overdamped diffusions governed by Fokker--Planck equations, we prove the inequality $\frac{2 V(t)^2}{A(t)} \leq \dot{e}_p(t),$ linking the squared net velocity $V(t)$, entropy production rate $\dot{e}_p(t)$, and dynamical activity $A(t)$. This relation captures a fundamental trade-off between transport, dissipation, and fluctuation intensity, valid far from equilibrium and without detailed balance. In addition, we introduce dimensionless thermodynamic ratios that quantify dissipation asymmetry, entropy extraction, and relaxation. These scaling laws unify discrete and continuous stochastic thermodynamics and provide experimentally accessible constraints on transport efficiency in nanoscale machines and active systems.