A 3-state Markov chain driven out of equilibrium displays persistent probability flux, violating detailed balance. The entropy production rate quantifies the thermodynamic cost of maintaining the cycle.
A 3-state system with rates {w₁₂, w₂₃, w₃₁} and reverse rates {w₂₁, w₃₂, w₁₃}. At equilibrium (w₁₂w₂₃w₃₁ = w₂₁w₃₂w₁₃), no net flux flows around the cycle. Breaking this Kolmogorov criterion drives a persistent current J. The entropy production rate σ = J·ln(w₁₂w₂₃w₃₁/w₂₁w₃₂w₁₃) ≥ 0 (second law). Particle trajectories trace out cycles with a preferred direction.