The Second Law of Thermodynamics as a Geometric Identity: A Contact-Geometric Derivation of Entropy Monotonicity

The second law of thermodynamics—the monotonic increase of entropy in isolated systems—has stood for over 150 years as an axiom of physics. Every standard formulation, from Clausius and Kelvin to Carathéodory and Lieb–Yngvason, postulates irreversibility rather than deriving it. We show that the second law follows from the contact structure of thermodynamic phase space combined with a sign condition on the dissipation rate that encodes thermodynamic stability. For any system whose equilibrium states form a Legendre submanifold of a contact manifold (T, η), the dissipation identity dH/dt = −λH—a purely geometric consequence of the contact Hamilton equations and the antisymmetry of dη—forces V = H² to serve as a Lyapunov function whose monotonic decay guarantees entropy monotonicity. We prove three theorems: (1) entropy monotonicity dS/dt ≥ 0 with an explicit derivation from the contact structure equations, (2) characterization of reversible processes as those tangent to the Legendre submanifold, and (3) Lyapunov stability of equilibrium via LaSalle’s invariance principle. We show that the sign condition λ = ∂H/∂S ≥ 0 is strictly weaker than the second law itself and equivalent to standard thermodynamic stability (concavity of entropy). The construction is verified analytically and numerically for ideal gases (including a detailed free expansion simulation), van der Waals systems, coupled heat reservoirs, and chemical reactions. The symplectification of the contact structure reveals macroscopic irreversibility as the projection of time-reversible Hamiltonian dynamics onto a subspace where the contact structure breaks the symmetry—providing a precise geometric account of the emergence of an arrow of time. All proofs use standard results from contact topology and require no statistical or probabilistic assumptions.