From Phase Space to Non-Equilibrium Dynamics: Exploring Liouville’s Theorem and Its Implications

We generalize Liouville’s theorem to incorporate entropy gradients in phase space, demonstrating that non-equilibrium systems exhibit compressible phase-space dynamics (dρ/dt≠0). This framework bridges Hamiltonian mechanics and thermodynamics, with applications in beam stacking, stochastic cooling, and quantum thermalization. Numerical simulations validate the theory, showing entropy stabilization at S≈1.382 (error <5×10−5) for N=104 particles.