From Phase Space to Non-Equilibrium Dynamics: Exploring Liouville’s Theorem and Its Implications - Mario J. Pinheiro (Applied Sciences)
We generalize Liouville’s theorem to incorporate entropy gradients in phase space, demonstrating that non-equilibrium systems exhibit compressible phase-space dynamics (<semantics>dρ/dt≠0</semantics>). 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 <semantics>S≈1.382</semantics> (error <semantics><5×10−5</semantics>) for <semantics>N=104</semantics> particles.