Reversibility and irreversibility

1. The thermodynamic description, which includes the principle that the entropy S of a thermally isolated system never decreases with time : if t1 < t2 then S(t1) ≤ S(t2) (1) 2. Macroscopic descriptions, such as the heat equation ∂e/∂t = K∇T (2) 3. Descriptions by kinetic equations such as Boltzmann’s equation ∂f/∂t+ v · ∇xf = I(f) (3) where I(f) is an integral expression representing the effect of collisions on the distribution function f(v,x, t) of particle velocities v at position x. 4. Microscopic representations as a system of particles obeying Newton’s equations of motion mdxi/dt = −∇iV (x1, · · · ,xN ), (4) where V is the potential energy, or Schrödinger’s equation for the same system