Boltzmann H-Theorem
Maxwell-Boltzmann equilibration — H(t) decreases monotonically
Parameters
N particles:
800
Temperature T:
1.0
Collision rate λ:
0.3
Initial distribution:
Monoenergetic (all same speed)
Bimodal (two peaks)
Uniform (flat)
Hot + Cold mix
Reset
Pause
H(t)
—
H_eq (Maxwell-B)
—
ΔH = H−H_eq
—
Time step
0
H-theorem (Boltzmann 1872):
H(t) = ∫ f(v) ln f(v) dv ≤ 0
dH/dt ≤ 0 always (from Boltzmann equation).
Equilibrium: f*(v) = √(m/2πkT)·exp(−mv²/2kT)
H is minimized at Maxwell-Boltzmann.
Loschmidt paradox: irreversibility from reversible dynamics via molecular chaos (Stosszahlansatz).