Any of these assumptions may be violated. For example, Boltzmann's collision can be represented as A v + A w ⟶ ⟶ A v ′ + A w ′ , where A v is a particle with velocity v. Under time reversal A v transforms into A − − v . Therefore, the collision is transformed into the reverse collision by the PT transformation, where P is the space inversion and T is the time reversal. Detailed balance for Boltzmann's equation requires PT -invariance of collisions' dynamics, not just T -invariance. Indeed, after the time reversal the collision A v + A w ⟶ ⟶ A v ′ + A w ′ , transforms into A − − v ′ + A − − w ′ ⟶ ⟶ A − − v + A − − w . For the detailed balance we need transformation into A v ′ + A w ′ ⟶ ⟶ A v + A w . For this purpose, we need to apply additionally the space reversal P. Therefore, for the detailed balance in Boltzmann's equation not T -invariance but PT -invariance is needed.