Reversible dynamics

For reversibility in thermodynamics, see reversible process (thermodynamics).

For further meanings of reversibility, see reversibility (disambiguation).

In mathematics, a dynamical system is reversible if the forward evolution is one-to-one, not many-to-one; so that for every state there exists a well-defined reverse-time evolution operator.

Stochastic systems are said to be reversible if their stationary states have the property of detailed balance.

In physics, the laws of motion are reversible if a stronger requirement holds: there must exist a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state, and the forward-time evolution of another corresponding state.

The following operator equation then holds:

$U_{-t}=\pi \,U_{t}\,\pi$

Classical mechanics has this property, if the operator also π reverses the conjugate momenta of all the particles of the system, p -> -p . (T-symmetry).

In quantum mechanical systems, one must also reverse all the charges and the parity of the spatial co-ordinates (C-symmetry and P-symmetry).