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.

A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process. More formally, for all sets of time increments {&tau_s}, where s = 1..k for any k, the joint probabilities $p(x_{t},x_{t+\tau _{1}},x_{t+\tau _{2}}..x_{t+\tau _{k}}=p(x_{t'},x_{t'-\tau _{1}},x_{t'-\tau _{2}}..x_{t'-\tau _{k}}$ A simple consequence for Markov processes is that they can only be reversible if their stationary distributions have the property p(x_t=i,x_t+1=j) = p(x_t=j,x_t+1=i). This is called the property of detailed balance.

In physics, laws of motion which are reversible are found to have the following property: there exists 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, given by the operator equation:

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

Classical mechanics has this property, if the operator 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).