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In mathematics, a dynamical system is invertible if the forward evolution is onetoone, not manytoone; so that for every state there exists a welldefined reversetime evolution operator.
The dynamics are timereversible if there exists a transformation (an involution) π which gives a onetoone mapping between the timereversed evolution of any one state, and the forwardtime evolution of another corresponding state, given by the operator equation:
Any timeindependent structures (for example critical points, or attractors) which the dynamics gives rise to must therefore either be selfsymmetrical or have symmetrical images under the involution π.
In physics, the laws of motion of classical mechanics have the above property, if the operator π reverses the conjugate momenta of all the particles of the system, p > p . (Tsymmetry).
In quantum mechanical systems, it turns out that the weak nuclear force is not invariant under Tsymmetry alone. If weak interactions are present, reversible dynamics are still possible, but only if the operator π also reverses the signs of all the charges, and the parity of the spatial coordinates (Csymmetry and Psymmetry).
A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for timereversed data from the same process. More formally, for all sets of time increments { τ_{s} }, where s = 1..k for any k, the joint probabilities
A simple consequence for Markov processes is that they can only be reversible if their stationary distributions have the property
This is called the property of detailed balance.
