- Reversible dynamics
Mathematics
In
mathematics , adynamical system isinvertible 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.The dynamics are time-reversible if 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::
Any time-independent structures (for example critical points, or
attractor s) which the dynamics gives rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.Physics
In
physics , thelaws of motion ofclassical mechanics have the above property, if the operator π reverses the conjugate momenta of all the particles of the system, " p -> -p ". (T-symmetry ).In quantum mechanical systems, it turns out that the
weak nuclear force is not invariant under T-symmetry 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 theparity of the spatial co-ordinates (C-symmetry andP-symmetry ).Stochastic processes
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 "{ τs }", 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 .See also
*
Reversible process
*Reversible computing
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