Time reversibility

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Lua error in package.lua at line 80: module 'strict' not found. Time reversibility is an attribute of some stochastic processes and some deterministic processes.

If a stochastic process is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which state arrived later.[clarification needed]

If a deterministic process is time reversible, then the time-reversed process satisfies the same dynamical equations as the original process (see reversible dynamics); in other words, the equations are invariant or symmetric under a change in the sign of time. Classical mechanics and optics are both time-reversible. Modern physics is not quite time-reversible; instead it exhibits a broader symmetry, CPT symmetry.[citation needed]

Time reversibility generally occurs when every process can be broken up into "elementary" sub-processes that undo each other's effects, and which have equal status, validity, likelihood, or rate. For example, in phylogenetics, a time-reversible nucleotide substitution model such as the generalised time reversible model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.

Time reversal in the field of acoustics and signal processing is a process in which the linear nature of waves is exploited to reverse a received signal; this signal is then re-emitted and a temporal compression occurs, resulting in a reverse of the initial excitation waveform being played at the initial source. Mathias Fink is credited with confirming acoustic time reversal in an experiment.

Stochastic processes

A formal definition of time-reversibility is stated by Tong[1] in the context of time-series. In general, a univariate stationary Gaussian process is time-reversible.

On the other hand, a process defined by a time-series model which computes values as a linear combination of past values and of present and past innovations (see autoregressive moving average model) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution (in which case the model is a Gaussian process).

A stationary Markov chain is reversible if the transition matrix {pij} and the stationary distribution {πj} satisfy

\pi_i p_{ij} =\pi_j p_{ji}, \,

for all i and j.[2] Such Markov chains provide examples of stochastic processes which are time-reversible but non-Gaussian.

Time reversal of numerous classes of stochastic processes have been studied including Lévy processes[3] stochastic networks (Kelly's lemma)[4] birth and death processes [5] Markov chains[6] and piecewise deterministic Markov processes.[7]

See also

Notes

  1. Tong(1990), Section 4.4
  2. Isham (1991), p 186
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References

  • Isham, V. (1991) "Modelling stochastic phenomena". In: Stochastic Theory and Modelling, Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. ISBN 978-0-412-30590-0.
  • Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP. ISBN 0-19-852300-9