Time reversibility
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[edit]
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
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[edit]
Notes[edit]
- ^ Tong(1990), Section 4.4
- ^ Isham (1991), p 186
- ^ Jacod, J.; Protter, P. (1988). "Time Reversal on Levy Processes". The Annals of Probability 16 (2): 620. doi:10.1214/aop/1176991776. JSTOR 2243828.
- ^ Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912.
- ^ Tanaka, H. (1989). "Time Reversal of Random Walks in One-Dimension". Tokyo Journal of Mathematics 12: 159. doi:10.3836/tjm/1270133555.
- ^ Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 0521633966.
- ^ Löpker, A.; Palmowski, Z. (2013). "On time reversal of piecewise deterministic Markov processes" (PDF). Electronic Journal of Probability 18. doi:10.1214/EJP.v18-1958.
References[edit]
-
- 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
