View source for Time reversibility

{{short description|Type of physical or mathematical process}}
A mathematical or physical process is '''time-reversible''' if the dynamics of the process remain well-defined when the sequence of time-states is reversed.

A [[Deterministic system|deterministic process]] is time-reversible if the time-reversed process satisfies the same [[dynamic equation (disambiguation)|dynamic equation]]s as the original process; in other words, the equations are [[Invariant (mathematics)|invariant]] or [[Symmetry|symmetrical]] under a change in the [[Sign (mathematics)|sign]] of time. 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.

== Mathematics ==
In [[mathematics]], a [[dynamical system]] is time-reversible if the forward evolution is [[one-to-one function|one-to-one]], so that for every state there exists a transformation (an [[Involution (mathematics)|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:

:<math>U_{-t} = \pi \, U_{t}\, \pi</math>

Any time-independent structures (e.g. [[critical point (mathematics)|critical points]] or [[attractor]]s) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.

== Physics ==
In [[physics]], the [[Newton's laws of motion|laws of motion]] of [[classical mechanics]] exhibit time reversibility, as long as the operator π reverses the [[conjugate momentum|conjugate momenta]] of all the particles of the system, i.e. <math>\mathbf{p} \rightarrow  \mathbf{-p} </math> ([[T-symmetry]]).

In [[quantum mechanics|quantum mechanical]] systems, however, 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 [[charge (physics)|charges]] and the [[parity (physics)|parity]] of the spatial co-ordinates ([[C-symmetry]] and [[P-symmetry]]). This reversibility of several linked properties is known as [[CPT symmetry]].

[[Thermodynamic process]]es can be [[Reversible process (thermodynamics)|reversible]] or [[irreversible process|irreversible]], depending on the change in [[entropy]] during the process. Note, however, that the fundamental laws that underlie the thermodynamic processes are all time-reversible (classical laws of motion and laws of electrodynamics),<ref>[http://www.isepp.org/Pages/01-02%20Pages/Albert.html David Albert on ''Time and Chance'']</ref> which means that on the microscopic level, if one were to keep track of all the particles and all the degrees of freedom, the many-body system processes are all reversible; However, such analysis is beyond the capability of any human being (or [[artificial intelligence]]), and the [[Thermodynamic state|macroscopic properties]] (like entropy and temperature) of many-body system are only ''[[State variable|defined]]'' from the [[Thermodynamic equilibrium|statistics of the ensembles]]. When we talk about such macroscopic properties in thermodynamics, in certain cases, we can see irreversibility in the time evolution of these quantities on a statistical level. Indeed, the second law of thermodynamics predicates that the entropy of the entire universe must not decrease, not because the probability of that is zero, but because it is so unlikely that it is a ''[[Large deviations theory|statistical impossibility]]'' for all practical considerations (see [[Crooks fluctuation theorem]]).

== Stochastic processes ==
A [[stochastic process]] is time-reversible if the joint probabilities of the forward and reverse state sequences are the same for all sets of time increments {&nbsp;''τ''<sub>''s''</sub>&nbsp;}, for ''s'' = 1,&nbsp;...,&nbsp;''k'' for any ''k'':<ref>Tong (1990), Section 4.4</ref>
: <math>p(x_t, x_{t+\tau_1}, x_{t+\tau_2}, \ldots , x_{t+\tau_k}) = p(x_{t'}, x_{t'-\tau_1}, x_{t'-\tau_2} , \ldots , x_{t'-\tau_k})</math>.

A univariate stationary [[Gaussian process]] is time-reversible. [[Markov process]]es can only be reversible if their stationary distributions have the property of [[detailed balance]]:
:<math>p(x_t=i,x_{t+1}=j) = \,p(x_t=j,x_{t+1}=i)</math>.

[[Kolmogorov's criterion]] defines the condition for a [[Markov chain]] or [[continuous-time Markov chain]] to be time-reversible.

Time reversal of numerous classes of stochastic processes has been studied, including [[Lévy processes]],<ref>{{Cite journal | last1 = Jacod | first1 = J. | last2 = Protter | first2 = P. | doi = 10.1214/aop/1176991776 | title = Time Reversal on Levy Processes | journal = The Annals of Probability | volume = 16 | issue = 2 | pages = 620 | year = 1988 | jstor = 2243828| doi-access = free }}</ref> [[stochastic network]]s ([[Kelly's lemma]]),<ref>{{Cite journal | last1 = Kelly | first1 = F. P. | author-link1 = Frank Kelly (mathematician)| title = Networks of Queues | journal = Advances in Applied Probability | volume = 8 | issue = 2 | pages = 416–432 | doi = 10.2307/1425912 | jstor = 1425912| year = 1976 | s2cid = 204177645 }}</ref> [[birth and death processes]],<ref>{{Cite journal | last1 = Tanaka | first1 = H. | doi = 10.3836/tjm/1270133555 | title = Time Reversal of Random Walks in One-Dimension | journal = Tokyo Journal of Mathematics | volume = 12 | pages = 159–174 | year = 1989 | doi-access = free }}</ref> [[Markov chain]]s,<ref>{{cite book | title = Markov Chains | first = J. R. | last= Norris | author-link = James R. Norris | publisher = Cambridge University Press | year =1998 | isbn = 978-0521633963}}</ref> and [[piecewise deterministic Markov processes]].<ref>{{Cite journal | last1 = Löpker | first1 = A. | last2 = Palmowski | first2 = Z. | doi = 10.1214/EJP.v18-1958 | title = On time reversal of piecewise deterministic Markov processes | journal = Electronic Journal of Probability | volume = 18 | year = 2013 | arxiv = 1110.3813| s2cid = 1453859 }}</ref>

==Waves and optics==
Time reversal method works based on the linear reciprocity of the [[wave equation]], which states that the time reversed solution of a [[wave equation]] is also a solution to the [[wave equation]] since standard wave equations only contain even derivatives of the unknown variables.<ref>{{cite journal|last1=Parvasi|first1=Seyed Mohammad|last2=Ho|first2=Siu Chun Michael|last3=Kong|first3=Qingzhao|last4=Mousavi|first4=Reza|last5=Song|first5=Gangbing|title=Real time bolt preload monitoring using piezoceramic transducers and time reversal technique—a numerical study with experimental verification|journal=Smart Materials and Structures|date=19 July 2016|volume=25|issue=8|pages=085015|doi=10.1088/0964-1726/25/8/085015|language=en|issn=0964-1726|bibcode=2016SMaS...25h5015P|s2cid=113510522 }}</ref> Thus, the [[wave equation]] is symmetrical under time reversal, so the time reversal of any valid solution is also a solution. This means that a wave's path through space is valid when travelled in either direction.

[[Time reversal signal processing]]<ref>Anderson, B. E., M. Griffa, C. Larmat, T.J. Ulrich, and P.A. Johnson, "Time reversal", ''Acoust. Today'', 4 (1), 5-16 (2008). https://acousticstoday.org/time-reversal-brian-e-anderson/</ref> is a process in which this property is used to reverse a received signal; this signal is then re-emitted and a temporal compression occurs, resulting in a reversal of the initial excitation waveform being played at the initial source.

==See also==
*[[T-symmetry]]
*[[Memorylessness]]
*[[Markov property]]
*[[Reversible computing]]

==Notes==
{{Reflist}}

==References==

:*Isham, V. (1991) "Modelling stochastic phenomena". In: ''Stochastic Theory and Modelling'', Hinkley, DV., Reid, N., [[Joyce Snell|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}}

[[Category:Dynamical systems]]
[[Category:Time series]]
[[Category:Symmetry]]