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 process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the 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, so that for every state 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 (e.g. critical points or attractors) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.
Physics
In physics, the laws of motion of classical mechanics exhibit time reversibility, as long as the operator π reverses the conjugate momenta of all the particles of the system, i.e. 
In 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 charges and the parity of the spatial co-ordinates (C-symmetry and P-symmetry). This reversibility of several linked properties is known as CPT symmetry.
Thermodynamic processes can be reversible or 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), 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 macroscopic properties (like entropy and temperature) of many-body system are only defined from the 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 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 { τs }, for s = 1, ..., k for any k:
.
A univariate stationary Gaussian process is time-reversible. Markov processes can only be reversible if their stationary distributions have the property of detailed balance:
.
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,stochastic networks (Kelly's lemma),birth and death processes,Markov chains, and piecewise deterministic Markov processes.
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. 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 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
- David Albert on Time and Chance
- Tong (1990), Section 4.4
- 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. S2CID 204177645.
- Tanaka, H. (1989). "Time Reversal of Random Walks in One-Dimension". Tokyo Journal of Mathematics. 12: 159–174. doi:10.3836/tjm/1270133555.
- Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 978-0521633963.
- Löpker, A.; Palmowski, Z. (2013). "On time reversal of piecewise deterministic Markov processes". Electronic Journal of Probability. 18. arXiv:1110.3813. doi:10.1214/EJP.v18-1958. S2CID 1453859.
- Parvasi, Seyed Mohammad; Ho, Siu Chun Michael; Kong, Qingzhao; Mousavi, Reza; Song, Gangbing (19 July 2016). "Real time bolt preload monitoring using piezoceramic transducers and time reversal technique—a numerical study with experimental verification". Smart Materials and Structures. 25 (8): 085015. Bibcode:2016SMaS...25h5015P. doi:10.1088/0964-1726/25/8/085015. ISSN 0964-1726. S2CID 113510522.
- 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/
References
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 process is time reversible if the time reversed process satisfies the same dynamic equations as the original process in other words the equations are invariant or symmetrical under a change in the 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 MathematicsIn mathematics a dynamical system is time reversible if the forward evolution is one to one so that for every state there exists a transformation an involution p 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 U t pUtp displaystyle U t pi U t pi Any time independent structures e g critical points or attractors which the dynamics give rise to must therefore either be self symmetrical or have symmetrical images under the involution p PhysicsIn physics the laws of motion of classical mechanics exhibit time reversibility as long as the operator p reverses the conjugate momenta of all the particles of the system i e p p displaystyle mathbf p rightarrow mathbf p T symmetry In 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 p also reverses the signs of all the charges and the parity of the spatial co ordinates C symmetry and P symmetry This reversibility of several linked properties is known as CPT symmetry Thermodynamic processes can be reversible or 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 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 macroscopic properties like entropy and temperature of many body system are only defined from the 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 statistical impossibility for all practical considerations see Crooks fluctuation theorem Stochastic processesA 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 ts for s 1 k for any k p xt xt t1 xt t2 xt tk p xt xt t1 xt t2 xt tk displaystyle 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 A univariate stationary Gaussian process is time reversible Markov processes can only be reversible if their stationary distributions have the property of detailed balance p xt i xt 1 j p xt j xt 1 i displaystyle p x t i x t 1 j p x t j x t 1 i 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 Levy processes stochastic networks Kelly s lemma birth and death processes Markov chains and piecewise deterministic Markov processes Waves and opticsTime 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 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 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 alsoT symmetry Memorylessness Markov property Reversible computingNotesDavid Albert on Time and Chance Tong 1990 Section 4 4 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 S2CID 204177645 Tanaka H 1989 Time Reversal of Random Walks in One Dimension Tokyo Journal of Mathematics 12 159 174 doi 10 3836 tjm 1270133555 Norris J R 1998 Markov Chains Cambridge University Press ISBN 978 0521633963 Lopker A Palmowski Z 2013 On time reversal of piecewise deterministic Markov processes Electronic Journal of Probability 18 arXiv 1110 3813 doi 10 1214 EJP v18 1958 S2CID 1453859 Parvasi Seyed Mohammad Ho Siu Chun Michael Kong Qingzhao Mousavi Reza Song Gangbing 19 July 2016 Real time bolt preload monitoring using piezoceramic transducers and time reversal technique a numerical study with experimental verification Smart Materials and Structures 25 8 085015 Bibcode 2016SMaS 25h5015P doi 10 1088 0964 1726 25 8 085015 ISSN 0964 1726 S2CID 113510522 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 ReferencesIsham 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
