Inmathematics andphysics,time-reversibility is theproperty of a process whose governing rules remain unchanged when the direction of its sequence of actions is reversed.

Adeterministic process is time-reversible if the time-reversed process satisfies the samedynamic equations as the original process; in other words, the equations areinvariant orsymmetrical under a change in thesign of time. Astochastic 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.

Inmathematics, adynamical system is time-reversible if the forward evolution isone-to-one, so that for every state there exists a transformation (aninvolution) 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:

${\displaystyle U_{-t}=\pi U_t\pi }$

Any time-independent structures (e.g.critical points orattractors) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.

Inphysics, thelaws of motion ofclassical mechanics exhibit time reversibility, as long as the operator π reverses theconjugate momenta of all the particles of the system, i.e.${\displaystyle \mathbf{p}\to \mathbf{-}\mathbf{p}}$ (T-symmetry).

Inquantum mechanical systems, however, theweak 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 thecharges and theparity of the spatial co-ordinates (C-symmetry andP-symmetry). This reversibility of several linked properties is known asCPT symmetry.

Thermodynamic processes can bereversible orirreversible, depending on the change inentropy 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),^[1] 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 (orartificial intelligence), and themacroscopic properties (like entropy and temperature) of many-body system are onlydefined from thestatistics of the ensembles. When we talk about such macroscopic properties in thermodynamics, in certain cases, we can see irreversibility in thetime evolution of these quantities on a statistical level. Indeed, thesecond 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 astatistical impossibility for all practical considerations (seeCrooks fluctuation theorem).

Astochastic process is time-reversible if the joint probabilities of the forward and reverse state sequences are the same for all sets of time increments { τ_s }, fors = 1, ..., k for anyk:^[2]

${\displaystyle p(x_t,x_{t+{\tau }_1},x_{t+{\tau }_2},\dots ,x_{t+{\tau }_k})=p(x_{t^{\prime }},x_{t^{\prime }-{\tau }_1},x_{t^{\prime }-{\tau }_2},\dots ,x_{t^{\prime }-{\tau }_k})}$.

Aunivariate stationaryGaussian process is time-reversible.Markov processes are reversible if and only if their stationary distributions have the property ofdetailed balance:

${\displaystyle p(x_t=i,x_{t+1}=j)=p(x_t=j,x_{t+1}=i)}$.

Kolmogorov's criterion defines the condition for aMarkov chain orcontinuous-time Markov chain to be time-reversible.

Time reversal of numerous classes of stochastic processes has been studied, includingLévy processes,^[3]stochastic networks (Kelly's lemma),^[4]birth and death processes,^[5]Markov chains,^[6] andpiecewise deterministic Markov processes.^[7]

Time reversal method works based on the linear reciprocity of thewave equation, which states that the time reversed solution of awave equation is also a solution to thewave equation since standard wave equations only contain even derivatives of the unknown variables.^[8] Thus, thewave 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^[9] 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.

• T-symmetry
• Memorylessness
• Markov property
• Reversible computing

• 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

• Dynamical systems
• Time series
• Symmetry

• Articles with short description
• Short description is different from Wikidata