Time reversibility

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:

$U_{-t} = \pi \, U_{t}\, \pi$

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 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. $\mathbf{p} \rightarrow \mathbf{-p}$ (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 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.

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 { τ_s }, for s = 1, ..., k for any k:^[1]

$p(x_t, x_{t+\tau_1}, x_{t+\tau_2}, \ldots , x_{t+\tau_k}) = p(x_{t'}, x_{t'-\tau_1}, x_{t'-\tau_2} , \ldots , x_{t'-\tau_k})$

A univariate stationary Gaussian process is time-reversible. Markov processes can only be reversible if their stationary distributions have the property of detailed balance:

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

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 Lévy processes,^[2] stochastic networks (Kelly's lemma),^[3] birth and death processes,^[4] Markov chains,^[5] and piecewise deterministic Markov processes.^[6]

Waves and optics

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 reverse of the initial excitation waveform being played at the initial source.

See also

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

Notes

1. Tong (1990), Section 4.4
2. Jacod, J.; Protter, P. (1988). "Time Reversal on Levy Processes". The Annals of Probability 16 (2): 620. doi:10.1214/aop/1176991776. JSTOR 2243828.
3. Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912.
4. Tanaka, H. (1989). "Time Reversal of Random Walks in One-Dimension". Tokyo Journal of Mathematics 12: 159. doi:10.3836/tjm/1270133555.
5. Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 0521633966.
6. 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

• 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