Time reversibility

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:

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

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 π.

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

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]

${\displaystyle p(x_{t},x_{t+\tau _{1}},x_{t+\tau _{2}},\ldots ,x_{t+\tau _{k}})=p(x_{t'},x_{t'-\tau _{1}},x_{t'-\tau _{2}},\ldots ,x_{t'-\tau _{k}})}$

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

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

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

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.[7] 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[8] 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.

• 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