# 时间可逆性

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 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.

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:

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:

= = 数学 = = 在数学中，如果正向演化是一对一的，那么动力系统是可逆的，因此对于每一个状态，都存在一个变换(对合) π，它给出了一个一对一的映射，在任何一个状态的时间反向演化和另一个相应状态的向前时间演化之间，由算符方程给出:

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

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. $\displaystyle{ \mathbf{p} \rightarrow \mathbf{-p} }$ (T-symmetry).

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).

= = 物理学 = = 在物理学中，经典力学的运动定律具有时间可逆性，只要运算符 π 反转系统中所有粒子的共轭动量，即:。(t 对称)。

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.

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.

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

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:Tong (1990), Section 4.4

= = = 随机过程 = = = a 随机过程是时间可逆的，如果正向和反向状态序列的联合概率对于所有的时间增量集{ τs }是相同的，对于 s = 1，... ，k 对于任意 k: Tong (1990) ，第4.4节

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

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

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.

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 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.[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 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[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.

Time reversal signal processingAnderson, 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/ 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

# = = =

• t 对称
• 无记忆
• 马尔可夫性
• 可逆计算

## 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: 159–174. doi:10.3836/tjm/1270133555.
5. Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 978-0521633963.
6. 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.
7. 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 (in English). 25 (8): 085015. Bibcode:2016SMaS...25h5015P. doi:10.1088/0964-1726/25/8/085015. ISSN 0964-1726.
8. 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/

# = 参考文献 =

• Isham, V. (1991) "Modelling stochastic phenomena". In: Stochastic Theory and Modelling, Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. .
• Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP.
• Isham, V. (1991) "Modelling stochastic phenomena". In: Stochastic Theory and Modelling, Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. .
• Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP.

• 伊沙姆(1991)“建模随机现象”。发表于: 《随机理论与建模》 ，欣克利，德国，里德，新泽西，斯内尔，e.j。(Eds).查普曼和霍尔。.:
• Tong，h. (1990)非线性时间序列: 一个动力系统的方法。牛津大学。

Category:Dynamical systems Category:Time series Category:Symmetry

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