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.

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.

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. (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. 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 (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).

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*:^{ [2] }

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,^{ [3] } stochastic networks (Kelly's lemma),^{ [4] } birth and death processes,^{ [5] } Markov chains,^{ [6] } and piecewise deterministic Markov processes.^{ [7] }

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.^{ [8] } 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 ^{ [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.

- ↑ 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/

In physics, a **Langevin equation** is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.

In mathematics, a **stochastic matrix** is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a **probability matrix**, **transition matrix**, **substitution matrix**, or **Markov matrix**. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices:

In information theory, the **asymptotic equipartition property** (**AEP**) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of data compression.

In simple terms, the Markov property states that the future states of a stochastic process are influenced only by the present, not the past, meaning that the past can be disregarded once the present is known.

In mathematics and statistics, a **stationary process** is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles, but overall it does not trend up nor down.

The **fluctuation theorem** (**FT**), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously *decrease*; the fluctuation theorem precisely quantifies this probability.

In physics, chemistry and related fields, **master equations** are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations – over time – of the probabilities that the system occupies each of the different states.

In probability theory, in particular in the study of stochastic processes, a **stopping time** is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a **stopping rule**, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time.

The principle of **detailed balance** can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.

A **partially observable Markov decision process** (**POMDP**) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics are determined by an MDP, but the agent cannot directly observe the underlying state. Instead, it must maintain a sensor model and the underlying MDP. Unlike the policy function in MDP which maps the underlying states to the actions, POMDP's policy is a mapping from the history of observations to the actions.

A **cyclostationary process** is a signal having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year.

In probability, a **discrete-time Markov chain** (**DTMC**) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past. For instance, a machine may have two states, *A* and *E*. When it is in state *A*, there is a 40% chance of it moving to state *E* and a 60% chance of it remaining in state *A*. When it is in state *E*, there is a 70% chance of it moving to *A* and a 30% chance of it staying in *E*. The sequence of states of the machine is a Markov chain. If we denote the chain by then is the state which the machine starts in and is the random variable describing its state after 10 transitions. The process continues forever, indexed by the natural numbers.

In probability theory, the **Gillespie algorithm** generates a statistically correct trajectory of a stochastic equation system for which the reaction rates are known. It was created by Joseph L. Doob and others, presented by Dan Gillespie in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power. As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of reagents is low and keeping track of the position and behaviour of individual molecules is computationally feasible. Mathematically, it is a variant of a dynamic Monte Carlo method and similar to the kinetic Monte Carlo methods. It is used heavily in computational systems biology.

The **Gittins index** is a measure of the reward that can be achieved through a given stochastic process with certain properties, namely: the process has an ultimate termination state and evolves with an option, at each intermediate state, of terminating. Upon terminating at a given state, the reward achieved is the sum of the probabilistic expected rewards associated with every state from the actual terminating state to the ultimate terminal state, inclusive. The index is a real scalar.

In queueing theory, a discipline within the mathematical theory of probability, an **M/D/1 queue** represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue.

In probability theory, **Kelly's lemma** states that for a stationary continuous-time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the forward-time process. The theorem is named after Frank Kelly.

In mathematics, a **continuous-time random walk** (**CTRW**) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. More generally it can be seen to be a special case of a Markov renewal process.

**Supersymmetric theory of stochastic dynamics** or **stochastics** (**STS**) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.

In stochastic processes, **Kramers–Moyal expansion** refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal.

The **separation principle** is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system

- 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

- Isham, V. (1991) "Modelling stochastic phenomena". In:

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