mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, time-reversibility is the

property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, re ...

of a process whose governing rules remain unchanged when the direction of its sequence of actions 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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

under a change in the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

of time. A

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...

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

, a

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

is time-reversible if the forward evolution is one-to-one, so that for every state there exists a transformation (an

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

) 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_ = \pi \, U_\, \pi

Any time-independent structures (e.g. critical points or

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

s) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.

Physics

, the laws of motion of

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

exhibit time reversibility, as long as the operator π reverses the conjugate momenta of all the particles of the system, i.e.

\mathbf \rightarrow  \mathbf

(

T-symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the futur ...

). In

quantum mechanical Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...

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 In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...

and

P-symmetry In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' Three-dimensional space, spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial co ...

). This reversibility of several linked properties is known as

CPT symmetry Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and ...

Thermodynamic process Classical thermodynamics considers three main kinds of thermodynamic processes: (1) changes in a system, (2) cycles in a system, and (3) flow processes. (1) A Thermodynamic process is a process in which the thermodynamic state of a system is c ...

es can be reversible or irreversible, depending on the change in

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

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), 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 Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...

), 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).

Stochastic processes

is time-reversible if the joint probabilities of the forward and reverse state sequences are the same for all sets of time increments , for ''s'' = 1, ..., ''k'' for any ''k'': :

p(x_t, x_, x_, \ldots , x_) = p(x_, x_, x_ , \ldots , x_)

. A univariate stationary

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...

is time-reversible.

Markov process In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...

es can only be reversible if their stationary distributions have the property of

detailed balance The principle of detailed balance can be used in Kinetics (physics), kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at Thermodynamic equilibrium, equilibrium, each elem ...

: :

p(x_t=i,x_=j) = \,p(x_t=j,x_=i)

. Kolmogorov's criterion defines the condition for a

Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...

continuous-time Markov chain A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a ...

to be time-reversible. Time reversal of numerous classes of stochastic processes has been studied, including Lévy processes, stochastic networks ( Kelly's lemma), birth and death processes,

s, and piecewise deterministic Markov processes.

Waves and optics

Time reversal method works based on the linear reciprocity of the

wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...

, which states that the time reversed solution of a

is also a solution to the

since standard wave equations only contain even derivatives of the unknown variables. Thus, the

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

Notes

References