Time reversibility
Encyclopedia
Time reversibility is an attribute of some stochastic process
es and some deterministic processes.
If a stochastic process
is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which state arrived later.
If a deterministic process is time reversible, then the time-reversed process satisfies the same dynamical equations as the original process, AKA reversible dynamics
; the equations are invariant or symmetric under a change in the sign of time. Classical mechanics
and optics
are both time-reversible. Modern physics is not quite time-reversible; instead it exhibits a broader symmetry, CPT symmetry
.
Time reversibility generally occurs when, within a process, it can be broken up into sub-processes which undo the effects of each other. For example, in phylogenetics
, a time-reversible nucleotide substitution model such as the generalised time reversible model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.
Time Reversal
, specifically in the field of acoustics
, is a process by which the linearity of sound waves 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. Mathias Fink is credited with proving Acoustic Time Reversal by experiment.
is time-reversible. The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations (see Autoregressive moving average model
) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution (in which case the model is a Gaussian process).
A stationary Markov Chain
is reversible if the transition matrix {pij} and the stationary distribution {πj} satisfy
for all i and j. Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es and some deterministic processes.
If a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which state arrived later.
If a deterministic process is time reversible, then the time-reversed process satisfies the same dynamical equations as the original process, AKA reversible dynamics
Reversible dynamics
- Mathematics :In mathematics, a dynamical system is invertible if the forward evolution is one-to-one, not many-to-one; so that for every state there exists a well-defined reverse-time evolution operator....
; the equations are invariant or symmetric under a change in the sign of time. Classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
and optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
are both time-reversible. Modern physics is not quite time-reversible; instead it exhibits a broader symmetry, CPT symmetry
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
.
Time reversibility generally occurs when, within a process, it can be broken up into sub-processes which undo the effects of each other. For example, in phylogenetics
Phylogenetics
In biology, phylogenetics is the study of evolutionary relatedness among groups of organisms , which is discovered through molecular sequencing data and morphological data matrices...
, a time-reversible nucleotide substitution model such as the generalised time reversible model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.
Time Reversal
Time Reversal Signal Processing
Time Reversal Signal Processing is a technique for focusing waves. A Time Reversal Mirror is a device that can focus waves using the time reversal method. TRMs are also known as time reversal mirror arrays, as they are usually arrays of transducers, but they do not have to be arrays...
, specifically in the field of acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, is a process by which the linearity of sound waves 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. Mathias Fink is credited with proving Acoustic Time Reversal by experiment.
Stochastic processes
A formal definition of time-reversibility is stated by Tong in the context of time-series. In general, a Gaussian processGaussian process
In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
is time-reversible. The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations (see Autoregressive moving average model
Autoregressive moving average model
In statistics and signal processing, autoregressive–moving-average models, sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.Given a time series of data Xt, the ARMA model is a...
) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution (in which case the model is a Gaussian process).
A stationary Markov Chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
is reversible if the transition matrix {pij} and the stationary distribution {πj} satisfy
for all i and j. Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.