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Time reversibility is an attribute of some stochastic processes 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 (see reversible dynamics); in other words, 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 every process can be broken up into "elementary" sub-processes that undo each other's effects, and which have equal status, validity, likelihood, or rate. 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 in the field of acoustics and signal processing is a process in which the linear nature of waves is exploited 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 confirming acoustic time reversal in an experiment. ==Stochastic processes== A formal definition of time-reversibility is stated by Tong〔Tong(1990), Section 4.4〕 in the context of time-series. In general, a univariate stationary Gaussian process is time-reversible. On the other hand, a process defined by a time-series model which computes 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 and the stationary distribution satisfy : for all ''i'' and ''j''.〔Isham (1991), p 186〕 Such Markov chains provide examples of stochastic processes which are time-reversible but non-Gaussian. Time reversal of numerous classes of stochastic processes have been studied including Lévy processes stochastic networks (Kelly's lemma) birth and death processes Markov chains and piecewise deterministic Markov processes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Time reversibility」の詳細全文を読む スポンサード リンク
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