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
A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...
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
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
is time-reversible if the forward evolution is
one-to-one
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual comm ...
, 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
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 ...
s) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.
Physics
In