Time evolution
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'''Time evolution''' is the change of state brought about by the passage of [[time]], applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be [[discrete time|discrete]] or even [[wiktionary:finite|finite]]. In [[classical physics]], time evolution of a collection of [[rigid body|rigid bodies]] is governed by the principles of [[classical mechanics]]. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by [[Newton's laws of motion]]. These principles can also be equivalently expressed more abstractly by [[Hamiltonian mechanics]] or [[Lagrangian mechanics]].
The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a [[Turing machine]] can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is discrete.
Stateful systems often have dual descriptions in terms of states or in terms of [[observable]] values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in [[quantum mechanics]] where the [[Schrödinger picture]] and [[Heisenberg picture]] are (mostly) equivalent descriptions of time evolution.
== Time evolution operators ==
Consider a system with state space ''X'' for which evolution is [[deterministic]] and [[reversible]]. For concreteness let us also suppose time is a parameter that ranges over the set of [[real number]]s '''R'''. Then time evolution is given by a family of bijective state transformations
:<math> \operatorname{F}_{t, s}: X \rightarrow X \quad \forall t,s \in \mathbb{R} </math>
F<sub>''t'', ''s''</sub>(''x'') is the state of the system at time ''t'', whose state at time ''s'' is ''x''. The following identity holds
:<math> \operatorname{F}_{u, t} (\operatorname{F}_{t, s} (x)) = \operatorname{F}_{u, s}(x). </math>
To see why this is true, suppose ''x'' ∈ ''X'' is the state at time ''s''. Then by the definition of F, F<sub>''t'', ''s''</sub>(''x'') is the state of the system at time ''t'' and consequently applying the definition once more, F<sub>''u'', ''t''</sub>(F<sub>''t'', ''s''</sub>(''x'')) is the state at time ''u''. But this is also F<sub>''u'', ''s''</sub>(''x'').
In some contexts in mathematical physics, the mappings F<sub>''t'', ''s''</sub> are called '''propagation operators''' or simply '''propagators'''. In [[classical mechanics]], the propagators or functions operate on the [[phase space]] of a physical system. In [[quantum mechanics]], the propagators are usually [[unitary operator]]s on a [[Hilbert space]]. The propagators can be expressed as [[time-ordered]] exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the [[S-matrix|scattering matrix]].
A state space with a distinguished propagator is also called a [[dynamical system]].
To say time evolution is '''homogeneous''' means that
:<math> \operatorname{F}_{u, t} = \operatorname{F}_{u - t,0} \quad \forall u,t \in \mathbb{R}.</math>
In the case of a homogeneous system, the mappings G<sub>''t''</sub> = F<sub>''t'',0</sub> form a one-parameter [[group (mathematics)|group]] of transformations of ''X'', that is
:<math> \operatorname{G}_{t+s} = \operatorname{G}_{t}\operatorname{G}_{s}. </math>
'''Non-reversibility'''. For non-reversible systems, the propagation operators F<sub>''t'', ''s''</sub> are defined whenever ''t'' ≥ ''s'' and satisfy the propagation identity
:<math> \operatorname{F}_{u, t} (\operatorname{F}_{t, s} (x)) = \operatorname{F}_{u, s}(x). \quad u \geq t \geq s. </math>
In the homogeneous case the propagators are exponentials of the Hamiltonian.
== References ==
{{Unreferenced|date=May 2008}}
[[Category:Dynamical systems]]
[[de:Zeitentwicklung]]
[[fr:Opérateur d'évolution]]