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''state lumping, aggregation''

Different names for a class of procedures that allows one to reduce the state of the system of equations describing the time evolution of a random or deterministic [[Dynamical system|dynamical system]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102901.png" /> denotes the state space of the system and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102902.png" /> denotes the (semi-) group describing its time evolution, one is interested in a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102903.png" /> and a (semi-) group (cf. [[Semi-group of operators|Semi-group of operators]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102904.png" /> such that the diagram

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102905.png" /></td> </tr></table>

is commutative.

Sometimes there is an additional structure to be preserved. For example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102906.png" /> acts on a class of measures defined on some [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102907.png" /> and is induced by the action of a transition semi-group of a [[Markov process|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102908.png" /> living in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c1102909.png" />. In this case the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029010.png" /> is induced by a measurable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029011.png" /> which reduces the  "size"  of the Markov process. See [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] for more. One also wants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029012.png" /> to be Markovian.

In actual applications, heuristic reasoning leads to mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029013.png" /> for which the existence of a reduced time evolution is not clear, and one is confronted with the problem of producing some sort of approximate (or reduced) flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029014.png" /> with which to replace the unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029015.png" />. Another outstanding problem is to obtain standard kinetic equations from exact microscopic dynamics, that is, given the (unknown) microscopic time flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029016.png" /> and the known macroscopic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029017.png" />, the problem is to find the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029018.png" />. In the physico-chemical literature there are numerous instances where the choice of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029019.png" /> is dictated by the heuristics of the situation, but regrettably the existence of the reduced dynamics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110290/c11029020.png" /> is similarly inferred, and/or the nature and quality of the approximation is missing. Nevertheless, there have been numerous efforts to make the results rigorous. See [[#References|[a4]]], and [[#References|[a5]]] or [[#References|[a6]]]. One of the most successful examples of these techniques in projecting out macroscopic transport equations from the kinetic equation is the derivation of the [[Boltzmann equation|Boltzmann equation]] or [[Navier–Stokes equations|Navier–Stokes equations]] from the [[Euler equation|Euler equation]] [[#References|[a7]]]. See [[#References|[a8]]] for an example of interest to physicists.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Syski,   "Passage times for Markov chains" , IOS , Amsterdam  (1991)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Glover,   "Markov functions"  ''Ann. Inst. H. Poincaré'' , '''27'''  (1991)  pp. 221–238</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Buchholz,   "Exact and ordinary lumpability in Markov chains"  ''J. Appl. Probab.'' , '''31'''  (1994)  pp. 59–75</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Balescu,   "Equilibrium and nonequilibrium statistical mechanics" , Wiley  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Spohn,   "Kinetic equations from Hamiltonian dynamics: Markovian limits"  ''Rev. Mod. Phys.'' , '''53'''  (1980)  pp. 569–616</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V. Korolyuk,   A. Turbin,   "Mathematical foundations of the state lumping of large systems" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Ellis,   M Pinsky,   "The projection of Navier--Stokes equations from the Euler equations"  ''J. Math. Pures Appl.'' , '''54'''  (1975)  pp. 157–182</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Francowicz,   C. Jedrzejek,   "Solution of Friedrichs model through exact master equations"  ''Acta Phys. Pol.'' , '''A54'''  (1978)  pp. 123–129</TD></TR></table>