Coarse graining

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.

If $ S $ denotes the state space of the system and $ {T ( t ) } : S \rightarrow S $ denotes the (semi-) group describing its time evolution, one is interested in a mapping $ R : S \rightarrow {S ^ \prime } $ and a (semi-) group (cf. Semi-group of operators) $ {T ^ \prime ( t ) } : {S ^ \prime } \rightarrow {S ^ \prime } $ such that the diagram

$$ \begin{array}{ccc} S & \rightarrow & S \\ {} _ {R} \downarrow &{} &\downarrow _ {R} \\ S ^ \prime & \rightarrow &S ^ \prime \\ \end{array} $$

is commutative.

Sometimes there is an additional structure to be preserved. For example, when $ T ( t ) $ acts on a class of measures defined on some measurable space $ E $ and is induced by the action of a transition semi-group of a Markov process $ X $ living in $ E $. In this case the mapping $ R $ is induced by a measurable mapping $ A : E \rightarrow {E ^ \prime } $ which reduces the "size" of the Markov process. See [a1], [a2], [a3] for more. One also wants $ T ^ \prime ( t ) $ to be Markovian.

In actual applications, heuristic reasoning leads to mappings $ R : S \rightarrow {S ^ \prime } $ 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 $ {Q ( t ) } : {S ^ \prime } \rightarrow {S ^ \prime } $ with which to replace the unknown $ T ^ \prime ( t ) $. Another outstanding problem is to obtain standard kinetic equations from exact microscopic dynamics, that is, given the (unknown) microscopic time flow $ T ( t ) $ and the known macroscopic $ T ^ \prime ( t ) $, the problem is to find the mapping $ R $. In the physico-chemical literature there are numerous instances where the choice of the mapping $ R $ is dictated by the heuristics of the situation, but regrettably the existence of the reduced dynamics $ T ^ \prime ( t ) $ 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 [a4], and [a5] or [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 or Navier–Stokes equations from the Euler equation [a7]. See [a8] for an example of interest to physicists.

References

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