Wednesday, December 14, 2016

dividing systems

This will be a bit different, and may well not be terribly original, but I want to think about some epistemological issues (perhaps with some practical values) associated with dividing up complex systems into parts.

In particular, suppose a complex system is parameterized by a (large) set of coordinates, and the equations of motion are such as to minimize an expression in the coordinates and their first time derivatives as is typical of Lagrangians in physics; I'll simply refer to it as the Lagrangian going forward, though in some contexts it might be a thermodynamic potential or a utility function or the like.  A good division amounts to separating the coordinates into three subsets, say A, B, and C, where A and C at least are nonempty and the (disjoint) union of the three is the full set of coordinates of the system.  Given values of the coordinates (and their derivatives) in A and B, we can calculate the optimal value of the Lagrangian by optimizing over values that C can take, and similarly given B and C I can optimize over A, and I get effective Lagrangians for (A and B) and (B and C) respectively.  Where this works best, though, is where the optimizing coordinates in C (in the first case) or A (in the second) depend only on the coordinates in B; conditional on B, A and C are independent of each other.  This works even better if B is fairly small compared to both A and C, and might in that case even be quite useful if the conditional independence is only approximate.

In general there will be many ways to write the Lagrangian as LA+LB+LC+LAB+LBC+LAC+LABC, with each term depending only on coordinates in that set, but it will be particularly useful to write a Lagrangian this way if the last two terms are small or zero. If we are "optimizing out" the C coordinates, the effective Lagrangian for A and B is LA+LB+LAB plus what we get from optimizing LC+LBC+LAC+LABC; this will depend only on B if the terms with both A and C are absent.  Thus on some level a good decomposition of a system is one in which the Lagrangian can be written as LAB+LB+LBC, where I've absorbed previous LA and LC terms into the first and last terms; for given evolutions of B variables, the A variables will optimize LAB and the C variables will optimize LBC and these two optimizations can be done separately.

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