Told ya!
Fra said:
I checked some papers and stuff relating to to Boltzmann and also discussion by Einstein, and it seems they indeed used the term complexions and complexion number
People sometimes seem to assume I am making the whole thing up!
Fra said:
Anyway, since I'm not seeing it from the point of pure math, boltzman and einstein seems to be using (my interpretation from context) it pretty much synonymous or a slight generalisation to the notion of the set of distinguishable microstates consistent with the macrostate
It will certainly help to extract the pure math. Suppose we have some set X. A
finite partition of X, written \pi, is a decomposition of X into r disjoint
blocks A_j \subset X such that X = \cup_{j=1}^r A_j (perhaps needless to say, "disjoint" means A_j \cap A_k = \emptyset, \; j \neq k). You can say that the elements of X are
microstates each of which gives rise to a unique
macrostate, with the macrostates corresponding to the blocks of the partition. Or you can say that the blocks are the preimages of some function f:X \rightarrow \mathbold{R}; for example, a function assigning an "energy" to each element (taking a different value on each block). My point is that none of this need have anything to do with physics.
Fra said:
and the complexion number is the number of possible distinguishable microstates or "possibilities" consistent with the constraints or macrostate. This is like the boltzmanns entropy except I suppose the microstate is generalised beyond the mechanical analogue.
Right, if you follow up my citation of the expository paper by Brian Hayes in
American Scientist, and if you recall the fundamental orbit-stabilizer relation from elementary group theory (see any good book on group theory, for example Neumann, Stoy, and Thompson,
Groups and Geometry), you should be able to see that the natural action by S_n on an n-set X induces an action on the set of partitions of X, and the size of the orbit of \pi is then
\frac{n!}{n_1! \, n_2! \dots n_r!}
while the orbit itself is the coset space
S_n/\left( S_{n_1} \times S_{n_2} \dots \times S_{n_r} \right)
where the partition is X = \cup_{j=1}^r A_j with |A_j| = n_j. Here, the stabilizer of \pi is a subgroup of S_n which is isomorphic to the external direct product S_{n_1} \times S_{n_2} \dots \times S_{n_r}; in other words, the stabilizer is a
Young subgroup, a kind of "internal direct product" of subgroups which are themselves symmetric groups.
If we are thinking of a function and we take \pi to be the partition of X into preimages, then the stabilizer consists of those permutations which respect the partition, i.e. don't map any point to a point lying in another preimage.
Here, the complexion is the coset space; that is, the orbit of \pi[/itex] under the induced action by S_n on the partitions of X. (This action can carry a given partition into any other partition with the same block sizes.) In Boltzmann's work the complexion of \pi is the set of microstates corresponding to a given macrostate (e.g. having a given energy value), and to get an "subadditive" measure of the "variety" of the sizes of the blocks, we use the logarithm of the size of the complexion as the Boltzmann entropy. Indeed, for finite complexions the logarithm of the sizes of the complexions is always a generalized Boltzmann entropy. As I already remarked, entropies are essentially dimensions, so we should expect that in another famously tractable case of group actions, finite dimensional Lie groups of diffeomorphisms, the dimension of the cosets (which are finite dimensional coset spaces) will behave as entropies, and they do.<br />
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From this point of view, the Boltzmann entropy of a function (or if you prefer, of the partition into preimages induced by this function) measures the "asymmetry" of the partition. If you are familiar with Polya enumeration theory, you are already familiar with the idea that among geometric configurations consisting of k points in some finite space, the more symmetrical configurations have <i>smaller</i> orbits under the symmetry group, while the more asymmetric configurations have <i>larger</i> orbits. A good example is D_n acting on a necklace strung with n beads.<br />
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Incidently, from the perspective of the theory of G-sets (sets equipped with an action by some specific group G; this category is analgous to the category of R-modules where R is some specific ring), the induced action on partitions is remarkable in that it satisfies an analogue of the primitive element theorem: every intersection of stabilizers of individual partitions, G_\pi, \, G_{\pi^\prime}, \dots, is the stabilizer of some partition. In general, it is certainly not true that every intersection of point stabilizers G_x, \, G_{x^\prime}, \dots is the stabilizer of some point! <br />
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In general, given any action by some group G on some set X, there are many interesting "induced actions" one can consider, several of which have "regularizing" properties, in the sense of improving the behavior in some respect. In the induced action on partitions we altered the set being acted on, but we can also alter the group which is acting. For example, it is easy to define the <i>wreath product</i> of two actions (by G on X and by H on Y) and the result is an action by G \wr H on X \times Y, in which we take copies of Y indexed by X, thinking of the copies of Y as <i>fibers</i> sitting over the <i>base</i> X, and let copies of H independently permute the copies of Y and let G permute these fibers. <br />
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In the case of finite permutation groups this gives a direct connection between Polya enumeration and complexions. Namely: the <i>pattern index</i> enumerates the conjugacy classes of pointwise stabilizers but forgets the lattice structure. For example, let us compare the natural permutation actions by the transitive permutation groups of degree five. Then C_5, \, D_5 both give pattern index 1,1,2,2,1,1 while F_{5:4}, \, A_5, \, S_5 give pattern index 1,1,1,1,1,1. These numbers correspond to the stabilizer lattice; for example the stabilizer lattice for C_2 \wr D_5 (now writing fiber first, as appropriate for right actions), acting in the wreath product action on the subsets of our 5-set, starts with G=C_2 \wr D_5 at the top, which covers a conjugacy class of five index ten subgroups (the stabilizers of the five points), which covers two classes of index four sugroups (two distinct types of five pairs of points), which each cover two conjugacy classes of index two subgroups (two distinct types of five triples of points), which each covers a single conjugacy class of index two subgroups (five quadruples of points), which covers a congacy class consisting of a unique index two subgroup (the trivial subgroup). Note that 10 \cdot 4 \cdot 2 \cdot 2 \cdot 2 = 320 = | C_2 \wr D_5 |. In more complicated cases, one really requires a Hasse graph to depict the stabfix lattice (modulo conjugacy). One thing I find useful is to attach to the edge from C down to C^\prime not only the stabilizer subgroup index, but also a symbol m/n indicating that each subgroup belonging to class C contains m subgroups belonging to class C^\prime, while each subgroup belonging to class C^\prime is contained in n subgroups belonging to class C. These integer ratios (not neccessarily in lowest terms!) together with the Hasse diagram describe the incidence relations among the fixsets.<br />
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you thus get around the issue of defining probability directly in terms of measurements, instead one has to be able to infer _distinguishable states_ microstructure from input, or the constructs is still uncelar (ie any "hidden" microstructures are not acceptable). This is to the limit of my ignorance not trivial either, in particular as the complexity and memory sizes vary - THIS is the keys where I think many interesting things. Oddly enough, I think this is very interesting, even for an amateur.
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I am not sure I see what you are getting at.<br />
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Chris will probably get upset if this is not what he meant, so I explicitly declare that this does not necessarily have any relation to it. I still wear my ignorance with pride :)
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Better to just say "if I understand you correctly" over and over, until we agree that you do understand correctly what I said. (This policy is symmetrical, of course.)<br />
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But it is more or less what I referred to in post 15. I don't follow Chris all steps. He is a matematician, I am not. This alone explains the communication issues.
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Well, my posts have only been sketches. There is a lot to say so if I tried to fill in all the background for a general audience and write out all the arguments, I'd quickly have a book (my notes on this stuff are in fact more extensive than my notes on gtr).
