By Richard Courant, Charles de Prima, John R. Knudsen
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Extra info for Advanced methods in applied mathematics; lecture course (1941)
Several particulars distinguish this approach. First is a draconian simplification which seeks to employ information theory concepts only as they directly relate to the basic limit theorems of the subject. That is, message uncertainty and information source uncertainty are interesting only because they obey the Coding, Source Coding, Rate Distortion, and related theorems. 'Information Theory' treatments which do not sufficiently center on these theorems are, from this view, far off the mark. Thus most discussion of 'complexity,' 'entropy maximization,' different definitions of 'entropy,' and so forth, just does not appear on the horizon.
Entropy is defined from free energy F by a Legendre transform - more of which follows below: where the Kj are appropriate system parameters. Neglecting volume problems (which will become quite important later), free energy can be defined from the system's partition function Z as () = \og[Z(K)]. The partition function Z, in turn, is defined from the system Hamiltonian defining the energy states - as where K is an inverse temperature or other parameter and the Ej are the energy states. 30) The terms dS/dKi are macroscopic driving 'forces' dependent on the entropy gradient.
Assume the piecewise, adiabatically memoryless ergodic information source (or sources) dual to cognitive process depends on three parameters, two explicit and one implicit. The explicit are K as above and, as a calculational device, an 'external field strength' analog J, which gives a 'direction' to the system. We will, in the limit, set J = 0. Note that many other approaches may well be possible, since renormalization techniques are more philosophy than prescription. The implicit parameter, r, is an inherent generalized 'length' characteristic of the phenomenon, on which J and K are defined.
Advanced methods in applied mathematics; lecture course (1941) by Richard Courant, Charles de Prima, John R. Knudsen