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| Title | Information Theory, Exact Statistical Mechanics and Quantum Systems |
| Speaker | Dr Robert Niven, Senior Lecturer, School of Aerospace, Civil and Mechanical Engineering, UNSW@ADFA |
| Date | Wednesday 03 August |
| Time | 11:00 - 12:00 |
| Venue | Building 15 - Rm 152 |
| Abstract |
This seminar first examines the three philosophical bases of the concept of entropy:
(i) information-theoretic, expressed in terms of "bits" of information, as developed
since the 1940s (an approach intrinisically linked to "Maxwell's demon"); (ii) axiomatic,
as developed by Shannon (1948) and Jaynes (1957); and (iii) combinatorial, as first
understood by Boltzmann (1877). It is argued that the combinatorial approach is the
most fundamental (most primitive) basis of entropy, and provides the means to analyse
systems which cannot be examined by the Shannon entropy or Kullback-Leibler cross-entropy.
Using the combinatorial method, the exact forms of the Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropies and probabilistic distributions are then derived, without the commonly used Stirling’s approximation. The new entropy measures are applicable to systems in which the total number of entities N and/or the number of entities in each state n[i] do not approach infinity, and thus form a superset of traditional (Stirling-approximate) MB, BE and FD statistical mechanics. The analysis is then used to determine the energy cost (in bits) of a “binary decision” for each statistical system, i.e. the cost of learning that a system, distributed over two equiprobable states, is in one such state. It is shown that a binary decision can be purchased for <1 bit in the three systems examined, if one has additional knowledge. Under exact BE statistics, a zero purchase cost is theoretically attainable. However, the cost of a binary decision is >1 bit in BE and FD systems in the absence of such knowledge, except in the Stirling limit. Accordingly, the observation of a BE or FD system is thermodynamically irreversible (requires an energy or information input); it therefore behaves as if it contains infinite entities and is infinitely degenerate until the moment of observation. The analysis provides a rational explanation for the quantum mechanical character of BE and FD systems, and for the “collapse of the wavefunction” in their observation. (This seminar relates to a recent publication (Niven, 2005, Physics Letters A, 342(4): 286-293), and will be presented to 3rd Int. Conf.: News, Expectations and Trends in Statistical Physics, Kolymbari, Crete, Greece, 13-18 August 2005.) |
