Abstract
If a message can have n different values and all values are equally probable, then the entropy of the message is log(n). In the present paper, we discuss the expectation value of the entropy, for an arbitrary probability distribution. We introduce a mixture of all possible probability distributions. We assume that the mixing function is uniform
-
either in flat probability space, i.e. the unitary n-dimensional hypertriangle
-
or in Bhattacharyya’s spherical statistical space, i.e. the unitary n-dimensional hyperoctant.
A computation is a manipulation of an incoming message, i.e. a mapping in probability space:
-
either a reversible mapping, i.e. a symmetry operation (rotation or reflection) in n-dimen sional space
-
or an irreversible mapping, i.e. a projection operation from n-dimensional to lower-dimensional space.
During a reversible computation, no isentropic path in the probability space can be found. Therefore we have to conclude that a computation cannot be represented by a message which merely follows a path in n-dimensional probability space. Rather, the point representing the mixing function travels along a path in an infinite-dimensional Hilbert space.
Similar content being viewed by others
References
A. Bhattacharyya, On a measure of divergence between two statistical populations defined by their probability distributions, Bulletin of the Calcutta Mathematical Society 35, 99 (1943).
C. Atkinson and A. Mitchell, Rao’s distance, Sankhyā, The Indian Journal of Statistics 43, 345 (1981).
W. Wootters, Statistical distance and Hilbert space, Physical Review D 23, 357 (1981).
F. Aherne, N. Thacker, and P. Rockett, The Bhattacharyya metric as an absolute similarity measure for frequency coded data, Kybernetika 32, 363 (1998).
L. Diósi and P. Salamon, From statistical distances to minimally dissipative processes, in: Thermodynamics of energy conversion and transport, S. Sieniutycz and A. De Vos, eds., Springer Verlag, New York, 2000, pp. 286–318.
D. Sommerville, An introduction to the geometry in n dimensions, Dover Publications, New York, 1958.
M. Kendall, A course in the geometry of n dimensions, Charles Griffin & co, London, 1961.
J. Avery, Hyperspherical harmonics — applications in quantum theory, Kluwer Academic Publishers, Dordrecht, 1989.
H. Teicher, On the mixture of distributions, The Annals of Mathematical Statistics 31, 55 (1960).
B. Everitt and D. Hand, Finite mixture distributions, Chapman and Hall, London, 1981.
A. De Vos, The expectation value of the entropy of a digital message, Open Sys. Information Dyn. 9, 97 (2002).
A. De Vos, The entropy of a mixture of probability distributions, Entropy 7, 15 (2005).
K. Jones, Entropy of random quantum states, J. Phys. A: General and Mathematical 23, L1247 (1990).
W. Wootters, Random quantum states, Foundations of Physics 20, 1365 (1990).
B. Andresen, Finite-time Thermodynamics, Københavns Universitet, København, 1983.
B. Andresen, Thermodynamics in finite time, Physics Today 37, September 1984, pp. 2–10.
S. Sieniutycz and P. Salamon, Finite-Time Thermodynamics and Thermoeconomics, Taylor & Francis, New York, 1990.
C. Wu, W. Chen, and J. Chen, Recent Advances in Finite-Time Thermodynamics, Nova Science, Commack, 1999.
R. Berry, V. Kazakov, S. Sieniutycz, Z. Szwast, and A. Tsirlin, Thermodynamic Optimization of Finite-Time Processes, Wiley, Chichester, 2000.
A. De Vos, Reversible computing, Progress in Quantum Electronics 23, 1 (1999).
A. De Vos, Fundamental limits of power dissipation in digital electronics, Proceedings of the 6 th International Conference on Mixed Design of Integrated Circuits and Systems, Kraklów, 17–19 June 1999, pp. 27–36.
A. De Vos, Lossless computing, Proceedings of the I.E.E.E. Workshop on Signal Processing, Poznaln, 10 October 2003, pp. 7–14.
R. Landauer, Irreversibility and heat generation in the computing process, I.B.M. Journal of Research and Development 5, 183 (1961).
C. Bennett and R. Landauer, The fundamental physical limits of computation, Scientific American 253, July 1985, pp. 38–46.
R. Landauer, Information is physical, Physics Today 44, May 1991, pp. 23–29.
B. Andresen, R. Berry, A. Nitzan, and P. Salamon, Thermodynamics in finite time — I. The step-Carnot cycle, Physical Review A 15, 2086 (1977).
J. Nulton, P. Salamon, B. Andresen, and Q. Anmin, Quasistatic processes as step equilibrations, J. Chem. Phys. 83, 334 (1985).
B. Andresen and P. Salamon, Distillation by thermodynamic geometry, in: Thermodynamics of energy conversion and transport, S. Sieniutycz and A. De Vos, eds., Springer Verlag, New York, 2000, pp. 319–331.
H. Ries and W. Spirkl: Optimal finite-time endoreversible processes — general theory and applications, in: Variational and extremum principles in macroscopic systems, S. Sieniutycz and H. Farkas, eds., Elsevier, Amsterdam, 2005, pp. 627–639.
M. Lemanska and Z. Jaeger, A nonlinear model for relaxation in exited closed physical systems, Physica D 170, 72 (2002).
G. Beretta, A nonlinear model dynamics for closed-system, constrained, maximal-entropygeneration relaxation by energy redistribution, arXiv:quant-ph/0501178 v1 (2005), in press at Physical Review E.
H. Callen, Thermodynamics, Wiley, New York, 1960, p. 202.
D. Tondeur and E. Kvaalen, Equipartition of entropy production. An optimality criterion for transfer and separation processes, Industrial & Engineering Chemistry Research 26, 50 (1987).
D. Tondeur, Equipartition of entropy production: a design and optimization criterion in chemical engineering, in: Finite-time thermodynamics and thermoeconomics, S. Sieniutycz and P. Salomon, eds., Taylor & Francis, New York, 1990, pp. 175–208.
E. Sauar, S. Kjelstrup Ratkje, and K. Lien, Equipartition of forces: a new principle for process design and optimization, Industrial & Engineering Chemistry Research 35 (1996), pp. 4147–4153.
A. De Vos and B. Desoete, Equipartition principles in finite-time thermodynamics, Journal of Non-Equilibrium Thermodynamics 25, 1 (2000).
W. Ebeling, On the relation between various entropy concepts and the valoric interpretation, Physica A 182, 108 (1992).
W. Ebeling, Entropy and information in processes of self-organization, uncertainty and predictability, Physica A 194, 563 (1993).
V. Buyarov, P. López-Artés, A. Martínez-Finkelshtein, and W. Van Assche, Information entropy of Gegenbauer polynomials, J. Phys. A: Mathematical and General 33, 6549 (2000).
Author information
Authors and Affiliations
Additional information
In honour of prof. dr. Henrik Farkas (Department of Chemical Physics, Technical University of Budapest) an outstanding scientist and most remarkable human being who unfortunately left us on 21 July 2005.
Rights and permissions
About this article
Cite this article
De Vos, A. Computing in Finite Time. Open Syst Inf Dyn 13, 179–195 (2006). https://doi.org/10.1007/s11080-006-8221-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11080-006-8221-1