Abstract
A generalization of Ehrenfest's urn model is suggested. This will allow usto treat a wide class of stochastic processes describing the changes ofmicroscopic objects. These processes are homogeneous Markov chains. Thegeneralization proposed is presented as an abstract conditional (relative)probability theory. The probability axioms of such a theory and some simpleadditional conditions, yield both transition probabilities and equilibriumdistributions. The resulting theory interpreted in terms of particles andsingle-particle states, leads to the usual formulae of quantum and classicalstatistical mechanics; in terms of chromosomes and allelic types, it allowsthe deduction of many genetical models including the Ewens sampling formula;in terms of agents' strategies, it gives a justification of the ``herdbehaviour'' typical of a population of heterogeneous economic agents.
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REFERENCES
Aoki, M.: 1996, New Approaches to Macroeconomic Modelling
Aoki, M.: 2002, Modelling Aggregate Behavior and Fluctuation in Economics
Ageno, M.: 1992, Le origini della irreversibilità
Boltzmann, L.: 1909, 1, 'Studien ueber das Gleichgewicht der lebendigen Kraft zwischen bewegten materielle Punkten' (1968), in F. Hasenoehrl (ed.), Wissenschaftliche Abhandlungen, Vol. I
Brillouin, L.: 1927, 'Comparaison des différentes statistiques appliqué aux problèmes des quanta', Ann. Phys. (Paris) VII
Carnap, R.: 1963, Logical Foundations of Probability
Carnap, R.: 1952, The Continuum of Inductive Methods
Cont, R. and J. P. Bouchaud: 2000, 'Herd Behaviour and Aggregate Fluctuations in Financial Markets', Jour. Macroeconomic Dynamics 4
Cook, R. and D. Costantini (eds): 1990, Statistics in Science
Costantini, D.: 1987 'Symmetry and the Indistinguishability of Classical Particles', Physics Letters A 123
Costantini, D. and U. Garibaldi: 1989, 'Classical and Quantum Statistics as Finite Random Processes', Foundations of Physics 19
Costantini, D. and U. Garibaldi: 1991, 'Una formulazione probabilistica del principio di esclusione di Pauli nel contesto delle inferenze predittive', Statistica LI
Costantini, D. and U. Garibaldi: 1997, 'A Probabilistic Foundation of Elementary Particle Statistics. Part I', Stud. Hist. Phil. Mod. Phys. 28
Costantini, D. and U. Garibaldi: 1998, 'A Probabilistic Foundation of Elementary Particle Statistics. Part II', Stud. Hist. Phil. Mod. Phys. 29
Costantini, D. and U. Garibaldi: 2000, 'A Purely Probabilistic Representation for the Dynamics of a Gas of Particles', Foundations of Physics 30
Costantini, D., U. Garibaldi, and M. A. Penco: 1996, 'Ludwig Boltzmann alla nascita della meccanica statistica', Statistica LVI
Costantini, D. and U. Garibaldi: 'Liouville Equation and Markov Chains: Epistemic and Ontological Probability” (to appear in Proc. of the Q. S. A. Conference, Cesena 2001).
Ehrenfest, P. and T. Ehrenfest: 1907, 'Ueber zwei bekannte Einwaende gegen Boltzmanns H-Theorem', Phys. Zeitschrift 8
Ewens, W. J.: 1972, 'The Sampling Theory of Selectively Neutral Alleles', Theor. Pop. Biol. 3
Fisher, R. A.: 1930, The Genetical Theory of Natural Selection
Garibaldi, U. and M. A. Penco: 2000 'Ehrenfest urn model generalized: a model for economic time series', Statistica applicata 12
Hacking, I.: 1975, The Emergence of Probability
Jancel, R.: 1963, Foundations of Classical and Quantum Statistical Mechanics
Jaynes, E. T.: 1996, Probability Theory: The Logic of Science
Johnson, N. L., S. Kotz, and N. Balakrishnam: 1997, Multivariate Discrete Distributions
Johnson, W. E.: 1932, 'Probability: The Relations of Proposal to Supposal; Probability: Axioms; Probability: The Deductive and the Inductive Problems', Mind 41
Kaĉ, M.: 1959, Probability and Related Topics in Physical Science
Kelly, F. P.: 1979, Reversibility and Stochastic Networks
Keynes, J. M.: 1962, A Treatise on Probability
Kingman, J. F. C.: 1978, 'The Representation of Partition Structures', Journal London Mathematical Society 18
Moran, P. A. P.: 1958, 'Random Processes in Genetics', Proc. Camb. Phil. Soc. 54
Penrose, O.: 1970, Foundations of Statistical Mechanics
Scardovi, I.: 1982, 'The Idea of Chance in the Statistical Intuition of Natural variability', Genus XXXVIII
Sudarshan E. C. G. and J. Mehra: 1970, 'Classical Statistical Mechanics of Identical Particles and Quantum Effects', Inter. J. Theoretical Phys. 3
Tolman, R. C.: 1979, The Principles of Statistical Mechanics
von Mises, R.: 1920, 'Ausschaltung der Ergodenhypothese in der physicalischen Statistik. I. Allgemainne Betrachtungen' and 'Ausschaltung der Ergodenhypothese in der physicalischen Statistik. II. Brownsche Bewegung als Beispiel', Physik. Zeitschr. XXI
von Mises, R.: 1957, Probability, Statistics and Truth
von Plato, J.: 1994, Creating Modern Probability
Wright, S.: 1931, 'Evolution in Mendelian Populations', Genetics 16
Zabell, S. L.: 1996, 'The Continuum of Inductive Methods Revisited', The Cosmos of Science
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Costantini, D., Garibaldi, U. The Ehrenfest Fleas: From Model to Theory. Synthese 139, 107–142 (2004). https://doi.org/10.1023/B:SYNT.0000021307.64103.b8
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DOI: https://doi.org/10.1023/B:SYNT.0000021307.64103.b8