# Thermodynamics, Statistical Mechanics and the Complexity of Reductions

Chapter

## Abstract

That reductions of theories and unifications of science frequently occur by means of identifications is a widely accepted hypothesis of methodology. Usually we are concerned with the micro-reductions of things, in which an entity is identified with a structured aggregate of smaller constituents. And, it is frequently alleged, we must also take into account further identifications in which the attributes of the macro-object, expressed by predicates in the macro-theory, are identified with attributes of the aggregate of micro-entities differently expressed by predicates of the micro- or reducing theory.

## Keywords

Statistical Mechanic Thermodynamic Limit Thermodynamic Quantity Thermodynamic Entropy Macroscopic Theory
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## Notes

- 3.A. Münster,
*Statistical Thermodynamics*, Vol. I, English edition, Springer-Verlag, Heidelberg (1969), sec. 2.11.Google Scholar - 4.A. Münster,
*op. cit*., chap. IV. See also D. Ruelle,*Statistical Mechanics, Rigorous Results*, Benjamin, New York (1969), chap. 5.Google Scholar - 5.A. Münster,
*op. cit*., p. 213.Google Scholar - 6.A. Münster,
*op. cit*., secs. 2.1-3.2 and 4.1-4.4. See also D. Ruelle,*op. cit*., chaps. 1-3.Google Scholar - 9.See H. Grad, ‘The Many Faces of Entropy’,
*Communications on Pure and Applied Mathematics***14**(1961), 323-354, esp. pp. 324-328. See also E. Jaynes, ‘Gibbs vs. Boltzmann Entropies’,*American Journal of Physics*,**33**(1965), 391-398, esp. sec. VI, pp. 397-398.Google Scholar - 10.On the failure of the one-particle distribution function to give the correct entropy when inter-particle forces are taken into account see E. Jaynes,
*op. cit*., pp. 391-394. See also his ‘Information Theory and Statistical Mechanics’, in K. Ford (ed.),*Brandeis University Summer Institute Lectures in Theoretical Physics*, 1962, vol. 3,’ statistical Physics’, pp. 181-218, esp. sec. 6, ‘Entropy and Probability’, pp. 212-217. For a discussion of how to move to the entropy defined by means of the two-particle distribution function and the generalization of this process see H. Grad,*op. cit., passim*.Google Scholar - 11.On the definition of the Gibbs entropy and its relation to the Boltzmann see R. Tolman,
*op. cit*., sec. 51, pp. 165-179, esp. (d) on pp. 174-177. There are in fact several different definitions for a Gibbs entropy, all of which “converge in the thermodynamic limit.” See J. Gibbs,*op. cit*., chap. XIV.Google Scholar - 12.See E. Jaynes, ‘Gibbs vs. Boltzmann Entropies’, sec. V, pp. 395-397.Google Scholar
- 13.On the limits of extending thermodynamics entropy to non-equilibrium cases see P. Landsberg,
*Thermodynamics*, Interscience, New York (1961), sec. 21, pp. 128-142. See also the work of Truesdell cited in note 19, below for important criticism of the “orthodox” view that thermodynamic quantities are “well defined” only in equilibrium situations.Google Scholar - 14.For a defense of the thesis that by use of the fine-grained entropy one is perfectly able to establish the statistical mechanical “analogue” of the Second Law, see E. Jaynes, ‘Gibbs vs. Boltzmann Entropies’, sec. IV, ‘The Second Law’, pp. 394-395. See also his ‘Information Theory and Statistical Mechanics’, sec. 6, pp. 212-217.Google Scholar
- 16.An introduction to coarse-graining can be found in N. van Kampen, ‘Fundamental Problems in Statistical Mechanics of Irreversible Processes’, in E. Cohen (ed.),
*Fundamental Problems in Statistical Mechanics*, Vol. I, North-Holland, Amsterdam (1961), pp. 173-202. See also O. Penrose,*Foundations of Statistical Mechanics*, Pergamon, Oxford (1970), Chap. I, ‘Basic Assumptions’, attempts a rationalization of coarse-graining. See esp. chap. I, sec. 3, ‘Observation’, where the “coarseness” of macro-observation is used to justify coarse-graining in statistical mechanics.Google Scholar - 18.Field, Hartry, ‘Theory Change and the Indeterminacy of Reference’,
*Journal of Philosophy***70**, No. 14, pp. 462-481, esp. p. 466.Google Scholar

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© D. Reidel Publishing Company, Dordrecht, Holland 1976