Riemannian geometric approach to critical points: General theory

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 57, Issue 5, 1998, Pages 5135-5145

  Ruppeiner, George ^a  

^a NEW COLLEGE OF FLORIDA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4243088744     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.5135     Document Type: Article

References (32)

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14. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).
15. My usage of the word “universal” is not exactly conventional. The spirit of the Riemannian geometric approach in two dimensions is that the critical exponents determine the scaled equation of state. However, in the modern theory of critical phenomena universality classes are differentiated by spatial and order parameter dimensionality. In this context, it is at least conceivable that different universality classes could have the same critical exponents but different scaled equations of state or, more likely, that a given equation of state represents more than a single critical fixed point, with the critical exponents of any one not sufficient to determine the full equation of state. This latter possibility may be the case beyond two variables and may force an eventual reassessment of the assumption about the universality of κ. However, the statement here seems at least a good point to start.
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20. In two and three dimensions, at least, two of these derivatives cancel due to a common factor of g.
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24. There appears to be no universally used notation for the critical exponents, beyond a few basic ones. My notation differs from Fisher’s 4. To make the correspondence with his, we would take a→2-α, b→Δ, and c→-θ.
25. V. Privman, P. C. Hohenberg, and A. Aharony, in Phase Transitions, edited by C. Domb and J. Lebowitz (Academic, New York, 1991), Vol. 14, p. 1.
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30. The latest exponents for the 3D Ising model are slightly different; see A. J. Liu and M. E. Fisher, Physica A 156, 35 (1989). I use the “historic” set of exponents, which lend themselves to convenient fractional representation. This should make little difference in my results.PHYADX
31. J. V. Sengers and J. Levelt-Sengers, Annu. Rev. Phys. Chem. 37, 189 (1986).ARPLAP
32. Note that my free energy has the opposite sign to the one in Refs. 56.