Abstract
Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension τ(n) (with respect to the components of the surface normaln) and the role of the step free energy τstep(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of τstep(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2τstep(m) is equal to the value of the jump of the derivative ∂τ/∂δ (when δ varies) at the point δ=0 [withn=(m 1 sin δ,m 2 sin δ, cos δ)]. Finally, using this fact, it is shown that the facet shape is determined by the function τstep(m).
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References
R. L. Dobrushin, R. Kotecký, and S. B. Shlosman,The Wulff Construction: A Global Shape from Local Interactions, (American Mathematical Society, Providence, Rhode Island, 1992).
A. Messager, S. Miracle-Sole, and J. Ruiz, Convexity properties of the surface tension and equilibrium crystals,J. Stat. Phys. 67:449 (1992).
R. L. Dobrushin and S. B. Shlosman, Thermodynamic inequalities and the geometry of the Wulff construction, inIdeas and Methods in Mathematical Analysis, Stochastics and Applications, S. Albeverio, S. E. Fenstad, H. Holden, and T. Lindstrom, eds. (Cambridge University Press, Cambridge, 1991).
R. L. Dobrushin, A Gibbs state describing the coexistence of phases for the three-dimensional Ising model,Theory Prob. Appl. 17:582 (1972).
J. Bricmont, J. L. Lebowitz, C. E. Pfister, and E. Olivieri, Non-translation invariant Gibbs states with coexisting phases. (I) Existence of a sharp interface for Widom-Rowlinson type lattice models in three dimensions,Commun. Math. Phys. 66:1 (1979); J. Bricmont, J. L. Lebowitz, and C. E. Pfister, (II) Cluster properties and surface tensions,Commun. Math. Phys. 66:21 (1979); (III) Analyticity properties,Commun. Math. Phys. 69:267 (1979).
P. Holicky, R. Kotecky, and M. Zahradnik, Rigid interfaces for lattice models at low temperatures,J. Stat. Phys. 50:755 (1988).
J. D. Weeks, G. H. Gilmer, and H. J. Leamy, Structural transition in the Ising model interface,Phys. Rev. Lett. 31:549 (1973).
H. van Beijeren, Interface sharpness in the Ising system,Commun. Math. Phys. 40:1 (1975).
J. Bricmont, J. R. Fontaine, and J. L. Lebowitz, Surface tension, percolation and roughening,J. Stat. Phys. 29:193 (1982).
J. Fröhlich and T. Spencer, The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas,Commun. Math. Phys. 81:527 (1981).
H. van Beijeren and I. Nolden, The roughening transition, inTopics in Current Physics, Vol. 43, W. Schommers and P. von Blackenhagen, eds. (Springer, Berlin, 1987).
D. B. Abraham, Surface structures and phase transitions, inCritical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz eds. (Academic Press, London, 1986).
R. Rottman and M. Wortis, Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions,Phys. Rep. 103:59 (1984).
R. Kotecky, Statistical mechanics of interfaces and equilibrium crystal shapes, inIX International Congress of Mathematical Physics, B. Simon, A. Truman, and I. M. Davies, eds. (Adam Hilger, Bristol, 1989).
J. Bricmont, A. El Mellouki, and J. Fröhlich, Random surfaces in statistical mechanics: Roughening, rounding, wetting,...,J. Stat. Phys. 42:743 (1986).
G. Gallavotti, Phase separation line in the two-dimensional Ising modelCommun. Math. Phys. 27:103 (1972).
R. Kotecky and S. Miracle-Sole, Roughening transition for the Ising model on a bcc lattice. A case in the theory of ground states,J. Stat. Phys. 47:773 (1987).
C. Rottman and M. Wortis, Equilibrium crystal shapes for lattice models with nearest-and next-nearest-neighbour interactions,Phys. Rev. B 29:328 (1984).
J. E. Avron, H. van Beijeren, L. S. Shulman, and R. K. P. Zia, Roughening transition, surface tension and equilibrium droplet shapes in a two-dimensional Ising system,J. Phys. A: Math. Gen. 15:L81 (1982).
S. Miracle-Sole, In preparation.
A. Messager, S. Miracle-Sole, J. Ruiz, and S. B. Shlosman, Interfaces in the Potts model. (II) Antonov's rule and rigidity of the order-disorder interface,Commun. Math. Phys. 140:275 (1991).
G. Gallavotti, A. Martin-Lof, and S. Miracle-Sole, Some problems connected with the description of coexisting phases at low temperatures in Ising models, inMathematical Methods in Statistical Mechanics A. Lenard, ed. (Springer, Berlin, 1973).
J. Bricmont, J. L. Lebowitz, and C. E. Pfister, On the local structure of the phase separation line in the two-dimensional Ising model,.J. Stat. Phys. 26:313 (1981).
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Miracle-Sole, S. Surface tension, step free energy, and facets in the equilibrium crystal. J Stat Phys 79, 183–214 (1995). https://doi.org/10.1007/BF02179386
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DOI: https://doi.org/10.1007/BF02179386