×

Isoperimetric phase transitions of two-dimensional droplets. (English) Zbl 0715.58054

Summary: We consider two-dimensional assemblies of particles governed by hamiltonians depending on the area and the perimeter of their convex hull. Provided the hamiltonian is quadratically homogeneous in the coordinates, we find an exact formula for the free energy. Phase transitions resulting from the competition between area and perimeter can easily be produced and explicitly dealt with. We illustrate those features by a simple example undergoing a second-order transition.

MSC:

58Z05 Applications of global analysis to the sciences
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Osserman, R.: Bull. A. M. S.84, 1182–1238 (1978) · Zbl 0411.52006 · doi:10.1090/S0002-9904-1978-14553-4
[2] Osserman, R.: Am. Math. Monthly86, 1–29 (1979) · Zbl 0404.52012 · doi:10.2307/2320297
[3] Bavaud, F.: J. Stat. Phys.57, 1059–1068 (1989) · Zbl 0714.60101 · doi:10.1007/BF01020048
[4] Bavaud, F.: Lett. Math. Phys.20, 75–84 (1990) · Zbl 0697.60018 · doi:10.1007/BF00417231
[5] Zia, R. K. P.: In: Proc. of the XXVIth Scottish Summer School in Physics 1984
[6] Rottman, C., Wortis, M.: Phys. Reports103, 59–79 (1984) · doi:10.1016/0370-1573(84)90066-8
[7] Coninck, J. De., Dunlop, F., Rivasseau, V.: Commun. Math. Phys.121, 401–419 (1989) · Zbl 0659.60140 · doi:10.1007/BF01217731
[8] Pfister, C. E., Penrose, O.: Commun. Math. Phys.115, 691–699 (1988) · doi:10.1007/BF01224133
[9] Shlosman, S. B.: Commun. Math. Phys.125, 81–90 (1989) · Zbl 0679.60099 · doi:10.1007/BF01217770
[10] Santaló, L. A.: Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley 1976 · Zbl 0342.53049
[11] Baddeley, A.: Adv. Appl. Prob.9, 824–860 (1977) · Zbl 0387.60019 · doi:10.2307/1426702
[12] Dupain, Y., Kamae, T., Mendès-France, M.: Arch. Rat. Mech. Anal.94, 155–166 (1986) · Zbl 0603.60100 · doi:10.1007/BF00280431
[13] Rényi, A., Sulanke, R.: Z. Wahrscheinlichkeitstheorie3, 138–147 (1964) · Zbl 0126.34103 · doi:10.1007/BF00535973
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.