Zero temperature limit for interacting Brownian particles. I: Motion of a single body. (English) Zbl 1121.82028

Summary: We consider a system of interacting Brownian particles in \(\mathbb R^d\) with a pairwise potential, which is radially symmetric, of finite range and attains a unique minimum when the distance of two particles becomes \(a>0\). The asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects. If the system is rigidly crystallized, namely if the particles are rigidly arranged in an equal distance a, the crystallization is kept under the evolution in macroscopic time scale. Then, assuming that the crystal has a definite limit shape under a macroscopic spatial scaling, the translational and rotational motions of such shape are characterized.
Part II, cf. Ann. Probab. 32, No. 2, 1228–1246 (2004; Zbl 1122.82029).


82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory


Zbl 1122.82029
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[1] Asimow, L. and Roth, B. (1978). The rigidity of graphs. Trans. Amer. Math. Soc. 245 279–289. · Zbl 0392.05026
[2] Asimow, L. and Roth, B. (1979). The rigidity of graphs. II. J. Math. Anal. Appl. 68 171–190. · Zbl 0441.05046
[3] Bodineau, T., Ioffe, D. and Velenik, Y. (2000). Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 1033–1098. · Zbl 0977.82013
[4] Funaki, T. (1995). The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 221–288. · Zbl 0834.60066
[5] Funaki, T. (1997). Singular limit for reaction-diffusion equation with self-similar Gaussian noise. In New Trends in Stochastic Analysis (K. D. Elworthy, S. Kusuoka and I. Shigekawa, eds.) 132–152. World Scientific, Singapore.
[6] Funaki, T. (2004). Zero temperature limit for interacting Brownian particles. II. Coagulation in one dimension. Ann. Probab. 32 1228–1246. · Zbl 1122.82029
[7] Funaki, T. and Nagai, H. (1993). Degenerative convergence of diffusion process toward a submanifold by strong drift. Stochastics Stochastics Rep. 44 1–25. · Zbl 0788.58059
[8] Hales, T. C. (1997). Sphere packings. I. Discrete Comput. Geom. 17 1–51. · Zbl 0883.52012
[9] Katzenberger, G. S. (1991). Solutions of a stochastic differential equation forced onto a manifold by a large drift. Ann. Probab. 19 1587–1628. JSTOR: · Zbl 0749.60053
[10] Kotani, M. and Sunada, T. (2000). Standard realizations of crystal lattices via harmonic maps. Trans. Amer. Math. Soc. 353 1–20. · Zbl 0960.58009
[11] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations . Cambridge Univ. Press. · Zbl 0743.60052
[12] Laman, G. (1970). On graphs and rigidity of plane skeletal structures. J. Engrg. Math. 4 331–340. · Zbl 0213.51903
[13] Landau, L. D. and Lifshitz, E. M. (1969). Mechanics , 2nd ed. Pergamon, New York. · Zbl 0081.22207
[14] Lang, R. (1979). On the asymptotic behaviour of infinite gradient systems. Comm. Math. Phys. 65 129–149. · Zbl 0394.60098
[15] McKean, H. P., Jr. (1969). Stochastic Integrals . Academic Press, New York. · Zbl 0191.46603
[16] Shigekawa, I. (1984). Transformations of the Brownian motion on a Riemannian symmetric space. Z. Wahrsch. Verw. Gebiete 65 493–522. · Zbl 0518.60087
[17] Whiteley, W. (1984). Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Amer. Math. Soc. 285 431–465. · Zbl 0518.52010
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