# Braid group, knot theory and statistical mechanics by M. L. Ge, C. N. Yang

By M. L. Ge, C. N. Yang

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377, 1 (1998). -N. Roux, Phys. Rev. E 61, 6802 (2000); C. F. Moukarzel, Granul. Matter 3, 41 (2001). [24] D. A. Head, A. V. Tkachenko, and T. A. Witten, Eur. Phys. J. E 6, 99 (2001). [25] S. F. Edwards and D. V. Grinev, Physica A 302, 162 (2001). [26] R. C. Ball and R. Blumenfeld, Phys. Rev. Lett. 88, 115505 (2002). [27] H. Makse, D. L. Johnson, and L. M. Schwartz, Phys. Rev. Lett. 84, 4160 (2000). [28] L. E. , Phys. Rev. E 65, 031304 (2001). [29] J. , Phys. Rev. Lett. 87, 035506 (2001). [30] S.

A second important though straightforward property of isostatic networks is that they enjoy an exact symmetry between force–force and displacement–displacement response functions: The excess stress induced on contact b by a vertical overload applied on sphere i (which is given by the stress–stress response) is exactly equal to the vertical displacement ∆ib that sphere i suffers when the interparticle contact b is stretched (defining the displacement response). This property has been recently used in experiments [50].

17] C. Kittel, Introduction to Solid State Physics (Wiley, 1956). [18] I. Goldhirsch and C. Goldenberg, Eur. Phys. J. E 9, 245 (2002). [19] R. Jackson, in Theory of Dispersed Multiphase Flow, edited by R. E. Meyer, pp. 291–337 (Academic Press, 1983). [20] P. A. Vermeer, in [3], pp. 163–195. [21] R. de Borst and L. J. Sluys, Comput. Meth. Appl. Mech. Eng. 90, 805 (1991). [22] P. C. Johnson, P. Nott, and R. Jackson, J. Fluid Mech. 210, 501 (1990); S. B. Savage, J. Fluid Mech. 377, 1 (1998). -N. Roux, Phys.