Abstract
We analyze the coupled non-linear integro-differential equations whose solution is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p−spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds.
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References
G. Ben Arous, A. Guionnet, Large deviations for Langevin spin glass dynamics. Prob. Th. Rel. Fields 102:455–509 (1995).
G. Ben Arous, A. Guionnet, Symmetric Langevin spin glass dynamics. Ann. Probab. 25:1367–1422 (1997).
G. Ben Arous, A. Dembo and A. Guionnet, Aging of spherical spin glasses. Probab. Theory Relat. Fields 120:1–67 (2001).
G. Ben Arous, Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians, Vol. III (Higher Ed. Press, Beijing, 2002), pp. 3–14.
G. Ben Arous, A. Dembo and A. Guionnet, Cugliandolo-Kurchan equations for dynamics of Spin-Glasses. Prob. Th. Rel. Fields 136:619–660 (2006).
J. Boucheaud, L. Cugliandolo, J. Kurchan and M. Mezard, Mode coupling approximations, glass theory and disordered systems. Physica A 226:243–273 (1997).
J. P. Bouchaud, L. F. Cugliandolo, J. Kurchan and M. Mezard, Out of equilibrium dynamics in spin-glasses and other glassy systems. Spin glass dynamics and Random Fields, A. P Young Editor (1997).
L. F. Cugliandolo, Dynamics of glassy systems. Les Houches (2002).
L. F. Cugliandolo and D. S. Dean, Full dynamical solution for a spherical spin-glass model. J. Phys. A: Math. Gen. 28:4213–4234 (1995).
L. F. Cugliandolo and J. Kurchan, Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Phys. Rev. Lett. 71:173 (1993).
L. F. Cugliandolo and P. Le Doussal, Large time nonequilibrium dynamics of a particle in a random potential. Phys. Rev. E, 53:1525–1552 (1996).
A. Guionnet and C. Mazza, Long time behaviour of non-commutative processes solution of a linear differential equation. Prob. Theory. Rel. Fields 131:493–518 (2005).
A. Guionnet, Dynamics for spherical models of sping glass and Aging. Proceedings of the Ascona meeting 2004.
W. Gotze, Liquids, Freezing and the Glass Transition Ed. J. P. Hansen, D. Levesque and J Zinn-Justin. Les Houches 1989 (North-Holland, 1991).
W. Gotze and L. Sjogren, Relaxation processes in supercooled liquids. Rep. Prog. Phys. 55:241–376 (1992).
R. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid. Mech. 5(4):497–543 (1959).
R. Kraichnan, Dynamics of nonlinear stochastic systems. J. Math. Phys. 2(1):124–148 (1961).
W. McComb, The Physics of Fluid Turbulence (Oxford University Press, 1990).
A. Monin and A. Yaglom, Statistical Fluids Mechanics, Vol. 2 (MIT Press, 1975).
M. Talagrand, Free energy of the spherical mean field model. To appear in Prob. Th. Rel. Fields (2007).
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AMS (2000) Subject Classification: Primary: 82C44, Secondery: 82C31, 60H10, 60F15, 60K35
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Dembo, A., Guionnet, A. & Mazza, C. Limiting Dynamics for Spherical Models of Spin Glasses at High Temperature. J Stat Phys 126, 781–815 (2007). https://doi.org/10.1007/s10955-006-9228-2
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DOI: https://doi.org/10.1007/s10955-006-9228-2