In these notes, we continue our investigation of classical toy models of disordered statistical mechanics, through techniques recently developed and tested mainly on the paradigmatic Sherrington-Kirkpatrick spin glass. Here, we consider the p-spin-glass model with Ising spins and interactions drawn from a normal distribution

$\mathcal {N}[0,1]$
N[0,1]⁠. After a general presentation of its properties (e.g., self-averaging of the free energy, existence of a suitable thermodynamic limit), we study its equilibrium behavior within the Hamilton-Jacobi framework and the smooth cavity approach. Through the former we find both the RS and the 1-RSB expressions for the free-energy, coupled with their self-consistent relations for the overlaps. Through the latter, we recover these results as irreducible expression, and we study the generalization of the overlap polynomial identities suitable for this model; a discussion on their deep connection with the structure of the internal energy and the entropy closes the investigation.

Topics
Magnetic materials, Magnetic ordering, Ising spins, Spin model, Entropy, Free energy, Statistical mechanics, Statistical thermodynamics, Gaussian processes, Stochastic processes
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Here, we name P our velocity, i.e., the velocity field coincides with the generalized time dependent momentum.
26.
Among the several many body theories developed in statistical mechanics along the years, the mean field two body ones are the most welcome as the Hamiltonian – even though no longer in classical space-time sense – are still quadratic forms such that linear response for the forces is still kept. However, especially in disordered system, many real features of glass forming dynamics seem better reproduced by violation of the linear response and in this sense by p-spin models.
© 2012 American Institute of Physics.
2012
American Institute of Physics
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