We present a simple derivation of the stochastic equation obeyed by the density function for a system of Langevin processes interacting via a pairwise potential. The resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved (Model A) and conserved (Model B) particle systems. The major feature is that the spatial white noise for this system appears not additively but multiplicatively. This simply expresses the fact that the density cannot fluctuate in regions devoid of particles. The steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in Models A and B.
We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N × N ) random matrix are positive (negative) decreases for large N as ∼ exp [−βθ(0)N 2 ] where the parameter β characterizes the ensemble and the exponent θ(0) = (ln 3)/4 = 0.274653 . . . is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number ζ, thus generalizing the celebrated Wigner semi-circle law. The density of states generically exhibits an inverse square-root singularity at ζ.
We present a detailed analysis for the Langevin dynamics of a spherical spinglass model (the spherical Sherrington-Kirkpatrick model). All the spins in the system are coupled by pairs via a random interaction matrix taken from the Gaussian ensemble.One nds that for a general initial con guration the system never reaches an equilibrium state and the theorems associated to`equilibrium dynamics' are violated. Only very particular initial conditions drive the system to equilibrium.The weak ergodicity breaking scenario is demonstrated for general`nonequilibrium' initial conditions. Two-time quantities such as the autocorrelation function explicitly depend on both times. When the time difference is short compared to the smaller time one nds stationary dynamics with time translation invariance and the uctuation-dissipation theorem satis ed. Instead, when the time di erence is of the order of the smaller time one nds non-stationary dynamics with aging phenomena, the system remembers the time spent after the initial time (the quench below the critical temperature). Interestingly enough the short time-di erence dynamics (FDT regime) for non-equilibrium initial conditions is identical to the relaxation within the equilibrium states obtained for the particular`equilibrium' initial conditions. This points to a self-similaririty of the energy landascape.In addition we analyse the e ect of temperature variations on the behaviour of the correlation function and energy density. In this way we are able to make some comparisons between experimentally observed results and exact calculations for the model. Somewhat surprisingly, this simple model captures some of the e ects seen in laboratory spin-glasses.
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