Mutual Information of Population Codes and Distance Measures in Probability Space

We studied the mutual information between a stimulus and a large system consisting of stochastic, statistically independent elements that respond to a stimulus. The Mutual Information (MI) of the system saturates exponentially with system size. A theory of the rate of saturation of the MI is developed. We show that this rate is controlled by a distance function between the response probabilities induced by different stimuli. This function, which we term the {\it Confusion Distance} between two probabilities, is related to the Renyi $\alpha$-Information.Comment: 11 pages, 3 figures, accepted to PR

Large time off-equilibrium dynamics of a manifold in a random potential

We study the out of equilibrium dynamics of an elastic manifold in a random potential using mean-field theory. We find two asymptotic time regimes: (i) stationary dynamics, (ii) slow aging dynamics with violation of equilibrium theorems. We obtain an analytical solution valid for all large times with universal scalings of two-time quantities with space. A non-analytic scaling function crosses over to ultrametricity when the correlations become long-range. We propose procedures to test numerically or experimentally the extent to which this scenario holds for a given system.Comment: 12 page

A Solvable Model of a Glass

An analytically tractable model is introduced which exhibits both, a glass--like freezing transition, and a collection of double--well configurations in its zero--temperature potential energy landscape. The latter are generally believed to be responsible for the anomalous low--temperature properties of glass-like and amorphous systems via a tunneling mechanism that allows particles to move back and forth between adjacent potential energy minima. Using mean--field and replica methods, we are able to compute the distribution of asymmetries and barrier--heights of the double--well configurations {\em analytically}, and thereby check various assumptions of the standard tunneling model. We find, in particular, strong correlations between asymmetries and barrier--heights as well as a collection of single--well configurations in the potential energy landscape of the glass--forming system --- in contrast to the assumptions of the standard model. Nevertheless, the specific heat scales linearly with temperature over a wide range of low temperatures.Comment: 11 pages, latex, including 5 figures, talk presented at the XIV Sitges Conferenc

Plasticity and learning in a network of coupled phase oscillators

A generalized Kuramoto model of coupled phase oscillators with slowly varying coupling matrix is studied. The dynamics of the coupling coefficients is driven by the phase difference of pairs of oscillators in such a way that the coupling strengthens for synchronized oscillators and weakens for non-synchronized pairs. The system possesses a family of stable solutions corresponding to synchronized clusters of different sizes. A particular cluster can be formed by applying external driving at a given frequency to a group of oscillators. Once established, the synchronized state is robust against noise and small variations in natural frequencies. The phase differences between oscillators within the synchronized cluster can be used for information storage and retrieval.Comment: 10 page

Phase variance of squeezed vacuum states

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number, $n$, as dictated by the Heisenberg limit, i.e., as $n^{-2}$, only for $N>4$. For $N\leq 4$ this fundamental scaling breaks down and it becomes $n^{-N/2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling $n^{-2}$ is recovered for large samples.Comment: Minor changes, version to appear in PRA, 8 pages, 2 figure

Schwinger-Keldysh Approach to Disordered and Interacting Electron Systems: Derivation of Finkelstein's Renormalization Group Equations

We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. We calculate within this formalism the perturbative RG equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein using replicas. The major simplifying feature of this approach, as compared to Finkelstein's is that instead of $N \to 0$ replicas, we only need to consider N=2 species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.Comment: 22 pages, 14 figure

Large times off-equilibrium dynamics of a particle in a random potential

We study the off-equilibrium dynamics of a particle in a general $N$-dimensional random potential when $N \to \infty$. We demonstrate the existence of two asymptotic time regimes: {\it i.} stationary dynamics, {\it ii.} slow aging dynamics with violation of equilibrium theorems. We derive the equations obeyed by the slowly varying part of the two-times correlation and response functions and obtain an analytical solution of these equations. For short-range correlated potentials we find that: {\it i.} the scaling function is non analytic at similar times and this behaviour crosses over to ultrametricity when the correlations become long range, {\it ii.} aging dynamics persists in the limit of zero confining mass with universal features for widely separated times. We compare with the numerical solution to the dynamical equations and generalize the dynamical equations to finite $N$ by extending the variational method to the dynamics.Comment: 70 pages, 7 figures included, uuencoded Z-compressed .tar fil

Langevin description of speckle dynamics in nonlinear disordered media

We formulate a Langevin description of dynamics of a speckle pattern resulting from the multiple scattering of a coherent wave in a nonlinear disordered medium. The speckle pattern exhibits instability with respect to periodic excitations at frequencies $\Omega$ below some $\Omega_{\mathrm{max}}$, provided that the nonlinearity exceeds some $\Omega$-dependent threshold. A transition of the speckle pattern from a stationary state to the chaotic evolution is predicted upon increasing nonlinearity. The shortest typical time scale of chaotic intensity fluctuations is of the order of $1/\Omega_\mathrm {max}$.Comment: 6 pages, 3 figure

Dynamics of relaxor ferroelectrics

We study a dynamic model of relaxor ferroelectrics based on the spherical random-bond---random-field model and the Langevin equations of motion. The solution to these equations is obtained in the long-time limit where the system reaches an equilibrium state in the presence of random local electric fields. The complex dynamic linear and third-order nonlinear susceptibilities $\chi_1(\omega)$ and $\chi_3(\omega)$, respectively, are calculated as functions of frequency and temperature. In analogy with the static case, the dynamic model predicts a narrow frequency dependent peak in $\chi_3(T,\omega)$, which mimics a transition into a glass-like state.Comment: 15 pages, Revtex plus 5 eps figure

Equilibrium and dynamical properties of the ANNNI chain at the multiphase point

We study the equilibrium and dynamical properties of the ANNNI (axial next-nearest-neighbor Ising) chain at the multiphase point. An interesting property of the system is the macroscopic degeneracy of the ground state leading to finite zero-temperature entropy. In our equilibrium study we consider the effect of softening the spins. We show that the degeneracy of the ground state is lifted and there is a qualitative change in the low temperature behaviour of the system with a well defined low temperature peak of the specific heat that carries the thermodynamic ``weight'' of the ground state entropy. In our study of the dynamical properties, the stochastic Kawasaki dynamics is considered. The Fokker-Planck operator for the process corresponds to a quantum spin Hamiltonian similar to the Heisenberg ferromagnet but with constraints on allowed states. This leads to a number of differences in its properties which are obtained through exact numerical diagonalization, simulations and by obtaining various analytic bounds.Comment: 9 pages, RevTex, 6 figures (To appear in Phys. Rev. E