1,397 research outputs found
Order Parameter Flow in the SK Spin-Glass I: Replica Symmetry
We present a theory to describe the dynamics of the Sherrington- Kirkpatrick
spin-glass with (sequential) Glauber dynamics in terms of deterministic flow
equations for macroscopic parameters. Two transparent assumptions allow us to
close the macroscopic laws. Replica theory enters as a tool in the calculation
of the time- dependent local field distribution. The theory produces in a
natural way dynamical generalisations of the AT- and zero-entropy lines and of
Parisi's order parameter function . In equilibrium we recover the
standard results from equilibrium statistical mechanics. In this paper we make
the replica-symmetric ansatz, as a first step towards calculating the order
parameter flow. Numerical simulations support our assumptions and suggest that
our equations describe the shape of the local field distribution and the
macroscopic dynamics reasonably well in the region where replica symmetry is
stable.Comment: 41 pages, Latex, OUTP-94-29S, 14 figures available in hardcop
Coupled dynamics of sequence selection and compactification in mean-field hetero-polymers
We study a simple solvable model describing the genesis of monomer sequences
for hetero-polymers (such as proteins), as the result of the equilibration of a
slow stochastic genetic selection process which is assumed to be driven by the
competing demands of functionality and reproducibility of the polymer's folded
structure. Since reproducibility is defined in terms of properties of the
folding process, one is led to the analysis of the coupled dynamics of (fast)
polymer folding and (slow) genetic sequence selection. For the present
mean-field model this analysis can be carried out using the finite-dimensional
replica method, leading to exact results for (first- and second-order)
transitions and to rich phase diagrams.Comment: 21 pages, 7 figure
Dynamics of a spherical minority game
We present an exact dynamical solution of a spherical version of the batch
minority game (MG) with random external information. The control parameters in
this model are the ratio of the number of possible values for the public
information over the number of agents, and the radius of the spherical
constraint on the microscopic degrees of freedom. We find a phase diagram with
three phases: two without anomalous response (an oscillating versus a frozen
state), and a further frozen phase with divergent integrated response. In
contrast to standard MG versions, we can also calculate the volatility exactly.
Our study reveals similarities between the spherical and the conventional MG,
but also intriguing differences. Numerical simulations confirm our analytical
results.Comment: 16 pages, 3 figures; submitted to J. Phys.
Phase Diagram and Storage Capacity of Sequence Processing Neural Networks
We solve the dynamics of Hopfield-type neural networks which store sequences
of patterns, close to saturation. The asymmetry of the interaction matrix in
such models leads to violation of detailed balance, ruling out an equilibrium
statistical mechanical analysis. Using generating functional methods we derive
exact closed equations for dynamical order parameters, viz. the sequence
overlap and correlation- and response functions, in the thermodynamic limit. We
calculate the time translation invariant solutions of these equations,
describing stationary limit-cycles, which leads to a phase diagram. The
effective retarded self-interaction usually appearing in symmetric models is
here found to vanish, which causes a significantly enlarged storage capacity of
, compared to \alpha_\c\sim 0.139 for Hopfield networks
storing static patterns. Our results are tested against extensive computer
simulations and excellent agreement is found.Comment: 17 pages Latex2e, 2 postscript figure
Magnon Localization in Mattis Glass
We study the spectral and transport properties of magnons in a model of a
disordered magnet called Mattis glass, at vanishing average magnetization. We
find that in two dimensional space, the magnons are localized with the
localization length which diverges as a power of frequency at small
frequencies. In three dimensional space, the long wavelength magnons are
delocalized. In the delocalized regime in 3d (and also in 2d in a box whose
size is smaller than the relevant localization length scale) the magnons move
diffusively. The diffusion constant diverges at small frequencies. However, the
divergence is slow enough so that the thermal conductivity of a Mattis glass is
finite, and we evaluate it in this paper. This situation can be contrasted with
that of phonons in structural glasses whose contribution to thermal
conductivity is known to diverge (when inelastic scattering is neglected).Comment: 11 page
Dynamical Replica Theory for Disordered Spin Systems
We present a new method to solve the dynamics of disordered spin systems on
finite time-scales. It involves a closed driven diffusion equation for the
joint spin-field distribution, with time-dependent coefficients described by a
dynamical replica theory which, in the case of detailed balance, incorporates
equilibrium replica theory as a stationary state. The theory is exact in
various limits. We apply our theory to both the symmetric- and the
non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes
the (numerical) experiments very well.Comment: 7 pages RevTex, 4 figures, for PR
Exact Solution of the Infinite-Range Quantum Mattis Model
We have solved the quantum version of the Mattis model with infinite-range
interactions. A variational approach gives the exact solution for the
infinite-range system, in spite of the non-commutative nature of the quantum
spin components; this implies that quantum effects are not predominant in
determining the macroscopic properties of the system. Nevertheless, the model
has a surprisingly rich phase behaviour, exhibiting phase diagrams with
tricritical, three-phase and critical end points.Comment: 14 pages, 11 figure
Double Criticality of the Sherrington-Kirkpatrick Model at T=0
Numerical results up to 42nd order of replica symmetry breaking (RSB) are
used to predict the singular structure of the SK spin glass at T=0. We confirm
predominant single parameter scaling and derive corrections for the T=0 order
function q(a), related to a Langevin equation with pseudotime 1/a. a=0 and
a=\infty are shown to be two critical points for \infty-RSB, associated with
two discrete spectra of Parisi block size ratios, attached to a continuous
spectrum. Finite-RSB-size scaling, associated exponents, and T=0-energy are
obtained with unprecedented accuracy.Comment: 4 pages, 5 figure
Dynamical Probability Distribution Function of the SK Model at High Temperatures
The microscopic probability distribution function of the
Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as
a function of time by a high-temperature expansion. The resulting formula to
the third order of the inverse temperature shows that an assumption made by
Coolen, Laughton and Sherrington in their recent theory of dynamics is
violated. Deviations of their theory from exact results are estimated
quantitatively. Our formula also yields explicit expressions of the time
dependence of various macroscopic physical quantities when the temperature is
suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published
in J. Phys. Soc. Jpn. 65 (1996) No.
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