1,486 research outputs found
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.
Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects
We develop further a recent dynamical replica theory to describe the dynamics
of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution
equations for macroscopic order parameters. We show how microscopic memory
effects can be included in the formalism through the introduction of a dynamic
order parameter function: the joint spin-field distribution. The resulting
formalism describes very accurately the relaxation phenomena observed in
numerical simulations, including the typical overall slowing down of the flow
that was missed by the previous simple two-parameter theory. The advanced
dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl
Cluster Derivation of the Parisi Scheme for Disordered Systems
We propose a general quantitative scheme in which systems are given the
freedom to sacrifice energy equi-partitioning on the relevant time-scales of
observation, and have phase transitions by separating autonomously into ergodic
sub-systems (clusters) with different characteristic time-scales and
temperatures. The details of the break-up follow uniquely from the requirement
of zero entropy for the slower cluster. Complex systems, such as the
Sherrington-Kirkpatrick model, are found to minimise their free energy by
spontaneously decomposing into a hierarchy of ergodically equilibrating degrees
of freedom at different (effective) temperatures. This leads exactly and
uniquely to Parisi's replica symmetry breaking scheme. Our approach, which is
somewhat akin to an earlier one by Sompolinsky, gives new insight into the
physical interpretation of the Parisi scheme and its relations with other
approaches, numerical experiments, and short range models. Furthermore, our
approach shows that the Parisi scheme can be derived quantitatively and
uniquely from plausible physical principles.Comment: 6 pages, 3 figures, proceedings of international conference on
"Disordered And Complex Systems", 10-14 July 2000 King's College Londo
Market response to external events and interventions in spherical minority games
We solve the dynamics of large spherical Minority Games (MG) in the presence
of non-negligible time dependent external contributions to the overall market
bid. The latter represent the actions of market regulators, or other major
natural or political events that impact on the market. In contrast to
non-spherical MGs, the spherical formulation allows one to derive closed
dynamical order parameter equations in explicit form and work out the market's
response to such events fully analytically. We focus on a comparison between
the response to stationary versus oscillating market interventions, and reveal
profound and partially unexpected differences in terms of transition lines and
the volatility.Comment: 14 pages LaTeX, 5 (composite) postscript figures, submitted to
Journal of Physics
Theory of agent-based market models with controlled levels of greed and anxiety
We use generating functional analysis to study minority-game type market
models with generalized strategy valuation updates that control the psychology
of agents' actions. The agents' choice between trend following and contrarian
trading, and their vigor in each, depends on the overall state of the market.
Even in `fake history' models, the theory now involves an effective overall bid
process (coupled to the effective agent process) which can exhibit profound
remanence effects and new phase transitions. For some models the bid process
can be solved directly, others require Maxwell-construction type
approximations.Comment: 30 pages, 10 figure
Dynamics of on-line Hebbian learning with structurally unrealizable restricted training sets
We present an exact solution for the dynamics of on-line Hebbian learning in
neural networks, with restricted and unrealizable training sets. In contrast to
other studies on learning with restricted training sets, unrealizability is
here caused by structural mismatch, rather than data noise: the teacher machine
is a perceptron with a reversed wedge-type transfer function, while the student
machine is a perceptron with a sigmoidal transfer function. We calculate the
glassy dynamics of the macroscopic performance measures, training error and
generalization error, and the (non-Gaussian) student field distribution. Our
results, which find excellent confirmation in numerical simulations, provide a
new benchmark test for general formalisms with which to study unrealizable
learning processes with restricted training sets.Comment: 7 pages including 3 figures, using IOP latex2e preprint class fil
Solvable Lattice Gas Models of Random Heteropolymers at Finite Density: II. Dynamics and Transitions to Compact States
In this paper we analyse both the dynamics and the high density physics of
the infinite dimensional lattice gas model for random heteropolymers recently
introduced in \cite{jort}. Restricting ourselves to site-disordered
heteropolymers, we derive exact closed deterministic evolution equations for a
suitable set of dynamic order parameters (in the thermodynamic limit), and use
these to study the dynamics of the system for different choices of the monomer
polarity parameters. We also study the equilibrium properties of the system in
the high density limit, which leads to a phase diagram exhibiting transitions
between swollen states, compact states, and regions with partial
compactification. Our results find excellent verification in numerical
simulations, and have a natural and appealing interpretation in terms of real
heteropolymers.Comment: 12 pages, 8 eps figures, revised version (to be published in EPJ
Generating functional analysis of Minority Games with real market histories
It is shown how the generating functional method of De Dominicis can be used
to solve the dynamics of the original version of the minority game (MG), in
which agents observe real as opposed to fake market histories. Here one again
finds exact closed equations for correlation and response functions, but now
these are defined in terms of two connected effective non-Markovian stochastic
processes: a single effective agent equation similar to that of the `fake'
history models, and a second effective equation for the overall market bid
itself (the latter is absent in `fake' history models). The result is an exact
theory, from which one can calculate from first principles both the persistent
observables in the MG and the distribution of history frequencies.Comment: 39 pages, 5 postscript figures, iop styl
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