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Register a new userTime evolution of the extremely diluted Blume-Emery-Griffiths neural network
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Added by
Rubem Erichsen Junior
Publication date
2002
fields
Physics
and research's language is
English
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The time evolution of the extremely diluted Blume-Emery-Griffiths neural network model is studied, and a detailed equilibrium phase diagram is obtained exhibiting pattern retrieval, fluctuation retrieval and self-sustained activity phases. It is shown that saddle-point solutions associated with fluctuation overlaps slow down considerably the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.
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The optimal capacity of a diluted Blume-Emery-Griffiths neural network is studied as a function of the pattern activity and the embedding stability using the Gardner entropy approach. Annealed dilution is considered, cutting some of the couplings referring to the ternary patterns themselves and some of the couplings related to the active patterns, both simultaneously (synchronous dilution) or independently (asynchronous dilution). Through the de Almeida-Thouless criterion it is found that the replica-symmetric solution is locally unstable as soon as there is dilution. The distribution of the couplings shows the typical gap with a width depending on the amount of dilution, but this gap persists even in cases where a particular type of coupling plays no role in the learning process.
The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied for arbitrary temperature. By employing a probabilistic signal-to-noise approach, a recursive scheme is found determining the time evolution of the distribution of the local fields and, hence, the evolution of the order parameters. A comparison of this approach is made with the generating functional method, allowing to calculate any physical relevant quantity as a function of time. Explicit analytic formula are given in both methods for the first few time steps of the dynamics. Up to the third time step the results are identical. Some arguments are presented why beyond the third time step the results differ for certain values of the model parameters. Furthermore, fixed-point equations are derived in the stationary limit. Numerical simulations confirm our theoretical findings.
The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature for arbitrary using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example, explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and compared with the thermodynamic approach. The analytic results find excellent confirmation in extensive numerical simulations.
We extend the Blume-Emery-Griffiths (BEG) model to a two-component BEG model in order to study 2D systems with two order parameters, such as magnetic superconductors or two-component Bose-Einstein condensates. The model is investigated using Monte Carlo simulations, and the temperature-concentration phase diagram is determined in the presence and absence of an external magnetic field. This model exhibits a rich phase diagram, including a second-order transition to a phase where superconductivity and magnetism coexist. Results are compared with experiments on Cerium-based heavy-fermion superconductors. To study cold atom mixtures, we also simulate the BEG and two-component BEG models with a trapping potential. In the BEG model with a trap, there is no longer a first order transition to a true phase-separated regime, but a crossover to a kind of phase-separated region. The relation with imbalanced fermi-mixtures is discussed. We present the phase diagram of the two-component BEG model with a trap, which can describe boson-boson mixtures of cold atoms. Although there are no experimental results yet for the latter, we hope that our predictions could help to stimulate future experiments in this direction.
The canonical phase diagram of the Blume-Emery-Griffiths (BEG) model with infinite-range interactions is known to exhibit a fourth order critical point at some negative value of the bi-quadratic interaction $K<0$. Here we study the microcanonical phase diagram of this model for $K<0$, extending previous studies which were restricted to positive $K$. A fourth order critical point is found to exist at coupling parameters which are different from those of the canonical ensemble. The microcanonical phase diagram of the model close to the fourth order critical point is studied in detail revealing some distinct features from the canonical counterpart.
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