A controversial problem: modified Ising model in a random field

Villain, Jacques; Semeria, B; Lançon, Frédéric; Billard, L.

doi:10.1088/0022-3719/16/32/009

Cited by 63 publications

(42 citation statements)

References 38 publications

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“…Equation (6) is in agreement with the above prediction b = y, and we speculate that in the case a-0 a profile with an arbitrary initial shape will tend to get close to (6).…”

Section: (I X I < L) 4l Is the Wavelength And Hit) Is The Amplitude supporting

confidence: 86%

“…A detailed calculation of n on ledges in the anisotropic case is given in Ref. 6. Now we come back to the decay of grooves.…”

Section: V-hoffz(x 9 Y)dxdy + Ffmentioning

confidence: 99%

“…1,5 This case is complicated and only a qualitative treatment can be given. 6 Here the equations of motion are written in the simpler case of evaporation dynamics (valid for longer wavelengths 1 ). The evaporation rate is proportional to the difference between the local chemical potential n at the surface and the chemical potential no of the vapor, assumed to be fixed.…”

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confidence: 99%

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Dynamics of a crystal surface below its roughening transition

Lançon

Villain

1990

Phys. Rev. Lett.

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The equations of the smoothening dynamics of a crystal surface below the roughening transition are written in the case of evaporation dynamics. It is assumed that only ledges separating smooth planes are mobile. The equations are applied to the classical experiment in which the decay of parallel grooves is studied. The formation of facets, which has been observed, is argued to be possible only if the average surface makes a small, but nonvanishing angle a with a high-symmetry direction. If a -0, the profile is predicted to have sharp maxima and minima with j singularities.

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“…Equation (6) is in agreement with the above prediction b = y, and we speculate that in the case a-0 a profile with an arbitrary initial shape will tend to get close to (6).…”

Section: (I X I < L) 4l Is the Wavelength And Hit) Is The Amplitude supporting

confidence: 86%

“…A detailed calculation of n on ledges in the anisotropic case is given in Ref. 6. Now we come back to the decay of grooves.…”

Section: V-hoffz(x 9 Y)dxdy + Ffmentioning

confidence: 99%

mentioning

confidence: 99%

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Dynamics of a crystal surface below its roughening transition

Lançon

Villain

1990

Phys. Rev. Lett.

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“…K. In particular, in the simplest case of the twoscale model K = 2 assuming 0 = ν 2 < ν 1 < 1 for the exponents, we obtain the probability distribution quoted in Eq. (18). The expressions for (y min , y max ) suffice to calculate the free energy expression in the thermodynamic limit.…”

Section: Analytical Results For N → ∞mentioning

confidence: 99%

Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces

Fyodorov¹,

Bouchaud²

2008

J. Phys. A: Math. Theor.

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We construct a N −dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N → ∞ the free energy of the system and overlap function are calculated exactly using the replica trick and Parisi's hierarchical ansatz. In the thermodynamic limit, we recover the most general version of the Derrida's Generalized Random Energy Model (GREM). The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. If the latter consists of K discrete values, the system is characterized by a K-step Replica Symmetry Breaking solution. We argue that our construction is in fact valid in any finite spatial dimensions N ≥ 1. We discuss the implications of our results for the singularity spectrum describing multifractality of the associated Boltzmann-Gibbs measure. Finally we discuss several generalisations and open problems, such as the dynamics in such a landscape and the construction of a Generalized Multifractal Random Walk.PACS numbers 64.60. De, 64.60.Al

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“…Methods inherited from spin glass theory also provided insight into many other problems, several of them originating from outside of physics. Prominent examples are various models of interfaces in random environment [71][72][73][74][75], granular media [76], combinatorial optimization (see [14] for an early review and [15] for a new development), game theory [77], protein and nucleic acid folding [78][79][80][81][82], and noise reduction in signal processing [83]. Last but not least, as we will expound it in the present paper, methods first introduced for describing the equilibrium properties of the SK model are of paramount importance in the statistical mechanical approach to neural networks.…”

Section: Sherrington-kirkpatrick Modelmentioning

confidence: 99%

Techniques of replica symmetry breaking and the storage problem of the McCulloch–Pitts neuron

Györgyi

2001

Physics Reports

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In this article we review the framework for spontaneous replica symmetry breaking. Subsequently that is applied to the example of the statistical mechanical description of the storage properties of a McCulloch-Pitts neuron, i. e., simple perceptron. It is shown that in the neuron problem the general formula appears that is at the core of all problems admitting Parisi's replica symmetry breaking ansatz with a one-component order parameter. The details of Parisi's method are reviewed extensively, with regard to the wide range of systems where the method may be applied. Parisi's partial differential equation and related differential equations are discussed, and a Green function technique introduced for the calculation of replica averages, the key to determining the averages of physical quantities. The Green functions of the Fokker-Planck equation due to Sompolinsky turns out to play the role of the statistical mechanical Green function in the graph rules for replica correlators. The subsequently obtained graph rules involve only tree graphs, as appropriate for a mean-field-like model. The lowest order Ward-Takahashi identity is recovered analytically and shown to lead to the Goldstone modes in continuous replica symmetry breaking phases. The need for a replica symmetry breaking theory in the storage problem of the neuron has arisen due to the thermodynamical instability of formerly given solutions. Variational forms for the neuron's free energy are derived in terms of the order parameter function x(q), for different prior distribution of synapses. Analytically in the high temperature limit and numerically in generic cases various phases are identified, among them one similar to the Parisi phase in long range interaction spin glasses. Extensive quantities like the error per pattern change slightly with respect to the known unstable solutions, but there is a significant difference in the distribution of non-extensive quantities like the synaptic overlaps and the pattern storage stability parameter. A simulation result is also reviewed and compared to the prediction of the theory. 07.05. Mh, 75.10Nr, 84.35.+i

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A controversial problem: modified Ising model in a random field

Cited by 63 publications

References 38 publications

Dynamics of a crystal surface below its roughening transition

Dynamics of a crystal surface below its roughening transition

Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces

Techniques of replica symmetry breaking and the storage problem of the McCulloch–Pitts neuron

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