A mechanical approach to mean field spin models

Genovese, Giuseppe; Barra, Adriano

doi:10.1063/1.3131687

Cited by 51 publications

(88 citation statements)

References 10 publications

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“…Let c > 1 + ν − 1 k * and let us transform the order parameters m according to the rotation defined by P as in Lemma (1), namely m = Pm, or, more explicitly, m µ a = ν k=1 P ak m µ k . Following [27,58,67,68,69] let us also introduce the interpolating function Φ N,ν (t, x), where the variables t ∈ R + and x ∈ R ν×P are meant, respectively, as generalized time and space: In order to lighten the notation we also introduce the interpolating Boltzmann factor B N,ν (t, x) such that Φ N,ν (t, x) = 1 N ln σ B N,ν (t, x) and the average performed with respect to B N,ν (t, x) shall be denoted as · (t,x) . The interpolating partition function Z N,ν (t, x) is defined analogously.…”

Section: Solution Of the Multi-species Hopfield Modelmentioning

confidence: 99%

Non-convex Multi-species Hopfield Models

2018

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In this work we introduce a multi-species generalization of the Hopfield model for associative memory, where neurons are divided into groups and both inter-groups and intra-groups pair-wise interactions are considered, with different intensities. Thus, this system contains two of the main ingredients of modern Deep neural network architectures: Hebbian interactions to store patterns of information and multiple layers coding different levels of correlations. The model is completely solvable in the low-load regime with a suitable generalization of the Hamilton-Jacobi technique, despite the Hamiltonian can be a non-definite quadratic form of the magnetizations. The family of multi-species Hopfield model includes, as special cases, the 3-layers Restricted Boltzmann Machine (RBM) with Gaussian hidden layer and the Bidirectional Associative Memory (BAM) model. Date: July 11, 2018. arXiv:1807.03609v1 [cond-mat.dis-nn] 22 Jun 2018 1 The random variable z is chosen symmetrically distributed and, typically, its probability density is taken as p(z) = [1 − tanh 2 (z)]/2.

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Section: Solution Of the Multi-species Hopfield Modelmentioning

confidence: 99%

Non-convex Multi-species Hopfield Models

2018

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“…[3]). An alternative mean field approach has been recently introduced in [4] that allows to rigorously establish a formal analogy between the van der Waals model and magnetic mean field models such as the Curie-Weiss model and its multi-component extensions [5,6,7,8]. Interestingly, the phenomenological approach has played a key role over the decades, in particular for chemical engineering applications [9,10,11,12,13] aimed at providing a more accurate description of composite systems, such as solutions and multi-phase systems, for which a statistical physics approach * contact: antonio.moro@northumbria.ac.uk and a mean field theory is not currently available.…”

Section: Introductionmentioning

confidence: 99%

Integrable extended van der Waals model

Giglio

Landolfi

Moro

2016

Physica D: Nonlinear Phenomena

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Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we propose a novel approach to the construction of multi-parameter generalisations of the van der Waals model. The theory of integrable nonlinear conservation laws still represents the inspiring framework. Starting from a macroscopic approach, a four parameter family of integrable extended van der Waals models is indeed constructed in such a way that the equation of state is a solution to an integrable nonlinear conservation law linearisable by a Cole-Hopf transformation. This family is further specified by the request that, in regime of high temperature, far from the critical region, the extended model reproduces asymptotically the standard van der Waals equation of state. We provide a detailed comparison of our extended model with two notable empirical models such as Peng-Robinson and Soave's modification of the Redlich-Kwong equations of state. We show that our extended van der Waals equation of state is compatible with both empirical models for a suitable choice of the free parameters and can be viewed as a master interpolating equation. The present approach also suggests that further generalisations can be obtained by including the class of dispersive and viscous-dispersive nonlinear conservation laws and could lead to a new type of thermodynamic phase transitions associated to nonclassical and dispersive shock waves.

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“…In the following we summarize the minimal assumptions needed when modelling chemical kinetics from the Statistical Physics perspective; for a more extensive treatment of this kind of modelling we refer to [21,26,28,29,41], while for a rigorous explanation of the underlying equivalence between Statistical Mechanics and Analytical Mechanics we refer to the seminal works by Guerra [42], dealing with the Sherrington-Kirkpatrick model (and then deepened in, e.g., [43][44][45][46]), and by Brankov and Zagrebnov in [47], dealing with the Husimi-Temperley model (and then deepened in, e.g., [48][49][50][51] …”

Section: Formulation Of the Problem: The Thermodynamical Freementioning

confidence: 99%

Complex Reaction Kinetics in Chemistry: A Unified Picture Suggested by Mechanics in Physics

et al. 2018

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Complex biochemical pathways can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. Among the elementary reactions so far extensively investigated, we recall the Michaelis-Menten and the Hill positivecooperative kinetics, which apply to molecular binding and are characterized by the absence and the presence, respectively, of cooperative interactions between binding sites. However, there is evidence of reactions displaying a more complex pattern: these follow the positive-cooperative scenario at small substrate concentration, yet negative-cooperative effects emerge as the substrate concentration is increased. Here, we analyze the formal analogy between the mathematical backbone of (classical) reaction kinetics in Chemistry and that of (classical) mechanics in Physics. We first show that standard cooperative kinetics can be framed in terms of classical mechanics, where the emerging phenomenology can be obtained by applying the principle of least action of classical mechanics. Further, since the saturation function plays in Chemistry the same role played by velocity in Physics, we show that a relativistic scaffold naturally accounts for the kinetics of the above-mentioned complex reactions. The proposed formalism yields to a unique, consistent picture for cooperative-like reactions and to a stronger mathematical control.

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A mechanical approach to mean field spin models

Cited by 51 publications

References 10 publications

Non-convex Multi-species Hopfield Models

Non-convex Multi-species Hopfield Models

Integrable extended van der Waals model

Complex Reaction Kinetics in Chemistry: A Unified Picture Suggested by Mechanics in Physics

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