Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions

1989 ◽  
Vol 78 (398) ◽  
pp. 0-0 ◽  
Author(s):  
D. A. Dawson ◽  
J. Gärtner
Keyword(s):  
Free Energy ◽  
Large Deviations ◽  
Field Model ◽  
Mean Field ◽  
Energy Functional ◽  
Mean Field Model ◽  
Free Energy Functional ◽  
Interacting Diffusions

scholarly journals FREE ENERGY IN THE GENERALIZED SHERRINGTON–KIRKPATRICK MEAN FIELD MODEL

2005 ◽  
Vol 17 (07) ◽  
pp. 793-857 ◽  
Author(s):  
DMITRY PANCHENKO
Keyword(s):  
Free Energy ◽  
Real Line ◽  
Prior Distribution ◽  
Field Model ◽  
Mean Field ◽  
Rigorous Proof ◽  
Bounded Subset ◽  
Mean Field Model ◽  
The Real ◽  
Parisi Formula

In [11], Talagrand gave a rigorous proof of the Parisi formula in the classical Sherrington–Kirkpatrick (SK) model. In this paper, we build upon the methodology developed in [11] and extend Talagrand's result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.

scholarly journals Large Deviations for a Mean Field Model of Systemic Risk

2013 ◽  
Vol 4 (1) ◽  
pp. 151-184 ◽  
Author(s):  
Josselin Garnier ◽  
George Papanicolaou ◽  
Tzu-Wei Yang
Keyword(s):  
Large Deviations ◽  
Systemic Risk ◽  
Field Model ◽  
Mean Field ◽  
Mean Field Model

The Role of the Diffusion Mechanism in Models of the Evolution of Microstructure During Phase Separation

1998 ◽  
Vol 529 ◽  
Author(s):  
T.T. Rautialnen ◽  
A.P. Sutton
Keyword(s):  
Monte Carlo ◽  
Phase Separation ◽  
Field Model ◽  
Diffusion Mechanism ◽  
Monte Carlo Model ◽  
Mean Field ◽  
Energy Functional ◽  
Square Lattice ◽  
Mean Field Model ◽  
Evolution Of Microstructure

AbstractWe have studied phase separation and subsequent coarsening of the microstructure in a two-dimensional square lattice using a stochastic Monte Carlo model and a deterministic mean field model. The differences and similarities between these approaches are discussed. We have found that a realistic diffusion mechanism through a vacancy motion in Monte Carlo simulations is cruicial in producing different coarsening mechanisms over a range of temperatures. This cannot be captured by the mean field model, in which the transformation is governed by the minimization of a free energy functional.

scholarly journals The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 881 ◽  
Author(s):  
Pierre Baudot
Keyword(s):  
Machine Learning ◽  
Free Energy ◽  
Network Architecture ◽  
Statistical Physics ◽  
Mean Field ◽  
Minimum Free Energy ◽  
Energy Functional ◽  
Local Minima ◽  
Mean Field Model ◽  
Energy Principle

Previous works established that entropy is characterized uniquely as the first cohomology class in a topos and described some of its applications to the unsupervised classification of gene expression modules or cell types. These studies raised important questions regarding the statistical meaning of the resulting cohomology of information and its interpretation or consequences with respect to usual data analysis and statistical physics. This paper aims to present the computational methods of information cohomology and to propose its interpretations in terms of statistical physics and machine learning. In order to further underline the cohomological nature of information functions and chain rules, the computation of the cohomology in low degrees is detailed to show more directly that the k multivariate mutual information ( I k ) are ( k - 1 ) -coboundaries. The ( k - 1 ) -cocycles condition corresponds to I k = 0 , which generalizes statistical independence to arbitrary degree k. Hence, the cohomology can be interpreted as quantifying the statistical dependences and the obstruction to factorization. I develop the computationally tractable subcase of simplicial information cohomology represented by entropy H k and information I k landscapes and their respective paths, allowing investigation of Shannon’s information in the multivariate case without the assumptions of independence or of identically distributed variables. I give an interpretation of this cohomology in terms of phase transitions in a model of k-body interactions, holding both for statistical physics without mean field approximations and for data points. The I 1 components define a self-internal energy functional U k and ( - 1 ) k I k , k ≥ 2 components define the contribution to a free energy functional G k (the total correlation) of the k-body interactions. A basic mean field model is developed and computed on genetic data reproducing usual free energy landscapes with phase transition, sustaining the analogy of clustering with condensation. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, providing a trivial topological expression of the second law of thermodynamics. The local minima of free energy, related to conditional information negativity and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of unsupervised deep learning where the neural network architecture is given by the chain complex and conclude by discussing future supervised applications.

TILT ANGLE AND THE TEMPERATURE SHIFTS CALCULATED AS A FUNCTION OF CONCENTRATION FOR THE AC* PHASE TRANSITION IN A BINARY MIXTURE OF LIQUID CRYSTALS

2011 ◽  
Vol 25 (13) ◽  
pp. 1791-1806 ◽  
Author(s):  
H. YURTSEVEN ◽  
M. KURT
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Liquid Crystals ◽  
Binary Mixture ◽  
Tilt Angle ◽  
Field Model ◽  
Mean Field ◽  
Temperature Shift ◽  
Mean Field Model ◽  
Mean Field Models

We study here the tilt angle and the temperature shifts as a function of concentration for the AC* phase transition in a binary mixture, using our mean field model with the biquadratic P2θ2 coupling — and also with the bilinear Pθ and P2θ2 couplings. By expanding the free energy in terms of the tilt angle and polarization, the tilt angle and the temperature shift are evaluated by using the coefficients given in the free energy expansion. By employing a concentration-dependent coefficient, the tilt angle and the temperature shift are calculated as a function of concentration of 10.O.4 for the SmAC* transition in a binary mixture of C7 and 10.O.4. Our calculated values of the tilt angle and the temperature shifts decrease as the concentration of 10.O.4 increases, as confirmed experimentally for the AC* transition in this binary mixture. This indicates that our mean field models studied here are satisfactory to explain the observed behavior of the AC* transition of the binary mixture of C7 and 10.O.4.

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