Titlebook: Statistical Field Theory for Neural Networks; Moritz Helias,David Dahmen Book 2020 Springer Nature Switzerland AG 2020 Statistical physics

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Perturbation Expansion,which can be systematically calculated via Taylor expansions and expectation values with regard to the solvable part of the theory. Diagrammatic rules for Gaussian solvable theories are presented to organize perturbation expansions using Feynman diagrams. The general rules are exemplified using a “.. + ..” theory.

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Perturbation Theory for Stochastic Differential Equations,part of the theory. We here want to show how to calculate perturbative corrections that arise from non-linearities in the stochastic differential equation, corresponding to the non-Gaussian part of the action.

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Probabilities, Moments, Cumulants,the cumulant-generating function. It, correspondingly, introduces moments and cumulants and their mutual connections. These definitions are key to the subsequent concepts, such as the perturbative computation of statistics.

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,Gaussian Distribution and Wick’s Theorem, theory boils down to a clever reorganization of Gaussian integrals. In this section we will therefore derive fundamental properties of this distribution. The diagrammatic perturbative methods to be developed in subsequent chapters rely on these elementary properties: The lines in Feynman diagrams r

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Perturbation Expansion,lvable part and a small perturbation. The general concept of an action allows the definition of an interaction potential that introduces corrections, which can be systematically calculated via Taylor expansions and expectation values with regard to the solvable part of the theory. Diagrammatic rules

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Functional Formulation of Stochastic Differential Equations,spects. First, it expresses the dynamical equations into a path-integral, where the dynamic equations give rise to the definition of an “action.” In this way, the perturbation expansion with the help of Feynman diagrams or the loopwise expansions to obtain a systematic treatment of fluctuations (in

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