Building Intuition for Dynamical Mean-Field Theory: A Simple Model and the Cavity Method
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916857197363200 |
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| author | Blumenthal, Emmy |
| author_facet | Blumenthal, Emmy |
| contents | Dynamical Mean-Field Theory (DMFT) is a powerful theoretical framework for analyzing systems with many interacting degrees of freedom. This tutorial provides an accessible introduction to DMFT. We begin with a linear model where the DMFT equations can be derived exactly, allowing readers to develop clear intuition for the underlying principles. We then introduce the cavity method, a versatile approach for deriving DMFT equations for non-linear systems. The tutorial concludes with an application to the generalized Lotka--Volterra model of interacting species, demonstrating how DMFT reduces the complex dynamics of many-species communities to a tractable single-species stochastic process. Key insights include understanding how quenched disorder enables the reduction from many-body to effective single-particle dynamics, recognizing the role of self-averaging in simplifying complex systems, and seeing how collective interactions give rise to non-Markovian feedback effects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16654 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Building Intuition for Dynamical Mean-Field Theory: A Simple Model and the Cavity Method Blumenthal, Emmy Disordered Systems and Neural Networks Biological Physics Dynamical Mean-Field Theory (DMFT) is a powerful theoretical framework for analyzing systems with many interacting degrees of freedom. This tutorial provides an accessible introduction to DMFT. We begin with a linear model where the DMFT equations can be derived exactly, allowing readers to develop clear intuition for the underlying principles. We then introduce the cavity method, a versatile approach for deriving DMFT equations for non-linear systems. The tutorial concludes with an application to the generalized Lotka--Volterra model of interacting species, demonstrating how DMFT reduces the complex dynamics of many-species communities to a tractable single-species stochastic process. Key insights include understanding how quenched disorder enables the reduction from many-body to effective single-particle dynamics, recognizing the role of self-averaging in simplifying complex systems, and seeing how collective interactions give rise to non-Markovian feedback effects. |
| title | Building Intuition for Dynamical Mean-Field Theory: A Simple Model and the Cavity Method |
| topic | Disordered Systems and Neural Networks Biological Physics |
| url | https://arxiv.org/abs/2507.16654 |