Fundamental problems in Statistical Physics XIV: Lecture on Correlation and response functions in statistical physics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912990736941056 |
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| author | Franosch, Thomas |
| author_facet | Franosch, Thomas |
| contents | In the first part of these short lecture notes, we will present an introduction on (auto-)correlation functions and linear-response functions in the language of a physicist. In particular, the fluctuation-dissipation theorem in classical physics is presented underlining the central role of correlation functions. The fundamental importance of (auto-)correlation functions raises the natural question on how they are characterized in general without referring to the concrete underlying dynamical laws. Perhaps unexpectedly -- despite being elegant and long established in the mathematical literature (Bochner's theorem for correlations; Herglotz-Nevanlinna representations for response) -- this answer is not widely appreciated in physics, partly because the requisite tools lie outside the standard curriculum.
In the second part we adopt a more rigorous viewpoint: we state the key structural properties of correlation functions and provide selected derivations of these results. Finally, we return to linear response and discuss general characterization results for response functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29481 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fundamental problems in Statistical Physics XIV: Lecture on Correlation and response functions in statistical physics Franosch, Thomas Statistical Mechanics In the first part of these short lecture notes, we will present an introduction on (auto-)correlation functions and linear-response functions in the language of a physicist. In particular, the fluctuation-dissipation theorem in classical physics is presented underlining the central role of correlation functions. The fundamental importance of (auto-)correlation functions raises the natural question on how they are characterized in general without referring to the concrete underlying dynamical laws. Perhaps unexpectedly -- despite being elegant and long established in the mathematical literature (Bochner's theorem for correlations; Herglotz-Nevanlinna representations for response) -- this answer is not widely appreciated in physics, partly because the requisite tools lie outside the standard curriculum. In the second part we adopt a more rigorous viewpoint: we state the key structural properties of correlation functions and provide selected derivations of these results. Finally, we return to linear response and discuss general characterization results for response functions. |
| title | Fundamental problems in Statistical Physics XIV: Lecture on Correlation and response functions in statistical physics |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2603.29481 |