{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T21:42:52Z","timestamp":1774042972207,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T00:00:00Z","timestamp":1703116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Italian Ministry of University and Research (MUR)","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]},{"name":"the Italian Ministry of University and Research (MUR)","award":["CUP: E53D23018070001"],"award-info":[{"award-number":["CUP: E53D23018070001"]}]},{"name":"A Unitary Mathematical Framework for Modelling Muscular Dystrophies","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]},{"name":"A Unitary Mathematical Framework for Modelling Muscular Dystrophies","award":["CUP: E53D23018070001"],"award-info":[{"award-number":["CUP: E53D23018070001"]}]},{"name":"Italian National Group of Mathematical Physics (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM)","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]},{"name":"Italian National Group of Mathematical Physics (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM)","award":["CUP: E53D23018070001"],"award-info":[{"award-number":["CUP: E53D23018070001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The exact, non perturbative, response theory developed within the field of non-equilibrium molecular dynamics, also known as TTCF (transient time correlation function), applies to quite general dynamical systems. Its key element is called the dissipation function because it represents the power dissipated by external fields acting on the particle system of interest, whose coupling with the environment is given by deterministic thermostats. This theory has been initially developed for time-independent external perturbations, and then it has been extended to time-dependent perturbations. It has also been applied to dynamical systems of different nature, and to oscillator models undergoing phase transitions, which cannot be treated with, e.g., linear response theory. The present work includes time-dependent stochastic perturbations in the theory using the Karhunen\u2013Lo\u00e8ve theorem. This leads to three different investigations of a given process. In the first, a single realization of the stochastic coefficients is fixed, and averages are taken only over the initial conditions, as in a deterministic process. In the second, the initial condition is fixed, and averages are taken with respect to the distribution of stochastic coefficients. In the last investigation, one averages over both initial conditions and stochastic coefficients. We conclude by illustrating the applicability of the resulting exact response theory with simple examples.<\/jats:p>","DOI":"10.3390\/e26010012","type":"journal-article","created":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T10:53:52Z","timestamp":1703156032000},"page":"12","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exact Response Theory for Time-Dependent and Stochastic Perturbations"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-8031-3451","authenticated-orcid":false,"given":"Leonardo","family":"Iannella","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Torino, Via Pietro Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4223-6279","authenticated-orcid":false,"given":"Lamberto","family":"Rondoni","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy"},{"name":"INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kubo, R., Toda, M., and Hashitsume, N. 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