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Derivation of the Langevin Equation from the Microcanonical Ensemble
KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
2024 (English)In: Entropy, E-ISSN 1099-4300, Vol. 26, no 4, article id 277Article in journal (Refereed) Published
Abstract [en]

When writing down a Langevin equation for the time evolution of a “system” in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker–Planck equation is established.

Place, publisher, year, edition, pages
Multidisciplinary Digital Publishing Institute (MDPI) , 2024. Vol. 26, no 4, article id 277
Keywords [en]
Brownian motion, driven diffusion, fluctuation–dissipation relation, Langevin equation, microcanonical ensemble
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kth:diva-346381DOI: 10.3390/e26040277ISI: 001210193400001Scopus ID: 2-s2.0-85191566162OAI: oai:DiVA.org:kth-346381DiVA, id: diva2:1857575
Note

QC 20240516

Available from: 2024-05-14 Created: 2024-05-14 Last updated: 2024-05-16Bibliographically approved

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