Out of time ordered effective dynamics of a Brownian particle

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He has presented the out of time-ordered (OTO) dynamics of a Brownian particle interacting with a large environment. To illustrate the features of these dynamics, He has described a toy model where the environment is a thermal bath comprising of two sets of harmonic oscillators coupled nonlinearly to the particle. Beginning with a Schwinger-Keldysh effective action of the particle, He has demonstrated its duality with non-linear Langevin dynamics. This Langevin dynamics or the equivalent Schwinger-Keldysh effective theory is, however, inadequate for determining the OTO correlators of the particle. Hr has shown that this limitation can be overcome by extending the effective theory to a path integral formalism defined on a contour with multiple time-folds. These extended effective dynamics has to satisfy some constraints due to microscopic reversibility and thermality of the environment. He has shown that, from the perspective of the Langevin dynamics, these constraints lead to a generalized fluctuation-dissipation relation between a non-Gaussianity in the noise distribution and a thermal jitter in the particle's damping coefficient.