Soon Hoe Lim
@su.se
Nordic Institute for Theoretical Physics
UNIVERSITY OF STOCKHOLM
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Soon Hoe Lim
IOP Publishing
Soon Hoe Lim, Ludovico Theo Giorgini, Woosok Moon, and J. S. Wettlaufer
AIP Publishing
We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.
Soon Hoe Lim, Jan Wehr, and Maciej Lewenstein
Springer Science and Business Media LLC
AbstractWe study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations for the slow degrees of freedom contain non-trivial drift correction terms and are driven by non-Markov noise processes. These results follow from a general homogenization theorem stated and proven here. We illustrate them using stochastic models of particle diffusion.
L. T. Giorgini, S. H. Lim, W. Moon, and J. S. Wettlaufer
IOP Publishing
In stochastic resonance, a periodically forced Brownian particle in a double-well potential jumps between minima at rare increments, the prediction of which pose a major theoretical challenge. Here, we use a path-integral method to predict these transitions by determining the most probable (or "{optimal}") space-time path of a particle. We characterize the optimal path using a direct comparison principle between the Langevin and Hamiltonian dynamical descriptions, allowing us to express the jump condition in terms of the accumulation of noise around the stable periodic path. In consequence, as a system approaches a rare event these fluctuations approach one of the deterministic minimizers, thereby providing a precursor for predicting the stochastic transition. We demonstrate the method numerically, which allows us to determine whether a state is following a stable periodic path or will experience an incipient jump. The vast range of systems that exhibit stochastic resonance behavior insures broad relevance of our framework, which allows one to extract precursor fluctuations from data.
Stefano Bo, Soon Hoe Lim, and Ralf Eichhorn
IOP Publishing
In stochastic thermodynamics standard concepts from macroscopic thermodynamics, such as heat, work, and entropy production, are generalized to small fluctuating systems by defining them on a trajec ...
Soon Hoe Lim and Jan Wehr
Springer Science and Business Media LLC
We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in Hottovy et al. (Commun Math Phys 336(3):1259–1283, 2015), whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.
Soon Hoe Lim, Jan Wehr, Aniello Lampo, Miguel Ángel García-March, and Maciej Lewenstein
Springer Science and Business Media LLC
We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal $$m = m_{0} \\epsilon $$m=m0ϵ, the reduced Planck constant to equal $$\\hbar = \\epsilon $$ħ=ϵ and the cutoff frequency to equal $$\\varLambda = E_{\\varLambda }/\\epsilon $$Λ=EΛ/ϵ, where $$m_0$$m0 and $$E_{\\varLambda }$$EΛ are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as $$\\epsilon \\rightarrow 0$$ϵ→0. We study the limit as $$\\epsilon \\rightarrow 0$$ϵ→0 of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.
Aniello Lampo, Soon Hoe Lim, Miguel Ángel García-March, and Maciej Lewenstein
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integrodifferential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.
Aniello Lampo, Soon Hoe Lim, Jan Wehr, Pietro Massignan, and Maciej Lewenstein
American Physical Society (APS)
Programa Masters d'Excel-lencia of the Fundacio Catalunya-La Pedrera; ERC; EU [323714]; Fundacio Cellex; Spanish MINECO [SEV-2015-0522, FIS2013-46768]; Generalitat de Catalunya [SGR 874]; "Ramon y Cajal" fellowship; NSF [MS 131271]