@article{SandevDomazetoskiKocarevetal.2022, author = {Sandev, Trifce and Domazetoski, Viktor and Kocarev, Ljupco and Metzler, Ralf and Chechkin, Aleksei}, title = {Heterogeneous diffusion with stochastic resetting}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {7}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac491c}, pages = {26}, year = {2022}, abstract = {We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation.}, language = {en} } @article{PetreskaPejovSandevetal.2022, author = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf}, title = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {2}, publisher = {MDPI AG, Fractal Fract Editorial Office}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6020088}, pages = {23}, year = {2022}, abstract = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.}, language = {en} } @article{ReichWeissmann2021, author = {Reich, Sebastian and Weissmann, Simon}, title = {Fokker-Planck particle systems for Bayesian inference: computational approaches}, series = {SIAM ASA journal on uncertainty quantification}, volume = {9}, journal = {SIAM ASA journal on uncertainty quantification}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {2166-2525}, doi = {10.1137/19M1303162}, pages = {446 -- 482}, year = {2021}, abstract = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.}, language = {en} } @article{MaoutsaReichOpper2020, author = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred}, title = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {8}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22080802}, pages = {35}, year = {2020}, abstract = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.}, language = {en} } @article{GodecMetzler2017, author = {Godec, Aljaž and Metzler, Ralf}, title = {First passage time statistics for two-channel diffusion}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa5204}, pages = {17}, year = {2017}, abstract = {We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Poisson-like behaviour if none of the phases has a vanishing diffusion coefficient. This stands in stark contrast to the standard (one-channel) Markov diffusion counterpart. We also discuss the relevance of our results in the context of cellular signalling.}, language = {en} }