Mathematical Physics (MP)http://hdl.handle.net/20.500.11824/162019-09-22T18:25:14Z2019-09-22T18:25:14ZSingle-trajectory spectral analysis of scaled Brownian motionSposini V.Metzler R.Oshanin G.http://hdl.handle.net/20.500.11824/10102019-09-09T01:00:12Z2019-06-01T00:00:00ZSingle-trajectory spectral analysis of scaled Brownian motion
Sposini V.; Metzler R.; Oshanin G.
A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, $T\to \infty $. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit $T\to \infty $ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion. We demonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent. We also compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
2019-06-01T00:00:00ZReduced description method in the kinetic theory of Brownian motion with active fluctuationsSliusarenko O.Sliusarenko Y.http://hdl.handle.net/20.500.11824/10062019-09-04T01:00:10Z2019-09-01T00:00:00ZReduced description method in the kinetic theory of Brownian motion with active fluctuations
Sliusarenko O.; Sliusarenko Y.
We develop a microscopic approach to the kinetic theory of many-particle systems with dissipative and potential interactions in presence of active fluctuations. The approach is based on a generalization of Bogolyubov–Peletminskii reduced description method applied to the systems of many active particles. It is shown that the microscopic approach developed allows to construct the kinetic theory of two- and three-dimensional systems of active particles in presence of non-linear friction (dissipative interaction) and an external random field with active fluctuations. The kinetic equations for these systems in case of a weak interaction between the particles (both potential and dissipative ones) and low-intensity active fluctuations are obtained. We demonstrate particular cases in which the derived kinetic equations have solutions that match the results known in the literature. In addition, analysis of particular solutions showed that in the case of a friction force linearly dependent on the speed of structural units, the manifestation of self-propelling properties in the process of evolution of an active medium may be due to the local nature of active fluctuations.
2019-09-01T00:00:00ZOn the propagation of nonlinear transients of temperature and pore pressure in a thin porous boundary layer between two rocks.Salusti E.Kanivetsky R.Droghei R.Garra R.http://hdl.handle.net/20.500.11824/10022019-08-07T01:00:13Z2019-01-01T00:00:00ZOn the propagation of nonlinear transients of temperature and pore pressure in a thin porous boundary layer between two rocks.
Salusti E.; Kanivetsky R.; Droghei R.; Garra R.
The dynamics of transients of fluid-rock temperature, pore pressure, pollutants in porous rocks are of vivid interest for fundamental problems in hydrological, volcanic, hydrocarbon systems, deep oil drilling. This can concern rapid landslides or the fault weakening during coseismic slips and also a new field of research about stability of classical buildings. Here we analyze the transient evolution of temperature and pressure in a thin boundary layer between two adjacent homogeneous media for various types of rocks. In previous models, this boundary was often assumed to be a sharp mathematical plane. Here we consider a non-sharp, physical boundary between two adjacent rocks, where also local steady pore pressure and/or temperature fields are present. To obtain a more reliable model we also investigate the role of nonlinear effects as convection and fluid-rock “frictions”, often disregarded in early models: these nonlinear effects in some cases can give remarkable quick and sharp transients. All of this implies a novel model, whose solutions describe large, sharp and quick fronts. We also rapidly describe transients moving through a particularly irregular boundary layer.
2019-01-01T00:00:00ZA Lê-Greuel type formula for the image Milnor numberNuño-Ballesteros J.J.Pallarés Torres I.http://hdl.handle.net/20.500.11824/9922019-07-12T01:00:12Z2019-02-01T00:00:00ZA Lê-Greuel type formula for the image Milnor number
Nuño-Ballesteros J.J.; Pallarés Torres I.
Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p\colon (\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g\colon (\mathbb{C}^{n-1},0)\to (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $\mu_I(f)+\mu_I(g)$ is equal to the number of critical points of $p|_{X_s}\colon X_s\to\mathbb{C}$ on all the strata of $X_s$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).
2019-02-01T00:00:00Z