Law of the Iterated Logarithm for some Markov operators

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).Comment: 23 page

New Class of Generalized Extensive Entropies for Studying Dynamical Systems in Highly Anisotropic Phase Space

Starting from the geometrical interpretation of the R\'enyi entropy, we introduce further extensive generalizations and study their properties. In particular, we found the probability distribution function obtained by the MaxEnt principle with generalized entropies. We prove that for a large class of dynamical systems subject to random perturbations, including particle transport in random media, these entropies play the role of Liapunov functionals. Some physical examples, which can be treated by the generalized R\'enyi entropies are also illustrated.Comment: 13 pages, 0 figure

Chaotic Mixing in Three Dimensional Porous Media

Under steady flow conditions, the topological complexity inherent to all random 3D porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a three-dimensional (3D) fluid mechanical analogue of the baker's map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW) which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with longitudinal advection, whereas the topological constraints associated with 2D porous media limits mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms.Comment: 36 page

Thresholds and Sediment Transport Dynamics in an Interbedded Shale and Limestone Controlled Urban Watercourse

Sediment transport is a fundamental component of research into river morphology and related engineering practices. The relationship between flow and sediment particle entrainment underpins many of the empirical models used to estimate sediment transport dynamics. The scientific literature reports a research gap specific to the thresholds of mobility of different sized particles in non-gravel bed systems, including those in bedrock channels. Particle tracer technology was used to study coarse sediment entrainment and transport dynamics in an urban, bedrock controlled stream channel in Toronto, Ontario, Canada. Passive integrated transponders were inserted in constrained and unconstrained particles within an incised reach of stream. The distribution of particles transport distances conformed to a two-parameter gamma distribution model, which assumes integrations of the travelled series of steps and rests. Size selective dependency of path length was found to increase for coarser clasts, as compared to observed conditions for gravel-bed systems. Coarser particles were also found to transport in an unconstrained mode, as compared to finer grains. A force exceedance model was applied to further test the performance of reported size selective transport relationships for the study site. Many particles were found to transport at critical shear ratios less than 1, when assuming a modified Shields’s based model for entrainment. Field data was then used to determine a reference shear based on the smallest magnitude competent storm. The results show that, when compared to alluvial gravel-bed conditions, finer particles require larger thresholds to mobilize and the inverse is true for coarser particles. Using the reference shear conditions, rates of sediment transport were calculated and compared to common models for coarse particle transport. The results confirm size selectivity by grain class and indicate differentiations between fine and coarse transport relationships for the site. This research confirms non-conformity of particle entrainment and transport relationships for the study site, when compared to common empirical model for gravel-bed rivers. The results may be used to obtain critical entrainment parameters and sediment transport relationships, which can then be used to inform design criteria for regional watercourses having like lithology and morphology

Stirring and mixing : 1999 Program of Summer Study in Geophysical Fluid Dynamics

The central theme of the 1999 GFD Program was the stirring, transport, reaction and mixing of passive and active tracers in turbulent, stratified, rotating fluids. The problem of mixing in fluids has applications in areas ranging from oceanography to engineering and astrophysics. In geophysical settings, mixing spans and unites a broad range of scales -- from micrometers to megameters. The mixing of passive tracers is of fundamental importance in environmental and industrial problems, such as pollution, and in determining the large-scale heat and salt balance of the worlds oceans. The transport of active tracers, on the other hand, such as vorticity, plays a key role in the turbulence that occurs in most geophysical and astrophysical fluids. William R. Young (Scripps Institution of Oceanography) gave a series of principal lectures, the notes of which as taken by the fellows, appear in this volume. Report of the projects of the student fellows makes up the second half of this volume.Funding was provided by the National Science Foundation under Grant No. OCE-9810647 and the Office of Naval Research under Grant No. NOO0l4-97-1-0934

Functional Limit Theorems for Volterra Processes and Applications to Homogenization

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the L\'evy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type It\^o SDE $dx_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with spatial parameters.Comment: 31 page

Foundations of Stochastic Thermodynamics

Small systems in a thermodynamic medium --- like colloids in a suspension or the molecular machinery in living cells --- are strongly affected by the thermal fluctuations of their environment. Physicists model such systems by means of stochastic processes. Stochastic Thermodynamics (ST) defines entropy changes and other thermodynamic notions for individual realizations of such processes. It applies to situations far from equilibrium and provides a unified approach to stochastic fluctuation relations. Its predictions have been studied and verified experimentally. This thesis addresses the theoretical foundations of ST. Its focus is on the following two aspects: (i) The stochastic nature of mesoscopic observations has its origin in the molecular chaos on the microscopic level. Can one derive ST from an underlying reversible deterministic dynamics? Can we interpret ST's notions of entropy and entropy changes in a well-defined information-theoretical framework? (ii) Markovian jump processes on finite state spaces are common models for bio-chemical pathways. How does one quantify and calculate fluctuations of physical observables in such models? What role does the topology of the network of states play? How can we apply our abstract results to the design of models for molecular motors? The thesis concludes with an outlook on dissipation as information written to unobserved degrees of freedom --- a perspective that yields a consistency criterion between dynamical models formulated on various levels of description.Comment: Ph.D. Thesis, G\"ottingen 2014, Keywords: Stochastic Thermodynamics, Entropy, Dissipation, Markov processes, Large Deviation Theory, Molecular Motors, Kinesi