Address:Alexander GrigoDepartment of Mathematics University of Oklahoma Norman, OK 73019 |
Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. Johann Wolfgang von Goethe (1749 - 1832) |
Dynamical systems, in particular ergodic theory, statistical properties of deterministic and random dynamical systems, probability theory, KAM, billiards, bifurcation theory, pattern formation, partial differential equations, and applications to non-equilibrium processes in statistical physics. My research mainly focuses on analytic results, but also involves numerical studies.
Abstract: We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to extend standard KAM theory to analyze stability of periodic orbits, because of the low smoothness of the system. In fact, the stadium has a $C^1$ boundary, and we show that $C^2$ smoothing results in appearance of elliptic periodic orbits.
Abstract: We derive Haff's cooling law for a periodic fluid consisting of two hard disks per unit cell by reducing it to a point particle moving inside a Sinai billiard with finite horizon with an inelastic collision rule. Indeed, our results also apply to general dispersing billiards with piece-wise smooth boundary with finite horizon and no cusps.
Abstract: In this paper we propose an algorithm to numerically simulate Markov processes of jump type. While these processes can naturally be generated as solutions to jump-SDEs the algorithm we propose is instead based on the interlacing construction of the process. We show that one can construct in the sense of strong convergence a high order numerical scheme based on high order ODE solvers. This result is in sharp contrast to the well known difficulty of constructing high-order numerical schemes for diffusion processes.
Abstract: We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).
Abstract: We review the current state of a fundamental problem of rigorous derivation of transport processes in classical statistical mechanics from classical mechanics. Such derivations for diffusion and momentum transport (viscosities) were obtained for minimal models of these processes involving one and two particles respectively. However, a minimal model which demonstrates heat conductivity contains three particles. Its rigorous analysis is currently out of reach for existing mathematical techniques. The gas of localized balls is widely accepted as a basis for a simplest model for derivation of Fourier's law. We suggest a modification of the localized balls gas and argue that this gas of localized activated balls is a good candidate to rigorously prove Fourier's law. In particular, hyperbolicity is derived for a reduced version of this model.
Abstract: Computation plays a key role in predicting and analyzing natural phenomena. There are two fundamental barriers to our ability to computationally understand the long-term behavior of a dynamical system that describes a natural process.The first one is unaccounted-for errors, which may make the system unpredictable beyond a very limited time horizon. This is especially true for chaotic systems, where a small change in the initial conditions may cause a dramatic shift in the trajectories. The second one is Turing-completeness. By the undecidability of the Halting Problem, the long-term prospects of a system that can simulate a Turing Machine cannot be determined computationally. We investigate the interplay between these two forces – unaccounted-for errors and Turing-completeness. We show that the introduction of even a small amount of noise into a dynamical system is sufficient to "destroy" Turing-completeness, and to make the system’s long-term behavior computationally predictable. On a more technical level, we deal with long-term statistical properties of dynamical systems, as described by invariant measures. We show that while there are simple dynamical systems for which the invariant measures are non-computable, perturbing such systems makes the invariant measures efficiently computable. Thus, noise that makes the short term behavior of the system harder to predict, may make its long term statistical behavior computationally tractable. We also obtain some insight into the computational complexity of predicting systems affected by random noise.
Abstract: A fundamental problem of non-equilibrium statistical mechanics is the derivation of macroscopic transport equations in the hydrodynamic limit. The rigorous study of such limits requires detailed information about rates of convergence to equilibrium for finite sized systems. In this paper, we consider the finite lattice $\{1, 2, \ldots, N\}$, with an energy $x_i \in (0,\infty)$ associated with each site. The energies evolve according to a Markov jump process with nearest neighbour interaction such that the total energy is preserved. We prove that for an entire class of such models the spectral gap of the generator of the Markov process scales as $O(N^{−2})$. Furthermore, we provide a complete classification of reversible stationary distributions of product type. We demonstrate that our results apply to models similar to the billiard lattice model considered in Gaspard and Gilbert (2009 J. Stat. Mech.: Theory Exp. 2009 24), and hence provide a first step in the derivation of a macroscopic heat equation for a microscopic stochastic evolution of mechanical origin.
Abstract: We demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing. More precisely, we construct billiard tables with arbitrary long free path away from a non-absolutely focusing component such that a nonlinearly stable periodic orbit exists. Therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing.
Abstract: We investigate the macroscopic description of a dilute, gas-like system of particles, which interact through binary collisions that conserve momentum and mass, but which dissipate energy, as in the case of granular media with inelastic collisions. Our starting point is on the mesoscopic level, through the Boltzmann equation. We deduce hydrodynamic equations for the macroscopic description that would reduce to the compressible Navier-Stokes equations if there were no energy dissipation. We do this in a regime where both the Knudsen number (the ratio of mesoscopic to macroscopic length scales) and the restitution deficit (which measures the inelasticity) are small but non-zero. In this regime, we show that for small values of the Knudsen number and small inelasticity it is possible to relate the actual dynamics to a reduced dynamics on a ``slow manifold'', which in the limit of zero inelasticity and zero Knudsen number is simply the ``manifold'' of local Maxwellians. Instead of expanding the Boltzmann equation itself, we expand this manifold in terms of these two small parameters. In this way, a number of ideas from the theory of dynamical systems, and especially geometric singular perturbation theory, enter our analysis. We discuss the resulting hydrodynamic equations, and compare them to those obtained by other researchers using other methods (suited to other regimes). As we explain here, the particular regime we investigate is especially interesting in the context of pattern formation in driven granular media.
Abstract: The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range $T \leq 30MeV$ and the baryon density range $\rho_B \leq 10^{14.2}g/cm^3$. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle ($1 \leq A \leq 4$) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and $\beta$ equilibrium is considered as a special case. The medium modification of the light clusters is described by self-energy and Pauli blocking effects, using an effective nucleon-nucleon interaction potential. Furthermore, the formation of quantum condensates is considered, in particular Cooper pairing in different (isospin singlet and triplet) channels as well as alpha-like quartetting. It is shown that the formation of light clusters and quantum condensates is of relevance in calculating thermodynamic properties of nuclear matter at moderate densities and temperatures, e.g., in context with the calculation of compact astrophysical objects.
Abstract: The equation of state (EOS) of nuclear matter at moderate temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range $T\leq 30MeV$ and the baryon density range $\rho_B \leq 10^{14.2}g/cm^3$. In addition to the mean-field effects, the formation of few-body correlations, in particular, the light bound clusters up to the alpha particle ($1 \leq A \leq 4$), is of interest. Calculation based on the relativistic mean-field theory with the parameter set TM1 is presented. We show results for different values of the asymmetry parameter, and $\beta$ equilibrium is considered as a special case.The medium modification of the light clusters is described by self-energy and Pauli blocking effects, using an effective nucleon-nucleon interaction potential.