BMe Research Grant
In this research, I investigate mathematical billiards - mainly Sinai billiards or dispersing billiards-, and some random walk models. Sinai billiard is a chaotic dynamical system (hyperbolic, ergodic, mixing, Bernoulli), thus in some sense it behaves as though it was random. In the last decades, after an immense progression of the theory - in such model of Newtonian mechanics - it has become possible to derive some laws of statistical physics in a mathematically rigorous way.
My research work is carried out at the Department of Stochastics, Mathematical Institute. There is a group of researchers affiliated at the Department of Stochastics and the Department of Differential Equations mainly consisting of professor Domokos Szász (vice president of the Hungarian Academy of Sciences) and his students. The successful seminars, Stochastics Seminar and Dynamical Systems Seminar, indicate the prestigious position of the research place.
The definition of the Sinai billiard is pretty simple. On the d dimensional torus (typically d=2), there are finitely many strictly convex disjoint sets with smooth boundary which are called scatterers. A massless particle moves freely on the complement of the scatterers until it hits the boundary of one of them, when it is simply reflected (with the angle of reflection being equal to the angle of incidence) (Figure 1). The periodic Lorentz process is defined the same way on the periodically extended scatterer configuration in the d dimensional space (Figure 2).
Figure 1 : A planar Sinai billiard table Figure 2: a trajectory of 15 collisions in a
with finite horizon. Animated version. Lorentz process with infinite horizon
Roughly speaking, the Brownian motion is the motion of a dust particle in a glass of water. Since the dust particle is bombarded by a vast number of molecules from different directions, the resulting movement is irregular, random. On the other hand, mathematicians often use the expressions Brownian motion and Wiener process (Figure 3) equivalently. The latter is a mathematical model for Brownian motion, a so-called stochastic process.
Figure 3: a realization of a one dimensional Wiener process
(here, the horizontal axis is time)
Hence in the dynamical theory of Brownian motion, every result proving the presence of the Wiener process in some deterministic Newtonian system, e.g. in Sinai billiard, is of crucial importance. The first results in these direction were [BS80] and [BSCh91]. While the state of the art result ([ChD09b]) also includes two interacting particles. Besides, nowadays one has real chance to discuss other phenomena from statistical physics (e.g. relaxation [Y98], thermal conductivity [GG08]).
My aim is to understand and prove stochastic properties of billiard models.
From this point of view one can distinguish two essentially different scatterer configurations: in the case of finite horizon, the free flight of the particle is bounded (as in Figure 1), while in infinite horizon it is unbounded. In the latter case, one can find an infinite line which is disjoint to all of the scatterers, see Figure 2.
In order to observe the previously mentioned convergence to the Wiener process (which statement is also called the invariance principle) one needs to use diffusive scaling in the case of finite horizon, i.e. to rescale the trajectory of length n by . In the case of infinite horizon, the scaling is superdiffusive, that is should be replaced by [SzV07, ChD09a].
The related questions we are interested in are the following.
As is was already noted, the subject of our research are random walks and Lorentz processes.
A random walk by definition is the random process in which a particle is wandering on the d dimensional lattice in discrete time. The steps are always assumed to be independent of the current position. Sometimes they are also independent of each other - this is called a simple walk. Here, of the non simple walks, we only treat the so called random walks with internal states. In this model, the distribution of steps are governed by an internal state (an element of a finite set, see Figure 4).
Figure 4: visited points by a random walk with internal
in the first 15 steps. Animated version.
In the asymptotic analysis of random walks, a basic tool is the Fourier transform which is also called characteristic function. The essence of this method is as follows. Let X, X1, X2, … be random variables (i.e. “random numbers”). Their characteristic functions are some functions defined by
where E denotes expectation, and i is imaginary unit. A nice property of the functions u is that Xn converges weakly to X (i.e. has approximately the same distribution) if and only if the functions un(t) converge to u(t). This, together with the fact that the characteristic function of a sum of independent random variables is the product of the individual characteristic functions, makes the computation of some limit distributions easy in the independent case. This method has deep generalizations. A simple one allows us to prove local limit theorems, i.e. we can approximate the probability that a random walker arrives back at the origin after many steps, whence we can analyses recurrence properties. A much deeper generalization renders the method suitable for random walk with internal states; in this case the Fourier transform is matrix valued. Finally, by introducing a one parameter family of operators acting on some function spaces, one can use the method even in the case of some deterministic systems.
In the case of Sinai billiard, it is usual to introduce a Poincaré section, i.e. to consider the dynamics only at the time instants of collisions. This way one obtains a two-dimensional phase space: one parameter is the position on the boundary of some scatterer, the other describes the velocity. A nice property of the discrete billiard map is the hyperbolicity (caused by the convexity of the scatterers) which means that the image of two nearby points are getting exponentially farther from each other in the course of iterations. On the other hand, grazing collisions imply the existence of singularities, which is an awkward property. Due to the hyperbolicity, there exist stable and unstable manifolds. An unstable manifold is expanded by the dynamics (hyperbolicity) and torn by the singularities (see Figure 5). The essence of the method of Chernov and Dolgopyat ([ChD09b], [Ch06]) is that in the so called growth lemma, they prove that the expansion predominates over fragmentation and then by coupling some measures on unstable manifolds, they can prove exponential decay of correlation and invariance principle (coupling lemma).
Figure 5: The phase space of a Sinai billiard; the image of an unstable manifold
is the union of some unstable manifolds
Figure 6: reflected Wiener process - with the realization of Figure 3.
Figure 7: quasi reflected Wiener process - with a realization of Figure 3.
In the mathematical theory of Lorentz processes with infinite horizon in high dimensions almost nothing is proven. At the moment, together with Domokos Szász and Tamás Varjú, we are working on distinguishing the diffusive and superdiffusive scatterer configurations. The conjecture - which was investigated by physicists, too - is that superdiffusion only appears in the case, when there is some affine subspace of maximal dimension, which is disjoint to the scatterers ([ZGNR86], [D12]). We want to prove the first two conjectures of Dettmann [D12].
We also plan to give a new proof (with the growth lemma and the martingale method) for the invariance principle in the case of planar infinite horizon. This might be useful for the treatment of local perturbations in the future.
[N11a] Nándori, P., Number of distinct sites visited by a random walk with internal states, Probability Theory and Related Fields 150 3 (2011), 373-403.
[N11b] Nándori, P., Recurrence properties of a special type of Heavy-Tailed Random Walk, Journal of Statistical Physics, 142, 2 (2011), 342-355.
[NSz12] Nándori, P., Szász, D., Lorentz Process with shrinking holes in a wall
Chaos: An Interdisciplinary Journal of Nonlinear Science 22, 2, 026115 (2012).
[NSzV12] Nándori, P., Szász, D., Varjú, T., A central limit theorem for time-dependent dynamical systems, Journal of Statistical Physics, 146, 6 (2012), 1213-1220.
Krámli András: The chaos from the viewpoint of a mathematician, Természet Világa 135, 7 (2004), ( in Hungarian)
Review paper on hyperbolic billiards by Chernov and Dolgopyat from the International Congress of Mathematicians, 2006
[BS80] Bunimovich L. A., Sinai, Ya. G., Statistical properties of Lorentz gas with
periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81), 479-
[BSCh91] Bunimovich L. A., Sinai, Ya. G., Chernov, N. I., Statistical properties of two-
dimensional hyperbolic billiards, Russ. Math. Surveys 46 (1991), 47-106.
[Ch06] Chernov, N., Advanced statistical properties of dispersing billiards, Journal of Statistical Physics 122 (2006), 1061-1094.
[ChD09a] Chernov, N., Dolgopyat, D., Anomalous current in periodic Lorentz gases with infinite horizon, Russian Mathematical Surveys 64 4 (2009), 651-699.
[ChD09b] Chernov, N., Dolgopyat. D., Brownian Brownian Motion–1, Memoirs AMS. 198, No. 927 (2009), pp 193.
[D12] Dettmann, C., New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis, Journal of Statistical Physics 146 (2012) 181-204.
[DSzV08] Dolgopyat, D., Szász, D., Varjú, T., Recurrence Properties of Planar Lorentz Process, Duke Mathematical Journal 142 (2008), 241-281.
[DSzV09] Dolgopyat, D., Szász, D., Varjú, T., Limit Theorems for Perturbed Lorentz Processes. Duke Mathematical Journal 148 (2009), 459-499.
[DE51] Dvoretzky, A., Erdős, P., Some Problems on Random Walk in Space, Proc. 2nd Berkeley Sympos. Math. Statis. Probab. (1951), 353-367.
[GG08] Gaspard, T., Gilbert, T., Heat conduction and Fourier's law in a class of many particle dispersing billiards, New Journal of Physics 10 103004 (2008).
[P09] Péne, F., Asymptotic of the number of obstacles visited by a planar Lorentz process, Discrete and Continuous Dynamical Systems, series A, 24 (2009), 567-588.
[S70] Sinai, Ya. G., Dynamical systems with elastic reflections: Ergodic properties of
dispersing billiards, Russ. Math. Surv. 25 (1970), 137–189.
[SzV07] Szász, D., Varjú, T., Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon, Journal of Statistical Physics 129 (2007), 59-80.
[Y98] Young L.–S. Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics 147 (1998), 585–650.
[ZGNR86] Zacherl A., Geisel T., Nierwetberg, J., Radons, G., Power spectra for anomalous diffusion in the extended Sinai billiard, Physical Letters 114A (1986) 317-321.