Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session H23: Focus Session: Brownian Motion and Stochastic Dynamics in the 100 Years Since Einstein |
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Sponsoring Units: GSNP DBP Chair: Peter Jung, Ohio University Room: LACC 410 |
Tuesday, March 22, 2005 8:00AM - 8:12AM |
H23.00001: Study of the dissipative Hofstadter model in the $\pi$-flux regime Eduardo Novais, Antonio Castro Neto, Francisco Guinea A generalization of the Caldeira-Leggett model, which describes a particle in a periodic potential and under an applied magnetic field, the Dissipative Hofstadter model, is analyzed. We build upon previous work$^{(1)}$, and study special points where the model is exactly soluble. Our solution describes a fixed point where the particle is neither fully localized nor completely free. In order to obtain the solution, we map the partition function of the DHM to the partition function onto that of a spin chain problem with non commuting terms.\\ (1) - C. Callan and D. Freed, Nucl. Phys. B {\bf 374}, 543 (1992). [Preview Abstract] |
Tuesday, March 22, 2005 8:12AM - 8:24AM |
H23.00002: The quantum description of Einstein's Brownian motion Francesco Petruccione, Bassano Vacchini A fully quantum treatment of Einstein's Brownian motion is described. The microscopic analysis is based on the two key features of Einstein's Brownian motion: the homogeneity of the background medium, reflected into the requirement of translational invariance and the connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to important restrictions both on the expression for possible interactions and on the structure of the completely positive generator of a quantum-dynamical semigroup, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, which is directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirements of translational invariance, further characterizing general structures of the reduced dynamics arising in the presence of symmetry under translations. [Preview Abstract] |
Tuesday, March 22, 2005 8:24AM - 8:36AM |
H23.00003: Preserving Positivity During Quantum Brownian Evolution Allan Tameshtit The conventional quantum Brownian propagator may be derived by considering a system of interest bilinearly coupled to and initially uncorrelated with a reservoir. Although possessing many attractive features, it is well known that this propagator does not preserve positivity of density operators, violating a basic tenet of quantum mechanics. (A density operator $\rho $ is positive means $\left\langle \psi \right|\rho \left| \psi \right\rangle \ge 0$ for all $\psi )$. In an effort to rectify this problem, workers have modified the propagator by the \textit{ad hoc }addition of extra terms to the corresponding generator. We show that no such terms need be added to the generator to preserve positivity provided one accounts for the rapid entanglement of the system of interest and the reservoir on a time scale too short for the conventional propagator to be valid. [Preview Abstract] |
Tuesday, March 22, 2005 8:36AM - 8:48AM |
H23.00004: Human dispersal on geographical scales Dirk Brockmann, Lars Hufnagel, Theo Geisel In order to account for various spatio-temporal phenomena in ecological systems, such as the geographical spread of epidemics, the knowledge of dynamical and statistical properties of human dispersal is of fundamental importance. However, these properties are difficult to assess mainly due to a lack of sufficiently large datasets. So far these properties could only be conjectured and the opinion that humans move diffusively still prevails in models. Based on a comprehensive dataset of over a million individual diplacements, collected over the past five years, we investigated the dynamical and statistical properties of human dispersal on geographical scales. We found that human dispersal is anomalous in two ways. First, the probability of finding a displacement of length $x$ decays as a power law, indicating that trajectories of humans are reminiscent of L\'{e}vy flights characterized by a L\'{e}vy exponent $\mu\approx 1/2$. Secondly, the waiting time distribution exhibits a heavy tail as well implying a subdiffusive influence. Effectively, the scaling behavior of distance with time, $X(t)\sim t^{p}$ is superdiffusive with an exponent $p$ near unity. We show that the stochastic dispersal can be accounted for in the continous time random walk framework yielding a bi-fractional Fokker-Planck equation. Our results represent the first solid and quantitative assessment of the properties of human dispersal on geographical scales. [Preview Abstract] |
Tuesday, March 22, 2005 8:48AM - 9:00AM |
H23.00005: Probability distribution of financial returns in a model of multiplicative Brownian motion with stochastic diffusion coefficient Antonio Silva, Richard Prange, Victor Yakovenko It is well-known that the mathematical theory of Brownian motion was first developed in the Ph.~D.\ thesis of Louis Bachelier for the French stock market before Einstein [1]. In Ref.\ [2] we studied the so-called Heston model, where the stock-price dynamics is governed by multiplicative Brownian motion with stochastic diffusion coefficient. We solved the corresponding Fokker-Planck equation exactly and found an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula interpolates between the exponential (tent-shaped) distribution for short time lags and the Gaussian (parabolic) distribution for long time lags. The theoretical formula agrees very well with the actual stock-market data ranging from the Dow-Jones index [2] to individual companies [3], such as Microsoft, Intel, etc. \\[4pt] [1] Louis Bachelier, ``Th\'eorie de la sp\'eculation,'' Annales Scientifiques de l'\'Ecole Normale Sup\'erieure, III-17:21-86 (1900).\\[0pt] [2] A. A. Dragulescu and V. M. Yakovenko, ``Probability distribution of returns in the Heston model with stochastic volatility,'' Quantitative Finance {\bf 2}, 443--453 (2002); Erratum {\bf 3}, C15 (2003). [cond-mat/0203046] \\[0pt] [3] A. C. Silva, R. E. Prange, and V. M. Yakovenko, ``Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact,'' Physica A {\bf 344}, 227--235 (2004). [cond-mat/0401225] [Preview Abstract] |
Tuesday, March 22, 2005 9:00AM - 9:36AM |
H23.00006: Stochastic bifurcation for a white-noise perturbed nonlinear oscillator Invited Speaker: Consider the vibrations of a thin beam excited by longitudinal white noise. The amplitude of the first mode of vibration evolves according to a 2-dimensional nonlinear stochastic differential equation. The white noise enters in a multiplicative fashion and so the origin is a fixed point for the system. When the noise intensity is sufficiently small relative to the coefficient of linear dissipation the origin is almost surely stable; however as the noise intensity is increased beyond a critical level the origin becomes almost surely unstable and the system evolves as a recurrent diffusion on the rest of the 2-dimensional space. We will discuss rigorous results on the changes in behavior of the system, and in particular the nature of its stationary measures, as the noise intensity passes through its critical level. In particular we identify the critical noise level, and the scaling of stationary moments just above the critical level. These results validate earlier numerical simulations of Wedig, Springer Lect. Notes Math., Vol 1486 (1991). The techniques used extend to a wide class of finite dimensional stochastic differential equations with a fixed point. [Preview Abstract] |
Tuesday, March 22, 2005 9:36AM - 9:48AM |
H23.00007: Excited Random Walk in One Dimension Tibor Antal, Sidney Redner We study the $k$-excited random walk, in which each site initially contains $k$ cookies, and a random walk that is at a site that contains at least one cookie eats a cookie and then hops to the right with probability $p$ and to the left with probability $q=1-p$. If the walk hops from an empty site, there is no bias. For the 1-excited walk on the half-line (each site initially contains one cookie), the probability of first returning to the starting point at time $t$ scales as $t^{-1-q}$. We also derive the probability distribution of the position of the leftmost uneaten cookie in the large time limit. For the infinite line, the probability distribution of the position of the 1-excited walk has an unusual anomaly at the origin and the distributions of positions for the leftmost and rightmost uneaten cookie develop a power-law singularity at the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime $p>3/4$, where the walk is transient, including a mean displacement that grows as $t^\nu$, with $\nu>\frac{1}{2}$ dependent on $p$, and a breakdown of scaling for the probability distribution of the walk. [Preview Abstract] |
Tuesday, March 22, 2005 9:48AM - 10:00AM |
H23.00008: Exact kinetic Monte Carlo approach for diffusion without the lattice hops Vasily Bulatov, Tomas Oppelstrup, George Gilmer, Malvin Kalos, Babak Sadigh, Wei Cai We present a new algorithm for kinetic Monte Carlo simulations applicable to a wide range of physical situations where multiple Brownian particles of finite dimensions diffuse, collide and react with each other. In its spirit, the new approach is reminiscent of the so-called event-based Monte Carlo algorithm (\textit{JERK}) developed over the years in SACLAY (France). Similar to \textit{JERK}, the algorithm alleviates the need to simulate every single diffusional hop but focuses on more significant changes in the system's configuration. Yet, unlike \textit{JERK}, the new algorithm is approximation-free and its accuracy is limited only by the quality of the diffusion and reaction rate coefficients. The new approach is based on the exact Green's function solutions obtainable in the time-dependent theory of first-passage processes. Applications of the new approach will be discussed, including Oswald ripening, semiconductor processing, defect microstructure evolution in fusion and fission reactor materials, and diffusion-controlled reactions in confined geometries. [Preview Abstract] |
Tuesday, March 22, 2005 10:00AM - 10:12AM |
H23.00009: Noise-induced escape of modulated systems: synchronization and scalings Dmitri Ryvkine, Mark Dykman We provide a complete solution of the Kramers problem of noise-induced escape in periodically modulated systems, including the previously discussed exponent\footnote{M.I. Dykman, B. Golding, and D. Ryvkine, Phys. Rev. Lett. {\bf 92}, 080602 (2004)} and the prefactor in the escape rate $W$. In the presence of modulation the prefactor is no longer given by an ``attempt frequency". It is a strongly nonmonotonic function of the modulation amplitude $A$ that displays several scaling regions. This behavior is related to the onset and ultimate loss of modulation-induced synchronization of escape events with increasing $A$. The loss of synchronization occurs as $A$ approaches the bifurcational value $A_c$ where the metastable state disappears. We identify a parameter range close to $A_c$ where the synchronization is still pronounced and $W$ scales as $W\propto (A_c-A)^{-1}$, whereas in the region where the synchronization is lost the scaling is totally different, $W\propto (A_c-A)^{1/2}$. In addition to the period-averaged escape rate we also find the instantaneous escape rate, which is given by the current away from the metastable state. The theory resolves the existing controversies. The results are supported by simulations. [Preview Abstract] |
Tuesday, March 22, 2005 10:12AM - 10:24AM |
H23.00010: Probing diffusion on the nanometer scale using electrochemistry Diego Krapf, Meng-Yue Wu, Henny W. Zandbergen, Cees Dekker, Serge G. Lemay Studying ion diffusion in liquid at the nanometer scale is experimentally very challenging. We report on an electrochemical approach to this problem: an electrode of nanometer dimensions is immersed in solution and biased so as to drive an electron transfer reaction with an ionic species in solution. The electrical current through the electrode provides a direct measure of the diffusive flux of ions to the electrode surface. Because the concentration gradient is localized in the immediate vicinity of the nanoelectrode, this provides very local information and high concentration gradients can be achieved. To carry out these experiments, we have recently developed a method for fabricating nanoelectrodes with a well-defined size and geometry. A pore is first drilled in an insulating membrane with a focused electron beam and it is then filled from one side using a noble metal. Conical electrodes as small as 1 nm in size are obtained. Measurements of the diffusive flux at such nanoelectrodes will be presented and their implications for ion diffusion on the nanometer scale will be discussed. [Preview Abstract] |
Tuesday, March 22, 2005 10:24AM - 11:00AM |
H23.00011: Constructive role of Brownian motion: Brownian motors and Stochastic Resonance Invited Speaker: Peter H\"anggi Noise is usually thought of as the enemy of order rather as a constructive influence. For the phenomena of Stochastic Resonance [1] and Brownian motors [2], however, stochastic noise can play a beneficial role in enhancing detection and/or facilitating directed transmission of information in absence of biasing forces. Brownian motion assisted Stochastic Resonance finds useful applications in physical, technological, biological and biomedical contexts [1,3]. The basic principles that underpin Stochastic Resonance are elucidated and novel applications for nonlinear classical and quantum systems will be addressed. The presence of non-equilibrium disturbances enables to rectify Brownian motion so that quantum and classical objects can be directed around on a priori designed routes in biological and physical systems (Brownian motors). In doing so, the energy from the haphazard motion of (quantum) Brownian particles is extracted to perform useful work against an external load. This very concept together with first experimental realizations are discussed [2,4,5]. \\[4pt] [1] L. Gammaitoni, P. H\"anggi, P. Jung and F. Marchesoni, {\it Stochastic Resonance}, Rev. Mod. Phys. {\bf 70}, 223 (1998).\\[0pt] [2] R. D. Astumian and P. H\"anggi, {\it Brownian motors}, Physics Today {\bf 55} (11), 33 (2002).\\[0pt] [3] P. H\"anggi, {\it Stochastic Resonace in Physics and Biology}, ChemPhysChem {\bf 3}, 285 (2002).\\[0pt] [4] H. Linke, editor, Special Issue on Brownian Motors, Applied Physics A {\bf 75}, No. 2 (2002).\\[0pt] [5] P. H\"anggi, F. Marchesoni, F. Nori, {\it Brownian motors}, Ann. Physik (Leipzig) {\bf 14}, xxx (2004); cond-mat/0410033. [Preview Abstract] |
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