Velocity spectrum for non-Markovian Brownian motion in a periodic potential

Non-Markovian Brownian motion in a periodic potential is studied by means of an electronic analogue simulator. Velocity spectra, the Fourier transforms of velocity autocorrelation functions, are obtained for three types of random force, that is, a white noise, an Ornstein—Uhlenbeck process, and a quasimonochromatic noise. The analogue results are in good agreement both with theoretical ones calculated with the use of a matrix-continued-fraction method, and with the results of digital simulations. An unexpected extra peak in the velocity spectrum is observed for Ornstein-Uhlenbeck noise with large correlation time. The peak is attributed to a slow oscillatory motion of the Brownian particle as it moves back and forth over several lattice spaces. Its relationship to an approximate Langevin equation is discussed.     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

Projector formalism of generalized Brownian motion theory applied to dissipative and noisy systems

The projector formalism of Zwanzig-Mori type is extended to obtain generalized Fokker-Planck and generalized nonlinear Langevin equations for coarse-grained variables when the underlying microscopic dynamics is dissipative and noisy (stochastic).     

Motion Control by ZV Shaper Synthesis Extended for Fractional Systems and Its Application to CRONE Control

Shaping command input or preshaping is used for reducing system oscillation in motion control. Desired systems inputs are altered so that the system finishes the requested move without residual oscillation. This technique, developed by N.C. Singer and W.P. Seering, is used for example in the aerospace field, in particular in flexible structure control. This paper presents the study of ZV shaper for explicit fractional derivative systems (generalized derivative systems). A robustness study of ZV shaper is then presented and applied to improve second generation CRONE control response time. Results from simulation and from a DC motor bench are also given.     

A field-sensitivity analysis of the relativistic motion of a single electron submitted to a perpendicular magnetic field

Sensitivity of the velocity of a single electron, in the relativistic case, to a perpendicular magnetic field is evaluated by defining a parameter which becomes useful in a number of cases related to dynamical systems. In particular, the ultrarelativistic case is examined.     

Constants of Motion for Several One-Dimensional Systems and Problems Associated with Getting Their Hamiltonians

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion.     

Effect of Pressure on Fluid Damping in MEMS Torsional Resonators with Flow Ranging from Continuum to Molecular Regime

High quality factor of dynamic structures at micro and nano scale is exploited in various applications of micro electro-mechanical systems (MEMS) and nano electro-mechanical system. The quality factor of such devices can be very high in vacuum. However, when vacuum is not desirable or not possible, the tiny structures must vibrate in air or some other gas at pressure levels that may vary from atmospheric to low vacuum. The interaction of the surrounding fluid with the vibrating structure leads to dissipation, thus bringing down the quality factor. Depending on the ambient fluid pressure or the gap between the vibrating and the fixed structure, the fluid motion can range from continuum flow to molecular flow giving a wide range of dissipation. The relevant fluid flow characteristics are determined by the Knudsen number which is the ratio of the mean free path of the gas molecule to the characteristic flow length of the device. This number is very small for continuum flow and reasonably big for molecular flow. In this paper, we study the effect of fluid pressure on the quality factor by carrying out experiments on a MEMS device that consists of a double gimbaled torsional resonator. Such devices are commonly used in optical cross-connects and switches. We only vary fluid pressure to make the Knudsen number go through the entire range of continuum flow, slip flow, transition flow, and molecular flow. We experimentally determine the quality factor of the torsional resonator at different air pressures ranging from 760 Torr to 0.001 Torr. The variation of this pressure over six orders of magnitude ensures required rarefaction to range over all flow conditions. Finally, we get the variation of quality factor with pressure. The result indicates that the quality factor, Q, follows a power law, Q ∝P ^–r , with different values of the exponent r in different flow regimes. In the second part of the paper, we propose the use of effective viscosity for considering velocity slip conditions in solving Navier–Stokes equation numerically. This concept is validated with analytical results for a simple case and then compared with the experimental results presented in this paper. The study shows that the effective viscosity concept can be used effectively even for the molecular regime if the air-gap to length ratio is sufficiently small (h ₀/L<0.01). As this ratio increases, the range of validity decreases.     

Levy area for the free Brownian motion: existence and non-existence

This note extends the work of Capitaine (J. Funct. Anal. 179 (1) (2001) 153) on the Levy area process for the free Brownian motion in two directions. First, we reprove that a Levy area for the Free Brownian motion exists in the Von Neumann tensor product, by exhibiting a non-commutative Burkholder-Davis-Gundy type inequality. Then, we show that there does not exist a Levy area in the projective tensor product.     

转动系统的相对论性分析力学理论

本文讨论了转动相对论力学理论，主要是建立转动系统的相对论性分析力学理论·构造转动系统的相对论性广义动能函数Ｔｒ＝∑ｎｉ＝１Ｉ０ｉΓｉ２（１－１－θ·２ｉ／Γｉ２）和广义加速度能量函数Ｓｒ＝１２∑ｎｉ＝１Ｉｉ（θ·ｉ·θ¨ｉ）２Γｉ２－θ·２ｉ＋θ¨２ｉ，给出其Ｈａｍｉｌｔｏｎ原理和三种不同形式的Ｄ′Ａｌｅｍｂｅｒｔ原理；对于完整约束系统，建立了转动系统的相对论性Ｌａｇｒａｎｇｅ方程、Ｎｉｅｌｓｅｎ方程、Ａｐｐｅｌ方程和Ｈａｍｉｌｔｏｎ正则方程；对于非完整约束系统，建立了转动系统的相对论性Ｒｏｕｔｈ方程、Чаплыгин方程、Ｎｉｅｌｓｅｎ方程和Ａｐｐｅｌ方程；并给出转动系统的相对论性Ｎｏｅｔｈｅｒ守恒律