# Browsing by Subject "Brownian motion"

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Item Brownian motion in liquids : theory and experiment(2017-08-04) Simha, Akarsh; Morrison, Philip J.; Raizen, Mark G.; Gamba, Irene M; Hazeltine, Richard D; Horton, Claude WShow more Since the theoretical work of Einstein [1905] and von Smoluckowski [1906], and the experiments of Perrin [1909], Brownian motion at long time-scales has been extensively studied for over a century. Short time-scale aspects of Brownian motion are however becoming increasingly relevant, as technology attempts to make smaller and faster devices. The subject matter of this dissertation is the study of short time-scale (typically ~µs) aspects of Brownian motion of microscopic particles in liquids, where the dynamics of the fluid medium are significant. We detail two recent experiments probing this regime: an experiment [Kheifets et. al., 2014] that measured the hydrodynamic instantaneous velocity of a dielectric particle in liquid medium and confirmed theories of Brownian motion based on hydrodynamics up to sub-microsecond time-scales, and a subsequent experiment [Mo et. al., 2015a] that verified the Maxwell-Boltzmann distribution of velocities well into the tails. In a liquid medium, the presence of a boundary near a particle has a significant impact on the characteristics of its Brownian motion, owing to the hydrodynamic coupling between the bounding walls and the particle. However, exact solutions to the hydrodynamic equations are not known even for the common situation of a flat wall. An approximate theory was developed by Felderhof [2005], using a point-particle approximation. Despite agreement with previous experiments, that work results in a drag coefficient with a spurious dependence on the particle's density. In this work, we describe a modification [Simha et. al., 2017] to the point-particle approximation that resolves this inconsistency. Moreover, Felderhof's approximation scheme neglects the size of the particle not only in comparison to the distance of the bounding wall, but also to the skin-depth of rotational flow it generates in the fluid. Since this skin-depth depends on the time-scale of the motion, it is not obvious that such an approximation scheme works at all time-scales. We use the formalism of boundary integral equations to set up a perturbative framework, and obtain the point-particle framework through a series of systematic approximations. This derivation explains why the theory works so well at all time-scales. An alternative calculation for a simple case of a no-slip sphere near a full-slip wall is presented, with results indicating that the point-particle approximation may not capture all non-perturbative terms. We then discuss an experiment [Mo et. al., 2015b] that probed the effects of a boundary on Brownian motion at short time-scales. The experiment agrees very well with the point-particle theory, demonstrating that the boundary significantly impacts Brownian motion down to a certain time-scale, and that the effects are diminished at shorter time-scales. Such effects allude to the possibility of using Brownian motion as a probe of the local environment.Show more Item Brownian motion of optically trapped microspheres and suspended diaphragms(2023-02-18) Xu, Yi, 1994-; Raizen, Mark G.; Keto, John W.; Kilic, Can; Caramanis, ConstantineShow more This dissertation details our experiments of measuring the mass of optically trapped microspheres and suspended diaphragms using their Brownian motion. The optically trapped microsphere is immersed in air and two different mass measurement methods are investigated based on fluctuations about thermal equilibrium. The first method is based on spectral analysis, which allows us to determine the relevant experimental parameters and also serves as a calibration step for the second method. The second method is based on the equipartition theorem and allows for rapid mass measurement. A new method for measuring the effective mass of suspended diaphragms is presented by combining laser Doppler vibrometry and Brownian motion power spectral density analysis. By analyzing the relation between the fundamental resonant frequencies, the mode shape and the total mass of the diaphragm can be determined without the need for prior knowledge of its mechanical properties. This technique provides a precise and efficient method for evaluating the mass of suspended diaphragms for real-life applications in a time-dependent way.Show more Item Fundamental tests of physics with optically trapped microspheres(2011-05) Li, Tongcang; Raizen, Mark G.; Heinzen, Daniel J.; Reid, Alan W.; Sitz, Greg O.; Gordon, VernitaShow more This dissertation details our experiments on studying the Brownian motion of an optically trapped microsphere with ultrahigh resolution, and cooling of its motion towards the quantum ground state. We have trapped glass microspheres in water, air and vacuum with optical tweezers. We developed a detection system that can monitor the position of a trapped microsphere with Angstrom spatial resolution and microsecond temporal resolution. We studied the Brownian motion of a trapped microsphere in air over a wide range of pressures. We measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the Maxwell-Boltzmann velocity distribution and the energy equipartition theorem for a Brownian particle. For short time scales, the ballistic regime of Brownian motion is observed, in contrast to the usual diffusive regime. We are currently developing a new detection system to measure the instantaneous velocity of a Brownian particle in water. In vacuum, we have used active feedback to cool the three center-of-mass vibration modes of a trapped microsphere from room temperature to millikelvin temperatures with a minimum mode temperature of 1.5 mK, which corresponds to the reduction of the root mean square (rms) amplitude of the microsphere from 6.7 nm to 15 pm for that mode. The mean thermal occupation number of that mode is reduced from about 6.8$\times 10^8$ at 297 K to about 3400 at 1.5 mK.Show more Item High frequency microrheology with optical tweezers(2015-08) Riegler, David; Raizen, Mark G.; Fink, ManfredShow more This thesis presents a method to measure the linear viscoelastic response of fluids by tracking and analyzing the thermal, Brownian motion of suspended tracer particles, known as passive microrheology. The particle is confined in a harmonic optical trap and its one dimensional trajectory is obtained by a home-built split beam detection system, which works similar but responds faster than position detection with commercial quadrant photodiodes. The theory which is necessary to convert the particle trajectory into the complex shear modulus is derived in detail, pointing out that the commonly used Mason-Weitz method needs to be modified in order to obtain correct results at high frequencies due to hydrodynamic effects of the fluid. It follows a detailed explanation of the data analysis procedure which is verified for water up to angular frequencies of 10⁷ rad/s in very good agreement with the theory. Finally, there is an outlook how to apply the method to actual complex fluids.Show more Item High-sensitivity tracking of optically trapped particles in gases and liquids : observation of Brownian motion in velocity space(2014-08) Kheifets, Simon; Raizen, Mark G.Show more The thermal velocity fluctuations of microscopic particles mediate the transition from microscopic statistical mechanics to macroscopic long-time diffusion. Prior to this work, detection methods lacked the sensitivity necessary to resolve motion at the length and time scales at which thermal velocity fluctuations occur. This dissertation details two experiments which resulted in velocity measurement of the thermal motion of dielectric microspheres suspended by an optical trap in gases and liquids. First, optical tweezers were used to trap glass microspheres in air over a wide range of pressures and a detection system was developed to track the trapped microspheres' trajectories with MHz bandwidth and <100 fm/rt(Hz) position sensitivity. Low-noise trajectory measurements allowed for observation of fluctuations in the instantaneous velocity of a trapped particle with a signal to noise ratio (SNR) of 26 dB, and provided direct verification of the equipartition theorem and of the Maxwell-Boltzmann velocity distribution for a single Brownian particle. Next, the detection technology was further optimized and used to track optically trapped silica and barium titanate glass microspheres in water and acetone with >50 MHz bandwidth and <3 fm/rt(Hz) sensitivity. Brownian motion in a liquid is influenced by hydrodynamic, time-retarded coupling between the particle and the fluid flow its motion generates. Our measurements allowed for instantaneous velocity measurement with an SNR of up to 16 dB and confirmed the Maxwell Boltzmann distribution for Brownian motion in a liquid. The measurements also revealed several unusual features predicted for Brownian motion in the regime of hydrodynamic coupling, including faster-than-exponential decay of the velocity autocorrelation function, correlation of the thermal force and non-zero cross-correlation between the particle's velocity and the thermal force preceding it.Show more Item On the optimal multiple stopping problem(2010-05) Ji, Yuhee, 1980-; Sîrbu, Mihai; Sîrbu, Mihai; Zitkovic, GordanShow more This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3.Show more Item On the optimal stopping of Brownian motion(2013) Miller, Christopher; Sîrbu, MihaiShow more In this thesis, first we briefly outline the general theory surrounding optimal stopping problems with respect primarily to Brownian motion and other continuous-time stochastic processes. In Chapter 1, we provide motivation for the type of problems encountered in this work, and illustrate their importance both mathematically and in terms of applications in science and engineering. In Chapter 2, we briefly outline many of the technical aspects of probability theory and stochastic analysis, highlighting important theorems that will be used throughout. Chapter 3, which is the main part of the thesis, presents an optimal stopping problem related to the maximum of a process. This chapter also illustrates how problems in this field are often transformed into equivalent problems in which standard techniques apply. Finally, in Chapter 4, we provide a new problem along these same lines, outline a solution to it, and discuss the interesting elements of the problem.Show more Item Optically trapped microspheres as sensors of mass and sound : Brownian motion as both signal and noise(2022-12-01) Hillberry, Logan Edward; Raizen, Mark G.; Alvarado, Jose; Morrison, Philip; Zheng, YuebingShow more Owing to their small size, advanced position detection possibilities, and accurate theoretical description, optically trapped microspheres have become a paradigmatic system for myriad sensing applications. This dissertation reports on two air-based experiments that leverage the unique properties of optically-trapped microspheres as measurement tools: inertial mass sensing and sound detection. We measure the mass of a microsphere in three ways. Careful error analysis allows quantitative comparison between our method and others appearing in the recent literature. As figures of merit, we focus on accuracy, precision, and speed. We find that monitoring the variance of the microsphere's velocity degree of freedom while undergoing equilibrium Brownian motion enables measurement of our microsphere's 25 pg mass with 4.3% accuracy and 1.6% precision across 14 vastly different trapping laser powers and using 10x less data than our most accurate (3.2%) and precise (0.9%) method. The more accurate method is a calibration step that must always be performed initially, but the microsphere's velocity variance may subsequently be monitored, thereby elevating mass to a dynamic measurement variable. For sound detection, we develop a model for the sensitivity of a microsphere's velocity to an external acoustic perturbation. In this case, the microsphere's Brownian motion is a noise source that must be overcome for a signal to be detectable. We validate our method by comparing measurements of pure-tone bursts between our system and two state-of-the-art, commercially-available acoustic sensors. We then demonstrate the microsphere's advantage in measuring high-frequency-content signals using impulsive sounds generated by laser ablation. We resolve an acoustic rise time of 1 µs on the same signal that our high-bandwidth microphone measures a 7 µs rise time. At the same time, our higher bandwidth resolves a nearly 3x larger peak pressure than the microphone. This dissertation builds toward these two experimental results by first contextualizing them in a non-technical historical review. Key technical background is then developed pedagogically, followed by details of the trapping and detection apparatus. After the experiments are reported, we conclude with a summary of the results and an outlook on the future of optically trapped microspheres as sensitive detectorsShow more Item Short timescale Brownian motion and applications(2015-08) Mo, Jianyong; Raizen, Mark G.; Downer, Mike; Fiete, Gregory A; Bengtson, Roger D; Kumar, PawanShow more This dissertation details our experiments and numerical calculations on short timescale Brownian motion and its applications. We test the Maxwell-Boltzmann distribution using micrometer-sized spheres in liquids at room temperature. In addition to that, we use Brownian particles as probes to study boundary effects imposed by a solid wall, viscoelasticities of complex fluids, slippage at solid-fluid interfaces, and fluid compressibility. The experiments presented in this dissertation relies on the use of tightly focused laser beams to both contain and probe the Brownian motion of microspheres in fluids. A dielectric sphere near the focus of a laser beam scatters some of the incident photons in a direction which depends on the particle's position. Changes in the particle's position are encoded in the spatial distribution of the scattered beam, which can be measured with high sensitivity. It is important to emphasize that the Brownian motion in this dissertation is exclusive for translational Brownian motion. We have reported shot-noise limited measurements of the instantaneous velocity distribution of a Brownian particle. Our system consists of a single micron-sized glass sphere held in an optical tweezer in a liquid in equilibrium at room temperature. We provide a direct verification of a modified Maxwell-Boltzmann velocity distribution and a modified energy equipartition theorem that account for the kinetic energy of the liquid displaced by the particle. Our measurements con rm the distribution over a dynamic range of more than six orders of magnitude in count-rate and five standard deviations in velocity. We have reported high-bandwidth, comprehensive measurements of Brownian motion of an optically trapped micrometer-sized silica sphere in water near an approximately at wall. At short distances, we observe anisotropic Brownian motion with respect to the wall. We find that surface confinement not only occurs in the long time scale diffusive regime but also in the short time scale ballistic regime, and the velocity autocorrelation function of the Brownian particle decays faster than that of particle in a bulk fluid. Furthermore, at low frequencies the thermal force loses its color due to the reflected flow from the no-slip boundary. The power spectrum of the thermal force on the particle near a no-slip boundary becomes at at low frequencies. We have numerically studied Brownian motion of a microsphere in complex fluids. We show that Brownian motion of immersed particles can be dramatically affected by the viscoelastic properties of the host fluids. Thus, this fact can be used to extract the properties of complex fluids via observing the motion of the embedded particles. This will be followed by two experimental demonstrations of obtaining the viscosities of water and acetone. We also study Brownian motion with partial and full slip boundary conditions both on the surface of a sphere and a boundary. We show that the motion of particles can be significantly altered by the boundary condition of fluid flow on a solid surface. We suggest that this fact can be used to measure the slippage, namely the slip length. Lastly, I will discuss the efforts to study fluid compressibility and nonequilibrium physics using a short duration pulsed laser. We expect to increase the postion sensitivity from current 10⁻¹⁵ m/[square root of Hz] to about 10⁻¹⁹ m/[ square root of Hz] by using a pulsed laser with a peak power of 10^8 W. With such a high position sensitivity, we expect to be able to resolve the compressibility of fluids. We will also discuss a few future experiments studying non-equilibrium physics.Show more