Brownian noise

Orientation angle, or polar spherical coordinate. Brownian noise function, (7.96). 

A monolithic three-axis accelerometer with three independent capacitive readout circuits on a single chip is described elsewhere [7] (Fig. 6.1.14). The circuit is similar to Fig. 6.1.10 and achieves 0.085 aF/v/Hz resolution with 100 fF sense capacitors. The noise is actually dominated by Brownian noise in the sensor itself, as tests in vacuum demonstrate. The actual capacitance resolution is therefore somewhat better than stated. This circuit uses correlated double sampling (CDS) for biasing and to reject flicker noise. 

Noise within an accelerometer is created from several sources. The noise comes from the switched capacitor design, inherent thermal noise within any devices, the flickeT noise of transistors, and the Brownian noise in the g-cell transducer due to random motion of atoms (see equation 5). Signal-to-noise ratio of 60 dB is common, but lower g devices with higher gains may be worse. 

Figure 12.7 CaLibration of trap stiffness by thermal noise analysis. A single 1.1 jim bead is held in the optical tweezers and data is collected at 2 kHz. (a) The graph shows the bead position vs time -solid lines denote 1 standard deviation of bead position, (b) The same data plotted as a histogram. The mean displacement is 0 nm and the variance is determined by the vibration due to brownian noise, (c) The trap stiffness can be determined from this information using the equipartition principle lf2Ktrsj, x ) = l/2ksT, where Ktrap trap stiffness, (x ) = variance, /fB= Boltzman constant and T= absolute temperature...

This method has a few disadvantages. The seismometer consumes more power since it is the forcing coil current that is counterbalancing the gravity vector now. In addition, inertial mass must be small to keep the current low and still achieve a wide tilt range, which results in higher levels of Brownian noise (see Principles of Broadband Seismometry ). 

Two noise sources common to all seismometers, active and passive, are Johnson and Brownian noise. Johnson noise is a noise source due to thermal agitation of the electrons that make up the current flow in a resistor. For a resistance R [Q] at temperature T [K], the expectation value of the voltage PSD across its terminals will be ... 

Brownian noise is similar to resistor noise, both in terms of the statistical mechanics which explain it, and in the white noise spectrum which results. In particular, the Brownian motion of the boom of a pendulum results in an equivalent acceleration PSD of the frame of that pendulum equal to ... 

Instrumentation (like Brownian noise, electronic and quantization noise)... 

As the glass transition is approached, the size of the cooperative groups of particles and the timescale of their motion increase (Kegel and van Blaaderen 2000, Weeks et al. 2000). Rearranging particle displacements are small, especially in glassy colloidal suspensions, and careful analysis is needed in order to distinguish these motions from the Brownian noise (Hunter and Weeks 2012). 

Thus the average velocity decays exponentially to zero on a time scale detennined by the friction coefficient and the mass of the particle. This average behaviour is not very interesting, because it corresponds to tlie average of a quantity that may take values in all directions, due to the noise and friction, and so the decay of the average value tells us little about the details of the motion of the Brownian particle. A more interesting... 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

In the most general case the diffusive Markov process (which in physical interpretation corresponds to Brownian motion in a field of force) is described by simple dynamic equation with noise source ... 

Initially, an overdamped Brownian particle is located in the potential minimum, say somewhere between x and X2- Subjected to noise perturbations, the Brownian particle will, after some time, escape over the potential barrier of the height AT. It is necessary to obtain the mean decay time of metastable state [inverse of the mean decay time (escape time) is called the escape rate]. 

It is known that when the transition of an overdamped Brownian particle occurs over a potential barrier high enough in comparison with noise intensity A<[>/kT 1, time evolution of many observables (e.g., the probability of... 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

In a case where F would contain a stochastic term (e.g., Brownian motion, noise), this equation would lead to the celebrated Fokker-Plank equation with a diffusion (second-order) term. 

Since the cantilevers are very small structures, they execute thermal (Brownian) motion. The longer the cantilever, the more sensitive it is for measuring surface stresses. However, increasing the length also increases the thermal vibrational noise of the cantilever, which from statistical physics is [18, 19]... 

In his seminal work [109], Kramers considered the noise-induced flux from a single metastable potential well i.e. he considered a Brownian particle... 

Here, m and rai are the mass and position vector of beads, respectively. is the friction tensor, which is assumed to be isotropic for simplicity in our simulation, that is, = Fl, where I is the unit dyad and r = 0.5t 1 (t = cr(m/ )° 5j (Grest, 1996). Further, f aj is the Brownian random force, which obeys the Gaussian white noise, and is generated according to the fluctuation—dissipation theorem ... 

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