Brownian fluctuations

Figure 11.1. Diagram contrasting a smoothly (but randomly) varying streampath with an erratically varying molecular pathway starting at about the same place. The molecular pathway is subject to rapid Brownian fluctuations. Dots represent sorption events on stationary particles, each of which delays the migration by a random time interval.

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

Metastable ( permanent ) foams, which have a life time of hours or days. These foams are capable of withstanding ordinary disturbances (thermal or Brownian fluctuations), but they may collapse from abnormal disturbances (evaporation, temperature gradients, etc). 

For convenience, foams have been classified into two extreme types unstable or transient foams with lifetimes of seconds and metastable (or pamanrait) foams with lifetimes measured in days. Metastable foams can withstand ordinary disturbances (Brownian fluctuations), but coUq)se from abnormal disturbances (e.g. evq)oration or temperature gradients). 

One clearly identifies a quite sharp adsorption transition that divides the projection of (zcm) in Fig. 13.11 (a) into an adsorbed (bright) regime and a desorbed (dark) regime. This transition appears as a straight line in the phase diagram and is parametrized by a T. Intuitively, this makes sense since at higher T the stronger Brownian fluctuation is more likely to overcome the surface attraction. 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

Peskin C S, Odell G M and Oster G F 1993 Cellular motions and thermal fluctuations the Brownian ratchet Biophys. J. 65 316... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

Figure 4. The Brownian ratchet model of lamellar protrusion (Peskin et al., 1993). According to this hypothesis, the distance between the plasma membrane (PM) and the filament end fluctuates randomly. At a point in time when the PM is most distant from the filament end, a new monomer is able to add on. Consequently, the PM is no longer able to return to its former position since the filament is now longer. The filament cannot be pushed backwards by the returning PM as it is locked into the mass of the cell cortex by actin binding proteins. In this way, the PM is permitted to diffuse only in an outward direction. The maximum force which a single filament can exert (the stalling force) is related to the thermal energy of the actin monomer by kinetic theory according to the following equation ...

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