Rayleigh distribution equation

Size distribution plays a major role in the microbubble stability, behavior in vivo, and the microbubble acoustic response. The Rayleigh-Plesset equation which describes the microbubble response to pressure waves suggests that ultrasound scattering is proportional to the sixth power of the microbubble diameter [46]. It is not possible, however, to inject large bubbles (e.g., 0.1 or 1 mm in diameter) in the bloodstream, because they would be immediately lodged in the vasculature as emboli, severely limiting the blood flow. Fortunately, microbubbles with the size of several micrometers are still quite echogenic in the ultrasound... 

Light scattered by small particles will be completely polarized if they are optically isotropic and distributed entirely at random. This polarization will take place in any plane normal to the scattered rays. In practice complete polarization by a suspension of particles is never realized, and consequently suitable corrections must be applied to Rayleigh s equation to account for the shape of the particles. 

The previous Sections started from the experimental validity of the DR equation and showed how this behaviour can be ascribed to a suitable distribution (the Rayleigh distribution) of adsorption energy. 

Equations (7.107) and (7.108) are the 2D Rayleigh and Rayleigh distributions, respectively, k is the correlation parameter. The relation between k and the correlation coefficient j HmHm+i) is... 

Prove that the mean of the Rayleigh distribution is given by Equation (2.45). 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

Example. The Rayleigh particle. The velocity V is governed by an M-equation with W(V V ) given by (VIII.4.14). The jumps in V are caused by collisions of the gas molecules and are therefore of order (m/M)v, where v is a typical velocity characterizing the velocity distribution F. They can be made small by choosing M large accordingly we take Q = M/m. The variable in which the jumps remain of the same size is the momentum P = MV. This is our variable X, while mV serves as intensive variable x. The transition probability in the variable X is... 

N. When this is combined with the light scattering equation of Rayleigh, and allowance made for a distribution in the particle size because of the stochastic nature of the process, one obtains (37)... 

Show that Equation 5.9, the Rayleigh-Jeans law, is identical to the Planck black-body distribution in the limit as h —> 0. 

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