Number-density fluctuations

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

Assuming that the entrance number density fluctuates in a sinusoidal fashion,... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

The phase Doppler method utilizes the wavelength of light as the basis of measurement. Hence, performance is not vulnerable to fluctuations in light intensity. The technique has been successfully appHed to dense sprays, highly turbulent flows, and combustion systems. It is capable of making simultaneous measurements of droplet size, velocity, number density, and volume flux. 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

If the number Z/v of electrons in each particle is constant, electron density fluctuation... 

The value of the peaks and troughs in the pair distribution function represent the fluctuation in number density. The peaks represent regions where the concentrations are in excess of the average value while the troughs represent a deficit. As the volume fraction is increased, the peaks and troughs grow, reflecting the increase in order with concentration. We... 

Baumgarten and Pigford (B2) have employed the y-ray method for their study of density fluctuations in fluidized beds. The method is laborious and time-consuming, and yields only approximate values based on a large number of bubbles. Because of this, the x-ray cinephotographic method is to be preferred for the study of the behavior of bubbles in fluidized beds. 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

It is therefore clear that particle number conservation considerations are not sufficient to determine S q, q) at very small but finite q. In the case of broken translational symmetry, as certainly occurs in the vicinity of a surface, the perfect screening of density fluctuation matrix elements, which is characteristic of homogeneous systems, does not hold due to nonconservation of momentum, and the small q limit of S(q, q) is nonuniversal even in the zero temperature case. 

A great advantage of this type of system is that it makes possible measurements of relatively small matrix elements such as SX4 the signal can be amphfied as necessary because only one component is proportional to SX4. NormaUzation by Sxx, which is done electronically, ehminates fluctuations in particle number density and in the Hght source. If Sx, is desired, the electronic servo can be turned off and the instrument used in the conventional way. 

It was thought for some time that central peaks were due to impurities, defects and other such extrinsic or intrinsic factors. A number of models and mechanisms based on entropy fluctuations, phonon density fluctuations, dielectric relaxation, molecular... 

Here, the quantities jn ° and ji are, respectively, the chemical potentials of pure solvent and of the solvent at a certain biopolymer concentration V is the molar volume of the solvent and n is the biopolymer number density, defined as n C/M, where C is the biopolymer concentration (% wt/wt) and M is the number-averaged molar weight of the biopolymer. The second virial coefficient has (weight-scale) units of cm mol g. Hence, the more positive the second virial coefficient, the larger is the osmotic pressure in the bulk of the biopolymer solution. This has consequences for the fluctuations in the biopolymer concentration in solution, which affects the solubility of the biopolymer in the solvent, and also the stability of colloidal systems, as will be discussed later on in this chapter. 

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