Spontaneous density fluctuations

Sharma, S.C. (1992). Positronium localization in spontaneous density fluctuations in molecular gases. Material Science Forum, 105-110 pp. 451-458. 

When this criterion is fulfilled the compound is stable with respect to the spontaneous development of inhomogeneities in the average atomic density. The phase is in other words stable with regard to infinitesimal density fluctuations. Equation (5.3) requires that the heat capacity is positive. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

The light-scattering spectrum which is related to 7 (q, /) by Eq. (3.3.3) consequently probes how a density fluctuation <5/ (q) spontaneously arises and decays due to the thermal motion of the molecules. Density disturbances in macroscopic systems can propagate in the form of sound waves. It follows that light scattering in pure fluids and mixtures will eventually require the use of thermodynamic and hydrodynamic models. In this chapter we do not deal with these complicated theories (see Chapters 9-13) but rather with the simplest possible systems that do not require these theories. Examples of such systems are dilute macromolecular solutions, ideal gases, and bacterial dispersions. ... 

Associated with any extensive property s/(like energy, and mass) is a specific property A(r, t) defined as the quantity of s/per unit volume (or the density of, s>/) at the space-time point (r, t). For a system removed from equilibrium or for an equilibrium system undergoing spontaneous thermal fluctuations, A(r, t) varies with space and time. In an arbitrary volume V, fixed with respect to the laboratory-fixed coordinate axes, the total quantity sfin this volume at time t is... 

The scaled particle theory of fluids developed by Reiss, Lebowitz, Helfand and Frisch > " need concern itself [in the case of hard spheres by virtue of Eq. (26)] only with calculating g a). To accomplish this we focus our attention on a spherical cavity of radius at least r centered about a fixed point in the fluid. A cavity is defined as a region of space devoid of molecular (hard sphere) centers (see Fig. 8). Such a cavity can be formed spontaneously in our fluid as a result of a local density fluctuation. 

When calculating the concentrational fluctuation spectral density, we must take into account that 1) fluctuations are isotropic so that spectral density can depend on the modulus but not on the direction of the wave-number vector 2) particles are solid spheres of finite volume and 3) decay of spontaneously originated fluctuations is governed by particle diffusion. As a result, we obtain the following formula ... 

There are further relations among critical-point exponents in whidi the dimensionality d appears explicitly. The equilibrium fluctuations of the density about its average value are coherent (that is, are of one sign) over distances of order These fluctuations that thus occur spontaneously in regions of linear dimension are the elementary density fluctuations with each of which is associated a free energy of order kT — k H. The free-energy density associated with equilibrium density (or composition) fluctuations near a critical point is thus of order But... 

We can also examine spontaneous equilibrium fluctuations by working in the fiequency domain and considering the spectral density (Fourier-Laplace transform) of equilibrium time correlation functions. For the case of momentum fluctuations, we write... 

Spontaneous thermal fluctuations of the density, p r,t), the momentum density, g(r,t), and the energy density, e(r,t), are dynamically coupled, and an analysis of their dynamic correlations in the limit of small wave numbers and frequencies can be used to measure a fluid s transport coefficients. In particular, because it is easily measured in dynamic light scattering. X-ray, and neutron scattering experiments, the Fourier transform of the density-density correlation function - the dynamics structure factor - is one of the most widely used vehicles for probing the dynamic and transport properties of liquids [56]. 

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