Entropy fluctuations

GEN.46.1. Prigogine, Entropie, fluctuations et dynamique (Entropy, fluctuations and dynamics), Sciences et Techniques, Janvier 1978, pp. 34-37. 

Application of this technique to measurements of the spectral distribution of tight scattered from a pure SF fluid at its critical point was present by Ford and Benedek The scattering is produced by entropy fluctuations which decay very slowly in the critical region. Therefore the spectrum of the scattered light is extremely narrow (10 - lO cps) and can only be observed by this light beating technique 240a)... 

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... 

Although these two exp.essions do not immediately look similar, they essentially denote quite similar contributions to the viscosity. Let us recall that the wavenumber-dependent specific heat is expressed in terms of the correlation between entropy fluctuations. The entropy fluctuation in turn is expressed as a linear combination of the energy and density fluctuation. Now, if we neglect energy fluctuation, then the wavenumber-dependent specific heat is nothing but the correlation between the density fluctuations and thus is proportional to the static structure factor. 

The anomalies of liquid water become more pronounced when it is supercooled. For example, the volume and entropy fluctuations of liquid water become more pronounced as the temperature decreases. This is in contrast to most other liquids, in which the volume and entropy fluctuations become smaller as the temperature is lowered. Furthermore, the volume and entropy fluctuations in water at less than 4°C are anticorrelated, that is, the increase in volume which occurs when water is cooled results in a decrease in entropy (Debenedetti, 2003). 

What is complexity There is no good general definition of complexity, though there are many. Intuitively, complexity lies somewhere between order and disorder, between regularity and randomness, between perfect crystal and gas. Complexity has been measured by logical depth, metric entropy, information content (Shannon s entropy), fluctuation complexity, and many other techniques some of them are discussed below. These measures are well suited to specific physical or chemical applications, but none describe the general features of complexity. Obviously, the lack of a definition of complexity does not prevent researchers from using the term. 

We now apply the above results to the energy function E(S, V,n, rit). To explore the effect of entropy fluctuations giving rise to slight departures from... 

The first term is connected with isobaric entropy fluctuation, which gives a diffusive component, and the second term is connected with an adiabatic pressure fluctuation, which gives rise to a high-frequency acoustic wave. The pressure wave is an acoustic standing wave oscillating with a period of Tac = A/v. This component decays by a mechanical acoustic damping or run out effect of the wave if the number of the fringes is limited. After the complete decay of this wave, the isobaric wave appears. This wave just stays where it is and decays by the thermal diffusion process as described in Section I1.B.2. This equation may be further expanded as... 

If a Flory-Huggins chemical potential is used the entropy fluctuation has the form... 

Eq. (14) would give the shape of a spectrum typical for a low-viscosity liquid, consisting of a central peak due to entropy fluctuations and frequency-shifted Stokes and anti-Stokes lines related to density fluctuations resulting in a bulk phonon. 

The qualitative idea that cooperativity starts at the crossover region and develops at lower temperatures [113-116], was able to be quantified with the development of heat-capacity spectroscopy [117] (HCS or 3calculated using the formula [49,118] ... 

Now the number of configurations available to a system at a given energy is measured by the entropy of the system. As discussed in Appendix 2.A, Cp is thus directly proportional to the entropy fluctuation in the system. 

For simple liquids, the volume of the system increases with temperature and thus p is always positive. Also ap decreases with decrease in temperature as the volume and entropy fluctuations in the system decrease. However, in the case of water it beeomes zero at the temperature where density is maximum (TMD) and then beeomes negative with further decrease in temperature. This suggests that below TMD the entropy inereases with decrease in volume. Like Cp and Kj, p also seems to diverge with a power law at low temperature as shown in Figure 2.4 [2]. 

The same argument can explain the increase of heat capacity with decreasing temperature, observed at very low temperatures. Heat capacity is related to entropy fluctuations. Regions of high density of bonded molecules are certainly more ordered, or have fewer distorted bonds, or have lower entropy than regions where the presence of fi ee or one-bond molecules dominates. 

Isobaric density fluctuations density fluctuations which do not depend on pressure fluctuations. Their evolution proceeds quite slowly and causes a much lower frequency shift on scattering. These fluctuations are related to the central component of scattered light called Rayleigh s component. 

In order to describe the static structure of the amorphous state as well as its temporal fluctuations, correlation functions are introdnced, which specify the manner in which atoms are distributed or the manner in which fluctuations in physical properties are correlated. The correlation fimctions are related to various macroscopic mechanical and thermodynamic properties. The pair correlation function g r) contains information on the thermal density fluctuations, which in turn are governed by the isothermal compressibility k T) and the absolute temperature for an amorphous system in thermodynamic equilibrium. Thus the correlation function g r) relates to the static properties of the density fluctuations. The fluctuations can be separated into an isobaric and an adiabatic component, with respect to a thermodynamic as well as a dynamic point of view. The adiabatic part is due to propagating fluctuations (hypersonic soimd waves) and the isobaric part consists of nonpropagating fluctuations (entropy fluctuations). By using inelastic light scattering it is possible to separate the total fluctuations into these components. 

Each thermodynamic response function is associated with microscopic fluctuations Kt = bV) )/k TV is proportional to volume (or density) fluctuations (Sy) and Cp to the entropy fluctuations (8S) at fixed pressure Cp = ((S5) )/A b whereas ap = S8V)/k TV reflects the S and V cross-correlations. In typical liquids, V and S fluctuations become smaller as the temperature decreases. In water, the fluctuations of these quantities become more pronounced as the temperature decreases. V and S in most liquids are positively correlated an increase in volume results in a corresponding increase in entropy, instead in water for... 

One particularly interesting result is the average value of the entropy fluctuations associated with the m independent variables a ... 

Thus, we see that the average value of entropy fluctuations due to m independent variables is given by the simple expression... 

As a simple example, let us consider entropy fluctuations due to r chemical reactions. As was shown in (14.1.9), we have... 

According to Landau and Placzdc (68,69), the first term represents local entropy fluctuations which do not propagate in normal liquids and are the source of the central (unshifted) component of the scattered light, while the second term represents the isentropic pressure fluctuations (Le., sound waves) which are the source of the Brillouin doublet. 

A diffuse layer is observed at the surface of a crystal which is growing into the melt. The building up of this layer and the dynamics of entropy fluctuations have been studied by quasi elastic light scattering. Water and salol have been used as test substances. In a stationary non equilibrium steady state the layer is several thousand lattice constants thick. The diffusion constant, which determines the dynamics of the entropy fluctuations in this layer, is Dj 3 10 cm /s. It is isotropic in space. During freezing fluctuations in order can not be separated from fluctuation of heat. Therefore we interpret D as J. Frenkel s constant of "structure diffusion". 

The index of refraction of the material in the layer can be estimated to be 1.333. The total intensity of the quasi elastically scattered light is proportional to the isothermal compressibility. From intensity measurements we estimate it to be about a factor 500 higher than the compressibility of water at O C. A diffusion constant, which describes the dynamics of the entropy fluctuations, has been determined to be about 3 10" cm /s. This value has to be compared with the thermal diffusivity in water a 10 cm /s or the constant for self diffusion in water Djj20 lO" cm /s. 

The entropy fluctuation AS obeys a diffusion equation similar to that for the temperature fluctuation ... 

Following the preceding analysis, we can see that the scattering caused by the entropy fluctuation does not shift the frequency. Instead, because of the exponentially decaying dependence, it broadens the light by an amount 5co= Fj. Again, since q = 2 A i sin(0/2), we have... 

This result is analogous to the equipartition formula for the average thermal energy of a system with m degrees of freedom, whose Hamilton function is quadratic in the attendant momenta and coordinates, cf. (5.85). Equation (7.17) shows that the average entropy fluctuation in a system containing m fluctuating quantities is contributed in increments of —ksl l by each of these quantities. 

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