Macroscopic fluctuations

Hirschfelder and co-workers (H6) give a detailed discussion of this question. It is proved that a liquid in tension such as that at temperature located on the line BC is unstable to macroscopic fluctuations in density but is stable (thermodynamically) to tiny fluctuations. Portion EF represents supersaturated vapor which also is stable (thermodynamically) to tiny fluctuations. The line CDE is proved to correspond to liquid or vapor which is thermodynamically unstable to density fluctuations of any magnitude whatever. This means that states corresponding to CDE are completely unattainable. ... 

In turbulent flow, the fluid particles do not travel in a well-ordered pattern. These particles possess velocities with macroscopic fluctuations at any point in the flow field. Even in steady turbulent flow, the local velocity components transverse to the main flow direction change in magnitude with respect to time. Instantaneous velocity consists of time-average velocity and its fluctuating component. When heat transfer is involved in turbulent flow, the instantaneous temperature is composed of the time-average temperature and its fluctuating components. 

Equation 6 applies to the analysis of fluid behavior in tribology and microchannel flows this kind of flow is called Stokes flow. In this case, the velocities are free of macroscopic fluctuations at any point in the flow field and the flow is defined as laminar. 

Two kinds of basic information can be derived from fluctuation analysis. One concerns the amplitude of the elementary events contributing to the mean macroscopic behaviour and to the macroscopic fluctuations of the system under study. This information is derived from measurements of the mean square amplitude of the fluctuations together with the mean value of the parameter under study. The second type of information concerns the kinetics of the transitions between the different states of the elementary components of the system. This information is obtained from the statistical correlation existing between fluctuations measured at different times. 

On the contrary, the predictions for the composition of the vapor phase are far less satisfactory. The failure of the theory to properly describe the macroscopic fluctuations that occur in the neighborhood of the critical point has a large effect. Particularly, the critical pressure predicted by the theory is more than twice as large... 

Oyy/Ais of the order of hT, as is Since a macroscopic system described by themiodynamics probably has at least about 10 molecules, the uncertainty, i.e. the typical fluctuation, of a measured thennodynamic quantity must be of the order of 10 times that quantity, orders of magnitude below the precision of any current experimental measurement. Consequently we may describe thennodynamic laws and equations as exact . 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

A system of interest may be macroscopically homogeneous or inliomogeneous. The inliomogeneity may arise on account of interfaces between coexisting phases in a system or due to the system s finite size and proximity to its external surface. Near the surfaces and interfaces, the system s translational synnnetry is broken this has important consequences. The spatial structure of an inliomogeneous system is its average equilibrium property and has to be incorporated in the overall theoretical stnicture, in order to study spatio-temporal correlations due to themial fluctuations around an inliomogeneous spatial profile. This is also illustrated in section A3.3.2. 

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

The small statistical sample leaves strong fluctuations on the timescale of the nuclear vibrations, which is a behavior typical of any detailed microscopic dynamics used as data for a statistical treatment to obtain macroscopic quantities. 

Molecules are in continuous random motion, and as a result of this, small volume elements within the liquid continuously experience compression or rarefaction such that the local density deviates from the macroscopic average value. If we represent by 6p the difference in density between one such domain and the average, then it is apparent that, averaged over all such fluctuations, 6p = 0 Equal contributions of positive and negative 6 s occur. However, if we consider the average value of 6p, this quantity has a nonzero value. Of these domains of density fluctuation, the following statements can be made ... 

We define the concentration of fluctuation domains at any instant by the symbol N. In addition, we assume that the polarizability associated with one of these domains differs from the macroscopic average value for the substance... 

The concentration [MB] constantly experiences tiny fluctuations, the duration of which can determine linewidths. It is also possible to adopt a traditional kinetic viewpoint and measure the time course of such spontaneous fluctuations directly by monitoring the time-varying concentration in an extremely small sample (6). Spontaneous fluctuations obey exactly the same kinetics of return to equiUbrium that describe relaxation of a macroscopic perturbation. Normally, fluctuations are so small they are ignored. The relative ampHtude of a fluctuation is inversely proportional to the square root of the number of AB entities being observed. Consequently, fluctuations are important when concentrations are small or, more usehiUy, when volumes are tiny. 

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

From the conceptual diagram in Fig. 15, it is obvious that if the radius of the nucleus exceeds the critical radius r the nucleus will grow into a macroscopically ruptured small pore. The passive film is more or less defective and the size of the defect will fluctuate from moment to moment. It is therefore reasonable to assume a certain probability that pore nuclei larger than the critical radius are formed in the film. 

In all these treatments, nonequilibrium fluctuation plays the most important role. This is defined as the fluctuation of a physical quantity that deviates from the standard state determined by the nonzero flux in a nonequilibrium state. Such fluctuation has a kind of symmetry in that the area average is equal to zero although the flux changes locally. Therefore, macroscopically, such fluctuation does not affect the flux itself. This means that the flux must be determined a priori and is indifferent to the fluctuations. 

As shown in Fig. 24, the mechanism of the instability is elucidated as follows At the portion where dissolution is accidentally accelerated and is accompanied by an increase in the concentration of dissolved metal ions, pit formation proceeds. If the specific adsorption is strong, the electric potential at the OHP of the recessed part decreases. Because of the local equilibrium of reaction, the fluctuation of the electrochemical potential must be kept at zero. As a result, the concentration component of the fluctuation must increase to compensate for the decrease in the potential component. This means that local dissolution is promoted more at the recessed portion. Thus these processes form a kind of positive feedback cycle. After several cycles, pits develop on the surface macroscopically through initial fluctuations. 

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