Microscopic fluctuations behavior

The macroscopic behavior of physical systems is determined by the microscopic behavior of these systems. Usually the microscopic fluctuations are averaged, and on larger scales the averaged values satisfy the classical equations. 

In thermodynamics the existence of extremum principles has an important consequence for the behavior of microscopic fluctuations. Since all macroscopic... 

The spontaneous switching between the stabilizing and destabilizing modes (i.e., the transition among solutions) was observed as nonperiodic and stochastic. As one of its possible origins, we suppose the combination of intrinsic microscopic fluctuations and spatiotemporal chaos characterized as nonperiodic but nonrandom behavior. Indeed, the system s behavior is chaotic as its time evolution is unstable and unreproducible. Its capability of successively searching for multiple solutions, however, is robustly maintained and qualitatively reproducible. This resembles the robustness of strange attractors of chaotic systems. 

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. 

In most problems involving boundary conditions, the boundary is assigned a specific empirical or deterministic behavior, such as the no-slip case or an empirically determined slip value. The condition is defined based on an averaged value that assumes a mean flow profile. This is convenient and simple for a macroscopic system, where random fluctuations in the interfacial properties are small enough so as to produce little noise in the system. However, random fluctuations in the interfacial conditions of microscopic systems may not be so simple to average out, due to the size of the fluctuations with respect to the size of the signal itself. To address this problem, we consider the use of stochastic boundary conditions that account for random fluctuations and focus on the statistical variability of the system. Also, this may allow for better predictions of interfacial properties and boundary conditions. 

In a study of this type, data are composed of single atom events. Statistical fluctuations of the data have to be carefully considered and a careful statistical analysis of the data has to be done before the conclusions derived can be statistically meaningful and reliable. Although there are many precautions which have to be taken in experiments of this type, because of the very well defined nature of these experiments the technique has now been developed to a high degree of reliability and field ion microscope studies of the behavior of single atoms are among the best established of all field ion experiments. 

This microscopic interaction model can be used to explain more specific interactions between drug molecules and lipids. Such specific interactions could be a selective coupling between a drag molecule and a particular chain conformation of the lipid (kink excitation). This could have a dramatic effect on the fluctuation system. The drug molecule would then control the formation of interfaces between lipid domains and bulk phase in the neighborhood of the transition. First results on an extended model of this type [50] have confirmed this view and demonstrated that the partition coefficient can develop non-classical behavior by displaying a maximum near the transition. And such a maximum has in fact been observed experimentally... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

The fact that the origin of the repulsion is different for plates and for rods means that the overall phase behavior can be different, because the interaction depends sensitively on the competition between the correlation-attraction and the repulsion. However, the mechanisms of attraction that have been proposed for rods are the same as for plates. Thus the ionic crystal picture for plates has been applied to rods [20-23], as has the thermal fluctuation picture (which was developed for rods more than 30 years ago by Oosawa [24-26]). In the case of rods, there are several versions of the ionic crystal model, which differ in microscopic details [21-23,27]. In the thermal fluctuation picture, fluctuations in the condensed counterion density along the rods lead to nonuniformities in the charge distribution, which can become correlated from one rod to another [26,28], leading to an attraction similar to the van der Waals interaction. We have introduced a third approach, called the charge fluctuation approach [29,30], which is an extension of the thermal fluctuation approach to ions of nonzero size, and which captures aspects of both the thermal fluctuation picture and the ionic crystal pictures. 

In solids, as in liquids, macroscopic behavior is a consequence of microscopic structure. Even more, in solids the structure defines the thermodynamic phase, and deviations from the nominal structure are true anomalies—defects. In contrast, defects are so prevalent in fluids that the term loses currency fluids are described instead by fluctuations, reflecting the diminished (albeit consequential) role of structure in fluid-phase behavior. Structure in solids is a much more cooperative and large-scale phenomenon than in liquids. This means that changes in the structure of solids do not happen incrementally or in isolation. Changes in structure are... 

For small curvatures, Eq. (6.15) shows that the curvature energy of a thin film is characterized by the three parameters k, k, and cq. The qualitative behavior of any system, including such properties such as the equilibrium shape, magnitude of thermal fluctuations, and any phase transitions, can of course be calculated as a function of these constants. However, the physics of the system can be radically different depending on the physical parameters e.g., a change in cq can induce shape changes in the system. It is thus of interest to relate the bending elastic moduli and the spontaneous curvature to the physics of the particular system of interest. This section first shows how these parameters are related to the pressure distribution in the membrane and then presents a simple but instructive microscopic model that relates k, and Co to more molecular properties. 

We have also seen that there are strong similarities between systems containing small amphiphilic molecules on the one hand and diblock copolymers on the other. In both cases, the amphiphiles contain within themselves the properties of the components of two mutually insoluble liquids. The theory of diblock copolymers is more advanced than that of the small molecular systems not only because a simple microscopic model describes most properties of the polymers very well, but also because these properties depend on large-scale behavior of the chains, not small-scale behavior of the monomers. Furthermore, the large polymerization index guarantees that thermal fluctuations are less important than in small molecular systems, so that mean-field theories give very reliable results. 

The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k km, where km denotes the first peak position of the static structure factor S k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the single relaxation time model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued. 

This equation is an example of a macroscopic reaction-transport equation that can be obtained in the long-time large-scale limit of mesoscopic equations. Recall that the mesoscopic approach is based on the idea that one can introduce mean-field equations for the particle density involving a detailed description of the movement of particles on the microscopic level. At the same time, random fluctuations around the mean behavior can be neglected due to a large number of individual particles. For example, we can obtain (3.1) from the mesoscopic integro-differential equation... 

The behavior of the transverse field relaxation rate A (see fig. 168) is quite interesting. It rises sharply when 7 is approached from above and drops markedly at T. It then remains about constant. The La compound exhibits a temperature-independent small relaxation rate which has its origin in the nuclear dipole fields of La ( 100% abundant). The variation of rate in Ceo,74Tho,26 mirrors the behavior of the dynamics of the local field produced by moments on Ce. Close to Tg critical slowing down occurs (for T > Ts), but motional narrowing again becomes effective for T < Tg. The microscopic origin of those fluctuations cannot be extracted from the [tSR data. The authors discuss normal... 

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