Fluctuation theory

Barnes and Hunter [290] have measured the evaporation resistance across octadecanol monolayers as a function of temperature to test the appropriateness of several models. The experimental results agreed with three theories the energy barrier theory, the density fluctuation theory, and the accessible area theory. A plot of the resistance times the square root of the temperature against the area per molecule should collapse the data for all temperatures and pressures as shown in Fig. IV-25. A similar temperature study on octadecylurea monolayers showed agreement with only the accessible area model [291]. 

Fox R F and Uhlenbeck G E 1970 Contributions to non-equilibrium thermodynamics. II. Fluctuation theory for the Boltzmann equation Rhys. Fluids 13 2881... 

Figure 38. Classification of nonequilibrium fluctuations. (Reprinted from M. Asanuma and R. Aogaki, Non-equilibrium fluctuation theory on pitting dissolution. I. Derivation of dissolution current equations." J. Chem. Phys. 106,9938,1997. Copyright 1997, American Institute of Physics.)...

Figure 40. Plot of the fluctuation-diffusion current J vs. iwr.91 id is the slope of the fluctuation-diffusion current given by Eq. (115). Solid and dotted lines correspond to the theoretical and experimental results, respectively. (NiCljJ = 0.1 mol nT3. [NsCl] = 7 mol m 3. V = 0.1 V, T= 300 K. (Reprinted from M. Asanuma and R. Aogaki, Nonequilibrium fluctuation theory on pitting dissolution, n. Determination of surface coverage of nickel passive film, J. Chem. Phys. 106, 9938, 1997, Fig. 8. Copyright 1997, American Institute of Physics.)...

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

AT DiBenedetto, DR Paul. Interpretation of gaseous diffusion through polymers by using fluctuation theory. J Polym Sci Part A 2 1001-1015, 1964. 

Smoluchowski (1908), Einstein (1910), Ornstein Zernike (1914, 1918). In a textbook on scattering HIGGINS Benoit ([136], Sect. 7.6) consider the fluctuation theory from a different point of view. 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory of Phase Transitions Pergamon Press, Oxford, 1979. 23 ... 

Peter Debye in 1944 further extended the work of Rayleigh and the fluctuation theory of Smoluchowski and Einstein to include the measurement of the scattering of light by macromolecular solutions for determining molecular size. 

Like there always exists a vapor under the water, there are excitations on the ground of any condensate. They appear due to quantum and thermal fluctuations. In classical systems and also at not too small temperatures in quantum systems, quantum fluctuations are suppressed compared to thermal fluctuations. Excitations are produced and dissolved with the time passage, although the mean number of them is fixed at given temperature. Pairing fluctuations are associated with formation and breaking of excitations of a particular type, Cooper pairs out of the condensate. Fluctuation theory of phase transitions is a well developed field. In particular, ten thousands of papers in condensed matter physics are devoted to the study of pairing fluctuations. At this instant we refer to an excellent review of Larkin and Varlamov [15]. 

To obtain a correct form of Eq. (22) allowing for thermodynamic non-ideality of the solution, fluctuation theory originally developed by Einstein, Zernicke, Smoluchowski and Debye has been adapted to polymer solutions. 

Fluctuation theory gives Eq. (32), which is converted to Eq. (33) via thermodynamic transpositions and approximation of partial molar volume of solvent to its molar volume Vj ... 

TNC. 14.1. Prigogine and P. Glansdorff, Variational properties and fluctuation theory, Physica 31, 1242-1256 (1965). 

Figure 9. Reduced equilibrium modulus of polyurethane networks from POP trlols and MDI in dependence on the sol fraction. networks from POP triol Mjj - 708, o networks from POP triol Mjj = 2630. C-) calculated dependence using Flory junction fluctuation theory for the value of the front factor A indicated. (Reproduced from Ref. 57. Copyright 1982 American Chemical Society.)...

For spin fluctuation systems, the best example is again UAl2 In fact as shown in Fig. 3 it is the unique example where T InT dependence at low temperature, predicted by spin fluctuation theory, has been in practice observed (very recently, such behaviour... 

The fluctuation theory of scattering by molecules is treated in books by Bhagavantam (1942), Fabelinskii (1968), and Chu (1974, Chap. 3). 

The fluctuation theory has received attention because it avoids some of the serious assumptions involved in the rate theory. The beginnings of fluctuation theory were presented by Einstein. Various workers since... 

The advantages of the fluctuation theory are that it does not require that clusters be spheres, they need not have sharply defined bounding surfaces, nor is an equilibrium between phases assumed. The disadvantage is a practical one how can the work term (defined later) be evaluated ... 

As pointed out, the value gh must be selected properly. Roughly speaking it will have a value such that the density of a cell when one molecule is in the cell will be equal to the vapor density. In any case it seems to be possible to select this value so that the distribution will predict the existence of nuclei, that is, cells which have the proper density and energy to cause spontaneous growth of a new phase. The evaluation of the interaction term, W>, is unsatisfactory. However the fluctuation theory cannot be dismissed. Light scattering measurements are strong proof that the assumed fluctuations are very real. 

Fig. 9.3. A schematic bell-shaped diagram of the distribution of electronic states of redox ions at an electrode/solution interface (fluctuation theory). (Reprinted with permission of the American Institute of Physics from J. O M. Bockris and S. U. M. Khan, Appl. Phys. Lett. 42 124 (1983).

G. Ruppeiner. Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605-59 (1995). 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

For a two-component system (macromolecule, component 2, dissolved in pure solvent, e.g., water, component 1), application of the fluctuation theory of scattering (15, 16) shows that in a volume element of volume SV (of the order of s z at the lowest angles of measurement), the intensity of radiation scattered by the solution in excess of the solvent, I(s) = /(s)solution — I(s)solvent, is proportional to the fluctuations in the number of electrons, n ... 

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