Internal fluctuation

Naturally, very long T,c values are expected for solids as viewed from the expected correlation time at the low temperature side of the T c minimum (0.1-0.2 s) as shown in Figure 1. Indeed, their values turns out to be the order of 10-30 s for carbon sites in the absence of internal fluctuations as in polysaccharides such as (1 — 3)-p-D-glucan and (1 —> 3)-p-D-xylan,46 8 fibrous proteins such as collagen49 and silk fibroin,50 free and metal-complexed ionophores,51 or in some instances up to 1000 s as in crystalline polyethylene.52... 

Internal fluctuations are caused by the discrete nature of matter. The density of a gas fluctuates because the gas consists of molecules fluctuations in a chemical reaction arise because the reaction consists of individual reactive collisions current fluctuations exist because the current is made up of electrons radioactive decay fluctuates owing to the individuality of the nuclei. Incidentally, this explains why the formulas for fluctuations in physical systems always contain atomic constants, such as Avogadro s number, the mass of a molecule, or the charge of an electron. 

Below the Curie temperature, K > 1, there are two stable stationary macrostates + external perturbations or by its internal fluctuations, it will move to either or — fia. This is called symmetry breaking although both the equations and the initial data are symmetrical the final macrostate is not. On the other hand, the mesostate pn(t) determined by the initial condition pn(0) = 5 0 remains symmetric at all t > 0. 

This is the quantum master equation describing internal fluctuations. The energy e enters only as a parameter and a stands for the entire set a, b,. 

These correlated fluctuations themselves ride on a further set of coherent fluctuations taking place at a much lower frequency scale and normally attributed to the phonons, the traditional exchange Bosons associated with superconductivity. Real systems are never devoid of ionic or nuclear motion, and at the very least it is now Hamiltonian (3) (and eventually its extension to alloys) that applies for a full discussion of superconductivity density fluctuations in the nuclear coordinates are omnipresent and of course their effects on electronic ordering have been evident for quite some time. An elementary estimate of the relative importance of (monopole) polarization arising from phonons and the (multipole) equivalents arising from internal fluctuations, primarily of a dipole character, can now be easily given. 

The state variables Xb. ... A, which are continuously subjected to either internal fluctuations or external perturbations, are represented by a column vector X... 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

Among these causes only those mentioned in point 3 are within the scope of this book. We shall discuss them in some detail in Section 6.5 in connection with present climatic variations. Of the other factors we shall clarify here only point 6, which is probably less known than the others. If in our climatic model (see later) the random modification of initial conditions results in a nonzero probability that the climate remains unchanged, then the climate is called stable. If the system has only one single stable climate (and possibly many unstable ones) it is said to be transitive (see more detail in SMIC, 1971). Otherwise, i is said to be intransitive. According to Lorenz (1968) the climate of the Earth is intransitive. This means that climate can go from one stable state to another without the modification of external or initial conditions, that is, variations are due to the internal fluctuations of the system. 

Excitable systems as considered here are many particle systems far from eqnilibrium. Hence variables as voltage drop (neurons), light intensity (lasers) or densities (chemical reactions) are always subject to noise and fluctations. Their sources might be of quite different origin, first the thermal motion of the molecules, the discreteness of chemical events and the quantum uncertainness create some unavoidable internal fluctuations. Bnt in excitable systems, more importantly, the crucial role is played by external sources of fluctuations which act always in nonequilibrium and are not counterbalanced by dissipative forces. Hence their intensity and correlation times and lengths can be considered as independent variables and, subsequently, as new control parameters of the nonlinear dynamics. 

The main conclusion from this exercise is that if a fluid is to exist in a stable equilibrium state, that is, an equilibrium state in which all small internal fluctuations will dissipate rather than grow, the fluid must be such that... 

The theoretical analysis of macromolecular diffusion in interacting solutions of flexible macromolecules is more difficult than that for hard spheres. The presence of internal fluctuations in chain configuration must lead to a softening of the interparticle force field and also means that the particle geometry may not be constant. Lee et al. (104) have recently published a theoretical analysis which describes the qualitative features... 

Note that the term in Ap allows for change in py as a function of pg due to exchange. The step function Pe(f) represents the position and density of exchangeable protons. There will in principle be another term Pef( ) which reflects the internal fluctuation of the density of exchangeable proton sites. This will modify Pp(r) in a manner dependent on Ap however, this is not considered further. 

The surface dynamics of bacteriorhodopsin has been studied by Barre et al. by solid-state NMR. They have measured site-specific H-C dipolar couplings in [3- C] Ala-labelled bacteriorhodopsin, which allowed an insight into internal fluctuation motions on the millisecond or microsecond timescale relevant to a variety of biological functions. 

In addition, the C NMR signals from the loops and several C NMR peaks from the transmembrane a-helices near to the surface, including Ala 39, 53, 168, and 215 were preferentially suppressed in these mutants as encountered when retinal-protein interactions are substantially modified as in the case of D85N mutant or they are absent as in bO. This is due to the acquired internal fluctuation with a correlation time of 10 s, which interferes with the proton decoupling frequency of 50 kHz. Further, it should be noted that the C CP-MAS... 

A second difference is in how the two systems respond to internal fluctuations or to externally imposed disturbances. Such responses are quantified by the thermodynamic response functions and, again, the natural choices for these two systems differ. For example, the first-order response to a change in temperature is given by (6.1.5), but the second-order response is given by a heat capacity the response for system 1 is given by Q, while that for system 2 is given by C. These two heat capacities differ ... 

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