Force spontaneously fluctuating

To establish the relationship between dissipation and fluctuation, consider a system left in thermal equilibrium, without any applied force. It is expected that spontaneous fluctuation Q is associated with a spontaneously fluctuating force. 

Why Because the frequencies at which charges spontaneously fluctuate are the same as those at which they naturally move, or resonate, to absorb external electromagnetic waves. This is the essence of the "fluctuation-dissipation theorem." It states that the spectrum (frequency distribution) over which charges in a material spontaneously fluctuate directly connects with the spectrum of their ability to dissipate (absorb) electromagnetic waves imposed on them. Computation of charge-fluctuation forces is essentially a conversion of observed absorption spectra. By its very nature, the measured absorption spectrum of a liquid or solid automatically includes all the interactions and couplings among constituent atoms or molecules. 

As emphasized elsewhere in this text, the physical act constituting an electrodynamic force is the correlated time-varying fluctuation of all component electric charges and electromagnetic fields in each material composing a system. Charge fluctuations at each point are either spontaneous or are in response to electric fields set up by fluctuations elsewhere. The dielectric permittivity is an experimental quantity that codifies not only the response of a material to an applied electric field but also the magnitude of spontaneous fluctuations. 

A particularly important variant of the optical force, interparticle forces, turns out to be crucial for SERS. This effect is similar to the attractive van der Waals force between small particles, which is due to interactions between spontaneously fluctuating dipoles, but the optical interaction is due to coupling between the actual particle dipoles induced by the trapping laser. Due to the interparticle optical forces, metal nanoparticles aggregate in an optical tweezers and produce hotspots, i.e., particle junctions with intense local fields for SERS. Raman probes can be excited either by the trapping laser or, preferably, by a separate low power beam that does not disturb the trapping. 

This potential force occurs in microstructured fluids like microemulsions, in cubic phases, in vesicle suspensions and in lamellar phases, anywhere where an elastic or fluid boundary exists. Real spontaneous fluctuations in curvature exist, and in liposomes they can be visualised in video-enhtuiced microscopy [59]. Such membrane fluctuations have been invoked as a mechanism to account for the existence of oil- or water-swollen lamellar phases. Depending on the natural mean curvature of the monolayers boimding an oil region - set by a mixture of surfactant and alcohol at zero -these swollen periodic phases can have oil regions up to 5000A thick With large fluctuations the monolayers are supposed to be stabilised by steric hindrance. Such fluctuations and consequent steric hindrance play some role in these systems and in a complete theory of microemulsion formation. 

If it is true that the rate of umbilical blood flow remains constant in spite of spontaneous fluctuations in 02 need and delivery, then there are important consequences for fetal homeostasis as recently discussed by Faber (50). A constant umbilical blood flow would assure that intravascular, hydrostatic pressures would remain constant in fetal placental capillaries, and the balance of maternal-fetal hydrostatic forces determining transplacental water movement would be maintained. The fetus would not gain or lose water as might otherwise happen if umbilical flow and pressure were to vary in response to different fetal 02 needs. The fetus could avoid becoming dehydrated during periods of increased 02 transport. 

They arise because of a transient polarization of the atom or molecule which will act on the surroundings to produce spontaneous fluctuations elsewhere. The causes of such interaction have been extensively reviewed by Kauz-mann and Jehle The electromagnetic properties of van der Waals forces were first shown by London in 1930 and are frequently referred to as the London-van der Waals forces. The extension of the theory of van der Waals attractive forces from the atomic or microscopic scale to bulk powders on the macroscopic scale was first carried out by Lifshitz in 19SS (ref. 16). 

The dispersion interactions, also called London-van der Waals forces, appear for all material bodies—in particular atoms and molecules due to spontaneous fluctuations in the electromagnetic field (two induced dipole interactions). As discussed in length in Refs 17, 18, 49, and 50, there exist essentially two main approaches aimed at calculating these interactions ... 

Polarizability is a general concept that quantifies the response of an electron cloud of an ion to the apphcation of a time-dependent electromagnetic field resulting in a frequency-dependent polarizability. Our strict concern is with static, or zero-frequency polarizability as variations of an electric field induced by thermal fluctuations of an electrolyte operate at timescales much larger than the timescales of inner dynamics of an electron cloud. Frequency-dependent polarizability leads to other interesting effects, such as the London forces [32], when spontaneous fluctuations of electronic structure of two molecules become correlated at close spacial separations. These interactions, however, play secondary role when compared to induced interactions that arise from static polarizability [33, 34]. 

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

We can run the cause-effect connection the other way. The natural motions of the charges within a material will necessarily create electric fields whose time-varying spectral properties are those known from how the materials absorb the energy of applied fields (the "fluctuation-dissipation theorem"). It is the correlations between these spontaneously occurring electric fields and their source charges that create van der Waals forces. At a deeper level, we can even think of all these charge or field fluctuations as results or distortions of the electromagnetic fields that would occur spontaneously in vacuum devoid of matter. 

Resonance frequencies or absorption frequencies occur when the natural frequencies of charge motion are close to the frequencies of the applied fields. It is no surprise that absorption frequencies are what show up in forces that depend on spontaneous charge fluctuation (see Fig. L2.23). 

As described in earlier sections, any two material bodies will interact across an intermediate substance or space. This interaction is rooted in the electromagnetic fluctuations— spontaneous, transient electric and magnetic fields—that occur in material bodies as well as in vacuum cavities. The frequency spectrum of these fluctuations is uniquely related to the electromagnetic absorption spectrum, the natural resonance frequencies of the particular material. In principle, electrodynamic forces can be calculated from absorption spectra. 

Some representative examples of common zero-temperature VER mechanisms are shown in Fig. 2b-f. Figures 2b,c describe the decay of the lone vibration of a diatomic molecule or the lowest energy vibrations in a polyatomic molecule, termed the doorway vibration (63), since it is the doorway from the intramolecular vibrational ladder to the phonon bath. In Fig. 2b, the excited doorway vibration 2 lies below large molecules or macromolecules. In the language of Equation (4), fluctuating forces of fundamental excitations of the bath at frequency 2 are exerted on the molecule, inducing a spontaneous transition to the vibrational ground state plus excitation of a phonon at Fourier transform of the force-force correlation function at frequency 2, denoted C( 2). 

The preceding conclusions about the stabiHty of stationary states near stable thermodynamic equilibrium are graphically interpreted in Figure 2.5. Indeed, if an incidental fluctuation of thermodynamic force X, around its stationary magnitude X, results in a minor deviation of the system from the stationary state near thermodynamic equilibrium, the internal trans formations must happen according to inequality (2.31), which wiU affect the value of X, and return the system again to its initial stationary state (see Figure 2.5A). Thus, if the system is near thermodynamic equilibrium in the stationary state, it cannot escape this state spontaneously due to... 

A final interesting observation is the existence of a frequency scale, 3x10 see in Eq. (2-39). This is the frequency at which the electronic cloud around an atom fluctuates it is therefore the rate at which the spontaneous dipoles fluctuate. Since the electromagnetic field created by these dipoles propagates at the speed of light c = 3 x lO cm/sec, only a finite distance c/v 100 nm is traversed before the dipole has shifted. Since the dispersion interaction is only operative when these dipoles are correlated with each other, and this correlation is dismpted by the time lag between the fluctuation and the effect it produces a distance r away, the dispersion interaction actually falls off more steeply than r when molecules or surfaces become widely separated. This effect is called the retardation of the van der Waals force. The effective Hamaker constant is therefore distance dependent at separations greater than 5-10 nm or so. 

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