Gaussian fluctuation

The assumption of Gaussian fluctuations gives the PY approximation for hard sphere fluids and tire MS approximation on addition of an attractive potential. The RISM theory for molecular fluids can also be derived from the same model. 

Levy, R. M., Belhadj, M., Kitchen, D. B. Gaussian fluctuation formula for electrostatic free energy changes. J. Chem. Phys. 95 (1991) 3627-3633... 

Fig. 10-3. Experimental proof that x-ray emission speetrography and radioactivity both conform to the unique Gaussian fluctuation curve based on N alone. Crosses = data of Rutherford and Geiger circles = x-ray emission data solid line = theoretical Gaussian curve. (Liebhafsky, Pfeiffer, and Zemany, Anal. Chem., 27, 1257.)...

If AU( 1) has Gaussian fluctuations, this means that the free energy is also a parabolic function of AU(1) itself. AU( 1) is known as the energy gap [52-54] we will denote it by p... 

Simonson, T., Gaussian fluctuations and linear response in an electron transfer protein, Proc. Natl Acad. Sci. USA 2002, 99, 6544-6549... 

L.133 Using two sets of backbone RDC data, collected in bacteriophage Pfl and bicelle media, they obtained order tensor parameters using a set of crystallographic coordinates for the structural model. This allowed the refinement of C -C bond orientations, which then provided the basis for their quantitative interpretation of C -H RDCs for 38 out of a possible 49 residues in the context of three different models. The three models were (A) a static xi rotameric state (B) gaussian fluctuations about a mean xi torsion and (C) the population of multiple rotameric states. They found that nearly 75% of xi torsions examined could be adequately accounted for by a static model. By contrast, the data for 11 residues were much better fit when jumps between rotamers were permitted (model C). The authors note that relatively small harmonic fluctuations (model B) about the mean rotameric state produces only small effects on measured RDCs. This is supported by their observation that, except for one case, the static model reproduced the data as well as the gaussian fluctuation model. 

To appreciate this important fact, let us study the diffusion process generated by the fluctuation c(t). This is a gaussian fluctuation that brings us back to gaussian case of Section VII and especially to Eq. (158). Using this equation, we write the time derivative of x2 t)) as follows ... 

Including Gaussian fluctuations around the classical minimum. Bender et al. [25] obtain... 

Because we must have 0.PB1, < 1 we obtain a stiff inequality on ft. Using Carr s result (based on a Gaussian fluctuation spectrum)... 

When the distribution being averaged over is dynamic [29] but classical with Gaussian fluctuations x(f), the gammas representing the homogeneous widths of different components should be omitted from the formulas and the average in Eq. (44) must be replaced by... 

Fredrickson and Helfand [58] used a selfconsistent Hartree approximation in Eq. (199) to study the Gaussian fluctuations around the solution, Eq, (200). While for ordinary second order transition the local function <4>2(r) > — << >>2 stays small even at Tc, this local fluctuation diverges here as T - Tc The reason is that normally the phase space for critical fluctuations is only the vicinity of a point in reciprocal space (the surroundings of q = 0 for a ferromagnet, the surroundings of a few discrete points qB at the Brillouin zone boundary for antiferromagnets, etc.), while here it is the vicinity of a sphere ( q = q ). Fluctuations lead here to a divergence of the mean square displacement of 4> similar as it happens due to phonons in one-dimensional crystals. 

In spin-boson models, the tunneling particle is described by a tight-binding picture and the solvent by a Gaussian fluctuating bath. Our QMC simulation starts by expressing a dynamical quantity of interest in the form of Eq. (2.2). The forward- and reverse-time propagators are then discretized in the usual way. The first crucial point to make is that... 

R. M. Levy, M. Belhadj, and D. B. Kitchen, ]. Ghem. Phys., 95, 3627 (1991). Gaussian Fluctuation Formula for Electrostatic Free Energy Changes in Solution. 

The Gaussian fluctuation (GF) Hre functional was initially developed by Chandler, Singh, and Richardson, for ID RISM and was adopted by Kovalenko and Hirata for the 3D RISM case [82,103] ... 

The first terra on the right side comes from a kind of surface tension and tends to smooth the surface, while the second term is a Gaussian fluctuating white noise satisfying the fluctuation dissipation theorem. Equation (31.6) leads for it = 2 to Ds = 1-5. Also numerically, this result is well verified for random deposition with surface diffusion [34—38]. For d = 3, we find from Eq. (31.6) = 3. In... 

We now elaborate the bistable SR model in the theoretical form that is conventionally used by the physicists. We now ask how an image pixel would transform if mean-zero Gaussian fluctuation noise rj(t) is added, so that the pixel is transferred from a weak-signal state to a strong-signal state, i.e. a binary-state transition occurs. Actually, such a discrete image pixel under noise can be modeled by a discrete particle under Brownian motion, the particle... 

As long as fluctuations are weak, they can be treated within the Gaussian approximation. One famous example of such a treatment is the random phase approximation (RPA) which describes Gaussian fluctuations in homogeneous phases. The RPA has been extended to inhomogeneous saddle points by Shi, Noolandi, and coworkers [66,67]. In this section, we shall re-derive this generalized RPA theory within the formalism developed in the previous sections. 

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