Gaussian fluctuation potential

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. 

Fig. 5.43. (a) Assumed Gaussian distribution of potentials in disordered material (b) Electron with energy E in a fluctuating potential finds fraction x(E) allowed, (c) Relative mobility according to Figure 5.42 assuming x(E) is Unear between percolation value x = 0.3 and x = 1. The mobility increases slowly above the percolation threshold E. After Kirkpatrick (1972). 

Chandler and co-workers " have proposed a self-consistent pair interaction, called the Gaussian fluctuation (GF) potential, of the form... 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

Physically, as we go to larger masses during the A integration, the widths of the Gaussians in the kinetic-energy piece of the sampling function become very narrow. This means that the distributions of the a/., are essentially Gaussian due to no influence from the potential on the scale of the very small particle fluctuations, the... 

If fluctuations are neglected, this system can be described with the celebrated model of resistively shunted junction [5], The normal conductor is a source of non-gaussian current fluctuations that instantly tilt the washboard potential and can lead to an escape of from the minimum. The escape gives rise to an observable voltage pulse. The escape rate in the same or similar systems has been studied for a variety of noise sources and potentials [6, 7, 8, 9],... 

In principle, since the potential V is not linear, Eqs. (4.1) can be used to simulate the non-Gaussian non-Markovian behavior of the variables < > and V as well as the rototranslational phenomena. Note that this non-Gaussian, non-Markovian behavior depends on the presence of the virtual body and the nonlinear nature of the potential V in spite of the Markovian-Gaussian character of the fluctuation-dissipation process governing the stodiastic torques and the stochastic forces Note also... 

To recover the ideal case of Eq. (1.1) we would have to assume that (u ), vanishes. The analog simulation of Section III, however, will involve additive stochastic forces, which are an unavoidable characteristic of any electric circuit. It is therefore convenient to regard as a parameter the value of which will be determined so as to fit the experimental results. In the absence of the coupling with the variable Eq. (1.7) would describe the standard motion of a Brownian particle in an external potential field G(x). This potential is modulated by a fluctuating field The stochastic motion of in turn, is driven by the last equation of the set of Eq. (1.7), which is a standard Langevin equation with a white Gaussian noise defined by... 

Here, m is the mass of the particles, V(r) the potential energy of the system, 7 the friction constant, and. F is a Gaussian random force uncorrelated in time that satisfies the fluctuation dissipation theorem [20]... 

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