Extended equation fluctuation

In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

The non-aqueous system of spherical micelles of poly(styrene)(PS)-poly-(isoprene)(PI) in decane has been investigated by Farago et al. and Kanaya et al. [298,299]. The data were interpreted in terms of corona brush fluctuations that are described by a differential equation formulated by de Gennes for the breathing mode of tethered polymer chains on a surface [300]. A fair description of S(Q,t) with a minimum number of parameters could be achieved. Kanaya et al. [299] extended the investigation to a concentrated (30%, PI volume fraction) PS-PI micelle system and found a significant slowing down of the relaxation. The latter is explained by a reduction of osmotic compressibihty in the corona due to chain overlap. 

When the constant field is weak and the fluctuating field is comparable to or even larger than the constant field, the above decomposition becomes meaningless. There is no way of distinguishing between the adiabatic and nonadiabatic effects. In order to obtain an understanding of this rather complex situation, we have examined a stochastic model,14 extending the theory in Section II. The stochastic equation of motion of a spin in a random local field is written as... 

Warning. The idea of using a nonlinear Fokker-Planck equation as a general framework for describing fluctuating systems has attracted many authors. Detailed balance, in its extended form, was a useful aid, but the link with the deterministic equation caused difficulties. It may therefore be helpful to emphasize three caveats. 

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. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

On the other hand, the stochastic treatment in terms of equation (2.2.2) is applied to the macroscopic values a(t) characterizing a system. As is generally-accepted, thermal fluctuations of macroscopic quantities being very small are not of great interest. The only exception when fluctuations appear to be important (which leads to an untrivial result) is the case when the solution of a set of equations F (c, ..., cs) = 0 finds itself near the bifurcation point [26, 34, 90], This is why we are going to consider now the spatially-extended systems [26, 67, 68],... 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

In the extended Lagrangian method, as applied to a fluctuating charge system,the charges are given a fictitious mass, M, and evolved in time according to Newton s equation of motion, analogous to Eq. [23],... 

The JE and Crooks fluctuation relations were originally developed to treat systems in canonical equilibrium states. Adib " extended the JE and thermodynamic integration schemes to obtain A>S of isoenergetic systems that originate in a microcanonical ensemble. The FR is obtained by designing constant energy equations of motion and is given by ... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

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