Langevin equation fluctuating force

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

In the limit of a very rapidly fluctuating force, the above equation can sometimes be approximated by the simpler Langevin equation... 

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 earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

This is the Langevin equation [271, 490]. The fluctuating force, f, varies so rapidly that it cannot be described as a function of time directly. Instead, only certain statistical properties can be defined. 

The velocity relaxation time is again f/rn and the mean square velocity (up = k T/m. Schell et al. [272] have used the Langevin equation to model recombination of reactants in solutions. Finally, from the properties of the fluctuating force (see above)... 

The plan of the article is as follows. First, we discuss the phenomenon of hydrodynamic interaction in general terms, and at the same time, we present some convenient notation. Then, we give the usual argument leading to the Fokker-Planck equation. After that we derive the Langevin equation that is formally equivalent to the Fokker-Planck equation, together with a statistical description of the fluctuating force. 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

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... 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

Linear Motion of a Brownian Particle. In the amplest case of Brownian motion, a massive particle is immersed in a mediiun of lighter partides whose rapid thermal motion produces a quickly fluctuating force on the massive Brownian particle. This force will be in part correlated with the motion of the Brownian particle itself. Langevin s simplifying hypothesis is that the correlated part of the force exerted by the medium is propor< tional and opposed to the velocity u of the particle. Langevin s equation of motion then has the form... 

The FPE has its genesis in the Langevin equations of motion of the particles, in which the influence of the bath particles is characterized by a friction and a fluctuating random force. Exact treatments lead to generalized Langevin equations when the solvent degrees of freedom are projected out from the classical equations of motion for the full particle-bath system In this case a frequency-dependent friction, or time-dependent memory kernel,... 

Both these Langevin equations are monofractal if the fluctuations are monofractal, which is to say, the time series given by the trajectory X(t) is a fractal random process if the random force is a fractal random process. 

Here I = 7.5 X 1015 g-cm2/mol is the moment of inertia of the ring about the torsional axis, 7/3 is the friction constant, k is the harmonic restoring force constant, and/(f) represents the random torques acting on the ring due to fluctuations in its environment. In using the Langevin equation, we implicitly assume that variations in/(f) occur on a much shorter time scale than do... 

The third contribution to the force on the particle is due to random fluctuations caused by interactions with solvent molecules. We will write this force as R(f). The Langevin equation of motion for a particle i can therefore be written" ... 

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