Langevin equation theory

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

K. Lindenberg, K. E. Shuler, V. Seshadri, and B. J. West, Langevin equations with multiplicative noise theory and applications to physical processes, in Probabilistic Analysis and Related Topics, Vol. 3, A. T. Bharucha-Reid (ed.), Academic Press, San Diego, 1983, pp. 81-125. 

C. C. Martens, Qualitative dynamics of generalized Langevin equations and the theory of chemical reaction rates, J. Chem. Phys. 116, 2516 (2002). 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

In the GH theory, it is assumed that the reaction barrier is parabolic in the neighborhood of x and that the solute reactive coordinate satisfies a generalized Langevin equation (GLE),... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

The second approach starts from the modified Langevin equation Eq. (37) and uses the equivalence of the Kramers theory to the multi-dimensional TST. It has been established by numerical comparison that there is agreement between the two approaches. 

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. 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

The rate theory of Grote and Hynes [149] included the non-Markovian (memory) effects by considering the following generalized Langevin equation (GLE) for the dynamics along the reaction coordinate ... 

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