Langevin equation, generalized

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

Zwanzig R 1973 Nonlinear generalized langevin equations J. Stat. Phys. 9 215-20... 

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... 

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),... 

Second, the classical dynamics of this model is governed by the generalized Langevin equation of motion in the adiabatic barrier [Zwanzig 1973 Hanggi et al. 1990 Schmid 1983],... 

This conclusion can be confirmed by an alternative derivation of Eq. (1.71). According to Mori [52], Eq. (1.71) may be obtained from a generalized Langevin equation ... 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

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),... 

The treatment of the solute-solvent system with the classical Generalized Langevin equation formalism [127], with especial attention to the present problem, has been examined by us [6] a wealth of information can be found in references [128-131],... 

In order to proceed now to a statistical mechanical description of the corresponding relaxation process, it is convenient to solve the equation of motion for the creation and destruction operators and cast them in a form ressembling a Generalized Langevin equation. We will only sketch the procedure. 

The set of equations (50) can be formally considered as generalized Langevin equations if the operator Fj(t) can be interpreted as a stochastic quantity in the statistical mechanical sense. If the memory function does not correlate different solute modes, namely, if Kjj =Sjj Kj, then a Langevin-type equation follows for each mode ... 

Cortes, E., West, B. J. and Lindenberg, K. On the generalized Langevin equation classical and quantum mechanical, J.Chem.Phys., 82 (1985), 2708-2717... 

Adelman, S. A. Generalized Langevin equations and many-body problems in chemical physics,... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

In Kramers classical one dimensional model, a particle (with mass m) is subjected to a potential force, a frictional force and a related random force. The classical equation of motion of the particle is the Generalized Langevin Equation (GLE) ... 

I. Generalized Langevin equation. Zwanzig s Hamiltonian, n. Evaluation of quantum rates for multi-dimensional systems, ni. Beyond the Langevin equation/quantum Kramers paradigm ... 

An alternative view of the same physical process is to model the interaction of the reaction coordinate with the environment as a stochastic process through the generalized Langevin equation (GEE)... 

Keywords stochastic dynamics, generalized Langevin equation, nonstationary and colored friction... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Use of this v (t) as the friction (t) in the generalized Langevin equation provides a complete specification of a nonlocal stationary stochastic dynamics with the exponential friction jo. 

The irreversible Generalized Langevin Equation (iGLE) described in Sec. II. is capable of modeling some of the nonstationary folding dynamics motivated in this section.Such an application is the subject of present work, but it has been mentioned here in order to further motivate the reader to assess the ubiquity of nonstationary phenomenon in physical problems. 

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