Langevin equation friction forces

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

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 interpretation of the Langevin equation presents conceptual difficulties that are not present in the Ito and Stratonovich interpretation. These difficulties are the result of the fact that the probability distribution for the random force rip(f) cannot be fully specihed a priori when the diffusivity and friction tensors are functions of the system coordinates. The resulting dependence of the statistical properties of the random forces on the system s trajectories is not present in the Ito and Stratonovich interpretations, in which the randomness is generated by standard Wiener processes Wm(f) whose complete probability distribution is known a priori. 

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) ... 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

In the adiabatic regime, the solvent relaxation time rc reaction coordinate. This limit corresponds to (t) = 5(t), so the power spectrum (Eq. (11.87)) is equal to , that is, to white noise . The GLE is reduced to the simple Langevin equation with a time-local friction force — x. Xr is found from Eq. (11.85) ... 

For 8=1, the noise spectral density is a constant (white noise), at least in the angular frequency range co oo, the Langevin force F(t) is delta-correlated, and the Langevin equation is nonretarded. The white noise case corresponds to Ohmic friction. The cases 0 < 8 < 1 and 8 > 1 are known respectively as the sub-Ohmic and super-Ohmic models. Here we will assume that 0 < 8 < 2, for reasons to be developed below [28,49-51]. 

This is an equation written, for example, for a Brownian particle of mass m moving along the x axis under the influence of a potential V x). Here (t) is a white-noise driving force (a stochastic variable) coming from the Brownian movement of the surroundings, and fx(l) is the systematic friction force. This equation can be solved exactly only in the known special cases V=0 and V=yx, where y is a coeflScient independent of time. Equation (3) is the Langevin equation equivalent to the Kramers equation ... 

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,... 

The important point is that the form of the Langevin equation in Eq. (2.5) is preserved under any canonical point transformation. For the case of Markovian random forces (constant friction),... 

---