Langevin equation memory term

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

Except for the term —mx(ti)y(t — f,), which we will discuss later, Eq. (6) is similar to a generalized Langevin equation, in which y(t) acts as a memory kernel and F(t) acts as a random force. By using, instead of y(t), the retarded memory kernel2 y(t) = 0(f)y(t), the upper integration bound of the integral in the left-hand side of Eq. (6) can be set equal to +oo. This latter equation can then be rewritten in the following equivalent form ... 

When the system dynamics depends on what occurred earlier—that is, the environment has memory—Eq. (141) is no longer adequate and the Langevin equation must be modified. The generalized Langevin equation takes this memory into account through an integral term of the form... 

Tsekov and Ruckenstein considered the dynamics of a mechanical subsystem interacting with crystalline and amorphous solids [39, 40]. Newton s equations of motion were transformed into a set of generalized Langevin equations governing the stochastic evolution of the atomic co-ordinates of the subsystem. They found an explicit expression for the memory function accounting for both the static subsystem—solid interaction and the dynamics of the thermal vibrations of the solid atoms. In the particular case of a subsystem consisting of a single particle, an expression for the fiiction tensor was derived in terms of the static interaction potential and Debye cut-off fi equency of the solid. 

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