Generalized Langevin Equation: An Introductory Review for Biophysicists

Abstract

An introductory, pedagogical review of the generalized Langevin equation (GLE) within the classical regime is presented. It is intended to be accessible to biophysicists with an interest in molecular dynamics (MD). Section 1 presents why the equation may be of interest within biophysical modeling. A detailed elementary first principles derivation of the (multidimensional) Kac–Zwanzig model is presented. The literature is reviewed with a focus on biophysical applications and representation by Markovian stochastic differential equations. The relationship with the Mori–Zwanzig formalism is discussed. The framework of model reduction and approximation is emphasized. Some open problems are identified.

Position-dependent memory kernel in generalized Langevin equations: theory and numerical estimation

“…As a conclusion, in this case, we can say that a 2-FDT holds in the sense that the covariance matrix of the noise in ( 19) is determined by some particular coefficients of the kernel K in (18). In practice, this means that the covariance of the noise can be computed through the Volterra equations (10).…”

“…Given an all-atom simulation (X(t)) t∈[0,T ] (or more generally a set of independent simulations), a practical issue is to parametrize the GLE (2), namely to estimate the coefficients f k and g k (s) in ( 8). This can be done using the Volterra equations (10), when the set E is finite. Indeed, the averages involved in (10) can be estimated along the trajectory, and then the equations can be numerically inverted using a discretization of the time integral [41].…”

“…This can be done using the Volterra equations (10), when the set E is finite. Indeed, the averages involved in (10) can be estimated along the trajectory, and then the equations can be numerically inverted using a discretization of the time integral [41]. In the following, we use the trapezoidal method of Ref.…”

“…A class of models frequently used to perform this task is given by the Generalized Langevin Equations (GLE) [6][7][8][9][10][11][12], which usually take the following form:…”

Likelihood-based non-Markovian models from molecular dynamics

“…However, many cases do not enter the validity range of this approximation, displaying memory effects ( 1 – 8 ). To go beyond the Markovian approximation, a popular class of processes is given by the generalized Langevin equation (GLE) ( 9 – 15 ) where x ( t ) is the value of the d -dimensional CV at time t ; v ( t ) is its time derivative; M is an effective mass, is a mean force, usually deriving from a potential V identified with the free energy; K is a memory kernel; and R ( t ) is a (colored) noise.…”