# Accurate and robust splitting methods for the generalized Langevin equation with a positive Prony series memory kernel

@inproceedings{Duong2021AccurateAR, title={Accurate and robust splitting methods for the generalized Langevin equation with a positive Prony series memory kernel}, author={Manh Hong Duong and Xiaocheng Shang}, year={2021} }

We study numerical methods for the generalized Langevin equation (GLE) with a positive Prony series memory kernel, in which case the GLE can be written in an extended variable Markovian formalism. We propose a new splitting method that is easy to implement and is able to substantially improve the accuracy and robustness of GLE simulations in a wide range of the parameters. An error analysis is performed in the case of a one-dimensional harmonic oscillator, revealing that all but one averages… Expand

#### One Citation

Non-Markovian systems out of equilibrium: Exact results for two routes of coarse graining

- Physics
- 2021

Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on… Expand

#### References

SHOWING 1-10 OF 57 REFERENCES

Efficient Numerical Algorithms for the Generalized Langevin Equation

- Mathematics, Computer Science
- ArXiv
- 2020

It is demonstrated in numerical experiments that the obtained GLE-based sampling scheme outperforms state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency. Expand

Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel.

- Mathematics, Physics
- The Journal of chemical physics
- 2013

This article derives a family of extended variable integrators for the Generalized Langevin equation with a positive Prony series memory kernel using stability and error analysis and implements the corresponding numerical algorithm in the LAMMPS MD software package. Expand

Trotter derivation of algorithms for Brownian and dissipative particle dynamics.

- Mathematics, Physics
- The Journal of chemical physics
- 2007

Using a method based on the exponentiation of time dependent operators, a numerical scheme for the Langevin dynamics is derived, which is equivalent to the proposal of Ermak and Buckholtz and not simply to the stochastic version of the velocity-Verlet algorithm. Expand

Adaptive Thermostats for Noisy Gradient Systems

- Mathematics, Physics
- SIAM J. Sci. Comput.
- 2016

A new numerical method is proposed for the adaptive Langevin/stochastic gradient Nos\'e--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. Expand

A stochastic Trotter integration scheme for dissipative particle dynamics

- Computer Science, Mathematics
- Math. Comput. Simul.
- 2006

The DPD-Trotter integrator demonstrates the inexistence of spurious spatial correlations in the radial distribution function for an ideal gas equation of state and compares the numerical integrator to other available DPD integration schemes. Expand

Anomalous Diffusion and the Generalized Langevin Equation

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2018

This work establishes a class of memory kernels for which the GLE is well-defined; it investigates the associated regularity properties of solutions; and it proves that large-time asymptotic behavior of the particle MSD is entirely determined by the tail behavior ofThe GLE's memory kernel. Expand

The generalized Langevin equation with power-law memory in a nonlinear potential well

- Mathematics
- 2018

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the velocity of microparticles in viscoelastic fluids. In this work, we consider… Expand

Fractional Kinetics in Kac–Zwanzig Heat Bath Models

- Mathematics
- 2004

We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data.… Expand

Data-driven parameterization of the generalized Langevin equation

- Mathematics, Computer Science
- Proceedings of the National Academy of Sciences
- 2016

A data-driven approach to determine the memory kernel and random noise in generalized Langevin equations, relying on a hierarchy of parameterized rational approximations in terms of the Laplace transform, which can be expanded to arbitrarily high order as necessary. Expand

Asymptotic analysis for the generalized Langevin equation

- Mathematics, Physics
- 2010

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs,… Expand