Error Analysis of Generalized Langevin Dynamics

This repository contains MATLAB code to reproduce the numerical experiments in

Quanjun Lang and Jianfeng Lu, Error Analysis of Generalized Langevin Dynamics with Approximated Memory Kernels.

The code implements and tests trajectory-wise error bounds for generalized Langevin equations (GLEs) with perturbed or approximated memory kernels. It covers both first-order (velocity-only) and second-order (position–velocity) stochastic Volterra equations with exponential and subexponential kernels, and compares empirical trajectory discrepancies with the theoretical rates derived in the paper.

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Repository structure

Top-level files and folders:

• addpaths.m
  Adds all subfolders of the repository to the MATLAB path.

• system_info.m
  Creates a structure with basic system parameters (dimension, damping, time step, final time, number of trajectories, etc.).

• update_sysInfo.m
  Helper to update or refine sysInfo (time grids, derived parameters).

• new_kernel_info_structure.m
  Helper for defining kernel families used in the numerical experiments.

• Schur_norm.m
  Computes the Schur-type norm of the memory kernels used in the analysis.

Kernels

• Kernels/kernel_info.m
  Defines memory kernel types and their parameters (e.g., exponential, power-law, oscillatory, cut-off, translated/dilated variants).

• Kernels/plot_kernels.m
  Visualization of kernels in different scales (linear, semilog, log–log).

Lyapunov distance

• Lyapunov/compute_Lyapunov_distance.m
  Computes the Lyapunov-type distance between trajectories.

• Lyapunov/Lyapunov_info.m
  Stores parameters for the Lyapunov function and related constants.

Potentials (for second-order GLEs)

• Potentials/potential_info.m
  Defines confining potentials and their gradients.

• Potentials/plot_potential_2d.m
  Plotting utility for potentials in two dimensions.

Trajectory generation and plotting

• Traj/get_noise.m
  Generates shared Brownian noise / random driving terms for trajectories.

• Traj/generate_data_first_order_highdim.m
  Simulator for high-dimensional first-order GLEs with given kernel and system parameters.

• Traj/generate_data_second_order_highdim.m
  Simulator for high-dimensional second-order GLEs (position–velocity formulation).

• Traj/plot_trajectory_decay_fit.m
  Fits and plots the decay of Lyapunov distances or mean-square errors.

• Traj/plot_velocity_highdim.m, Traj/plot_trajectory_highdim.m
  Visualization of sample trajectories and velocities.

Section 5.1 – Differential–integral inequality

Folder: Sec_5_1_diff_int_ineq/

• first_order_loop_exp.m
  Experiment loop for the exponential-kernel case.

• first_order_loop_powerlaw.m
  Experiment loop for the power-law kernel case.

• exp_loop.mat, powerlaw_loop.mat, powerlaw_loop_large.mat
  Saved data for the corresponding experiments.

• power_law_decay_ex.pdf
  Example figure for the power-law decay.

Section 5.2 – First-order GLE with exponential kernels

Folder: Sec_5_2_first_order_GLE/

Representative files:

• main_first_order_exponential.m
  Main script to reproduce the first-order GLE experiment with exponential kernels.

• main_first_order_exp.m
  Alternative entry script (same section, slightly different setup).

• main_first_order_powerlaw_old.m
  Older script for power-law experiments (kept for reference).

• four_est_kernel_loop.m
  Kernel perturbation loop used to compare four types of approximations.

• ensemble_test_S_c2_C2.m
  Ensemble experiments for constants and rates in the error bounds.

• new_kernel_info.m, exp_data_Oct17.mat, README_first_order_GLE.txt (if present)
  Additional configuration and stored data for this section.

Section 5.3 – Second-order GLE

Folder: Sec_5_3_second_order_GLE/

Representative files:

• main_second_order_exp_loop.m
  Main loop for second-order GLE experiments (various kernel perturbations).

• ensemble_test_second_order.m
  Ensemble experiments for second-order systems.

• second_order_exp_data.mat, *_loop.mat, *_data*.mat
  Data files generated or used by the scripts.

Section 5.4 – First-order GLE with power-law kernels

Folder: Sec_5_4_first_order_GLE/

Representative files:

• main_first_order_polwer_law_loop.m
  (Typo in name is intentional, kept for backwards compatibility.) Main loop for power-law kernel experiments in first-order GLEs.

• ensemble_test_first_order.m
  Ensemble tests comparing empirical and theoretical decay rates.

• *_loop.mat, *_data*.mat
  Data files for this section.

Theoretical rates

Folder: Theoretical_rates/

• fit_traj_rate.m
  Fits decay rates from trajectory data (e.g., via log–linear regression in time).

• get_theoretical_rate_exponential.m
  Computes theoretical exponential decay rates predicted by the theory.

• get_theoretical_threshold_powerlaw.m
  Computes theoretical thresholds and power-law exponents from the analytical bounds.

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Requirements

• MATLAB (tested on recent versions such as R2023a/R2024a).
• No nonstandard toolboxes are required beyond base MATLAB.

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Getting started

Clone or download the repository

git clone https://github.com/QuanjunLang/error_analysis_GLE.git
cd error_analysis_GLE

Set up the MATLAB path

In MATLAB, change the working directory to the repository root and run:

addpaths;   % add all subfolders to the MATLAB path

Run a core example

Examples:

% Section 5.1: differential–integral inequality, exponential kernel
cd('Sec_5_1_diff_int_ineq');
first_order_loop_exp;

% Section 5.1: differential–integral inequality, power-law kernel
first_order_loop_powerlaw;

% Section 5.2: first-order GLE with exponential kernels
cd('../Sec_5_2_first_order_GLE');
main_first_order_exponential;

% Section 5.3: second-order GLE
cd('../Sec_5_3_second_order_GLE');
main_second_order_exp_loop;

% Section 5.4: first-order GLE with power-law kernels
cd('../Sec_5_4_first_order_GLE');
main_first_order_polwer_law_loop;

Each script generates trajectories, computes Lyapunov distances or mean-square errors, and produces plots that correspond to figures in the paper. Some scripts load precomputed .mat files; these are included in the corresponding section folders.

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Modifying experiments

• Adjust global system parameters in system_info.m and update_sysInfo.m.
• Change kernel types and perturbations in Kernels/kernel_info.m (and new_kernel_info_structure.m when used).
• For second-order systems, modify potentials in Potentials/potential_info.m.
• Re-run the relevant main_*.m script to regenerate trajectories and plots.

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Reproducing figures and tables

Rough mapping between paper sections and code:

• Section 5.1 (differential–integral inequality):
  Sec_5_1_diff_int_ineq/first_order_loop_exp.m and first_order_loop_powerlaw.m.

• Section 5.2 (first-order GLE with exponential kernels):
  Sec_5_2_first_order_GLE/main_first_order_exponential.m (and main_first_order_exp.m) together with four_est_kernel_loop.m and ensemble_test_S_c2_C2.m.

• Section 5.3 (second-order GLE):
  Sec_5_3_second_order_GLE/main_second_order_exp_loop.m and ensemble_test_second_order.m.

• Section 5.4 (first-order GLE with power-law kernels):
  Sec_5_4_first_order_GLE/main_first_order_polwer_law_loop.m and ensemble_test_first_order.m.

The scripts in Theoretical_rates/ and Traj/plot_trajectory_decay_fit.m are used across sections to fit empirical rates and compare them to the theoretical predictions.

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Citation

If you use this code in your research or teaching, please cite:

Quanjun Lang and Jianfeng Lu, Error Analysis of Generalized Langevin Dynamics with Approximated Memory Kernels.

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License

This project is released under the MIT License. See the LICENSE file for the full text.