[Paper Review] How to produce discreet Gaussian sequences: Algorithm and code

Sparisoma Viridi, Veinardi Suendo|arXiv (Cornell University)|Jul 17, 2011

Nuclear reactor physics and engineering3 references3 citations

TL;DR

This paper presents an algorithm and C++ code to generate discreet Gaussian-distributed sequences by discretizing the continuous Gaussian distribution into M groups with representative values, using integer rounding to ensure particle counts. The method achieves high correlation (R² ≈ 1) with N ≥ 1,000,000 and M ≤ 5,000, or N ≥ 1,000 and M = 10, demonstrating accurate discrete approximation of Gaussian statistics for simulations.

ABSTRACT

Algorithm and code to produce sequences whose members obey Gaussian distribution function is reported. Discreet and limited number of groups are defined in the distribution function, where each group is represented only with one value instead of a range of value. The produced sequences are also checked back whether they still fit the discreet distribution function. Increasing of number of particles N increases the value of correlation coefficient R^2, but increasing number of groups M reduces it. Value R^2 = 1 can be found for N = 1000000 at least with M = 5000$ and for M = 10 at least with N = 1000.

Motivation & Objective

• To develop a reproducible method for generating discrete sequences that follow a Gaussian distribution function.
• To address the challenge of simulating large particle systems with continuous distributions using finite, discrete groupings.
• To ensure generated sequences maintain statistical fidelity to the target Gaussian distribution through normalization and error correction.
• To provide a computationally efficient and reproducible algorithm using standard C++ functions for random permutation and integer truncation.
• To evaluate the accuracy of the discrete approximation using the coefficient of determination R² as a metric.

Proposed method

• Discretize the continuous Gaussian distribution f(z) into M equally spaced groups using z_min and z_max where N(z) ≈ 1 - ε.
• Define each group’s representative value as z_i = z_min + (i - 0.5)Δz, ensuring midpoint representation of each interval.
• Compute the number of particles per group as N_i = (N / N') * int[N(z_i)], where N' is the sum of all N_i to preserve total particle count.
• Normalize the final N_i values using a rescaling factor to match the total particle count N, correcting for integer truncation loss.
• Generate the zeroth sequence by listing particles in increasing z_i order, then produce randomized sequences via random shuffling using C++'s std::random_device and std::swap.
• Evaluate accuracy using R² = 1 - (SS_err / SS_tot), comparing discrete N_i to the fitted discrete distribution N_d(z_i).

Experimental results

Research questions

• RQ1 How can a discrete, finite sequence of particles be generated to approximate a continuous Gaussian distribution with high statistical fidelity?
• RQ2 What is the impact of the number of particles N and the number of groups M on the correlation coefficient R² between the discrete sequence and the target Gaussian distribution?
• RQ3 How does integer truncation of particle counts per group affect the total particle count and distribution accuracy?
• RQ4 Can the generated sequences be randomized while preserving the underlying Gaussian distribution and enabling reproducibility via seed control?
• RQ5 What is the optimal balance between N and M to achieve R² ≈ 1 in the discrete approximation?

Key findings

• The correlation coefficient R² increases with larger N, approaching 1 when N ≥ 1,000,000.
• R² decreases with increasing M, indicating reduced accuracy for finer discretization.
• R² = 1 can be achieved with N ≥ 1,000,000 and M ≤ 5,000, or with N ≥ 1,000 and M = 10.
• The discrete distribution requires a rescaled normalization constant N_d = (N * N') / ΣN(z_i) to match the total particle count.
• The zeroth sequence (ordered by group) and randomized sequences (via seed-controlled shuffling) both preserve the target Gaussian distribution.
• The method enables reproducible, high-fidelity discrete Gaussian sequences suitable for molecular dynamics and statistical physics simulations.