径向分布函数g(r)代表了球壳内的平均数密度

为离中心分子距离为r，体积为 的球壳内的瞬时分子数。
具体参见李如生，《平衡和非平衡统计力学》科学出版社：1995

CODE:

这样的代码看着不够明了。。。。。。

伪代码：

for (int i = 0; i < TOTN - 1; ++i)
  for (int j = i + 1; j < TOTN; ++j) {
    double dij = sqrt( pow(Pos[0]-Pos[j][0], 2) + pow(Pos[1]-Pos[j][1], 2) + pow(Pos[2]-Pos[j][2], 2));
    int kbin = func(dij); // dij所对应的bin的序号
    g(kbin) += 2;
  }
  // normalize
  for (int k = 0; k < NBIN; ++k)
    g(k) /= 4.0 * PI * r(k) * r(k) * dr * RHO; // r 为第k个bin所对应的距离值

calculate radial distribution function in molecular dynamics （转载科学网樊哲勇）

Here are the computer codes for this article:

md_rdf.cpp

find_rdf.m

test_rdf.m

         Calculating radial distribution function in molecular dynamics

First I recommend a very good book on molecular dynamics (MD) simulation: the book entitled "Molecular dynamics simulation: Elementary methods" by J. M. Haile. I read this book 7 years ago when I started to learn MD simulation, and recently I enjoyed a second reading of this fantastic book. If a beginner askes me which book he/she should read about MD, I will only recommend this. This is THE BEST introductory book on MD. It tells you what is model, what is simulation, what is MD simulation, and what is the correct attitude for doing MD simulations.

In my last blog article, I have presented a Matlab code for calculating velocity autocorrelation function (VACF) and phonon density of states (PDOS) from saved velocity data. In this article, I will present a Matlab code for calculating the radial distribution function (RDF) from saved position data. The relevant definition and algorithm are nicely presented in Section 6.4 and Appendix A of Haile's book. Here I only present a C code for doing MD simulation and a Matlab code for calculating and plotting the RDF. We aim to reproduce Fig. 6.22 in Haile's book!

Step 1.

Use the C code provided above to do an MD simulation. Note that I have used a different unit systems than that used in Haile's book (he used the LJ unit system). This code only takes 14 seconds to run in my desktop. Here are my position data (there are 100 frames and each frame has 256 atoms):

r.txt

Step 2.

Write a Matlab function which can calculate the RDF for one frame of positions:

function [g] = find_rdf(r, L, pbc, Ng, rc)

% determine some parameters

N = size(r, 1);         % number of particles

L_times_pbc = L .* pbc; % deal with boundary conditions

rho = N / prod(L);      % global particle density

dr = rc / Ng;           % bin size

% accumulate

g = zeros(Ng, 1);

for n1 = 1 : (N - 1)                               % sum over the atoms

for n2 = (n1 + 1) : N                          % skipping half of the pairs

r12 = r(n2, :) - r(n1, :);                  % position difference vector

r12 = r12 - round(r12 ./L ) .* L_times_pbc; % minimum image convention

d12 = sqrt(sum(r12 .* r12));                % distance

if d12 < rc                                 % there is a cutoff

index = ceil(d12 / dr);                % bin index

g(index) = g(index) + 1;                % accumulate

end

end

end

% normalize

for n = 1 : Ng

g(n) = g(n) / N * 2;           % 2 because half of the pairs have been skipped

dV = 4 * pi * (dr * n)^2 * dr; % volume of a spherical shell

g(n) = g(n) / dV;              % now g is the local density

g(n) = g(n) / rho;             % now g is the RDF

end

Step 3.

Write a Matlab script to load the position data, call the function above, and plot the results:

clear; close all;

load r.txt; % length in units of Angstrom

% parameters from MD simulation

N = 256;                % number of particles

L = 5.60 * [4, 4, 4]; % box size

pbc = [1, 1, 1];        % boundary conditions

% number of bins (number of data points in the figure below)

Ng = 100;

% parameters determined automatically

rc = min(L) / 2;     % the maximum radius

dr = rc / Ng;        % bin size

Ns = size(r, 1) / N; % number of frames

% do the calculations

g = zeros(Ng, 1); % The RDF to be calculated

for n = 1 : Ns

r1 = r(((n - 1) * N + 1) : (n * N), :); % positions in one frame

g = g + find_rdf(r1, L, pbc, Ng, rc);    % sum over frames

end

g = g / Ns;                                 % time average in MD

% plot the data

r = (1 : Ng) * dr / 3.405;

figure;

plot(r, g, 'o-');

xlim([0, 3.5]);

ylim([0, 3.5]);

xlabel('r^{\ast}', 'fontsize', 15)

ylabel('g(r)', 'fontsize', 15)

set(gca, 'fontsize', 15);

Here is the figure I obtained:

Does it resemble Fig. 6. 22 in Haile's book?