9. Structure, diffusion and time correlations
Position in the course: Lesson 9 of 11. Complete the preceding derivation and use the explained exercises to check understanding.
1. Define an observable before analyzing a trajectory
A radial distribution function measures static pair structure. Mean-square displacement measures motion over a lag. Their convergence requirements differ: a liquid can show plausible short-range order while rare hops remain poorly sampled. Assume an isotropic homogeneous 3D fluid for the scalar formulas below. Interfaces, anisotropic crystals and mixtures require component-resolved definitions.
2. Normalize a radial distribution
For species \(A\) and \(B\) with bulk number density \(\rho_B\), the expected number of \(B\) particles in a spherical shell around one \(A\) is
A histogram count must be divided by the number of reference particles, frames, target density and shell volume. For finite bins, use \(4\pi(r_{out}^3-r_{in}^3)/3\), not an arbitrary constant. Exclude a particle paired with itself. Finite-\(N\) same-species normalization may use \((N-1)/V\); record the convention. In a homogeneous ideal gas, \(g\) approaches one under consistent normalization. A first minimum can define a coordination shell, but this convention is not a unique chemical bond assignment.
3. Derive Einstein diffusion
For unwrapped positions, \(\Delta r(t)=\int_0^t v(s)ds\). In a stationary isotropic system with zero mean drift,
The factor two follows by combining the two halves of the square double integral. Differentiate and take a long-time limit where the velocity correlation is integrable:
At short times, \(v\) has not decorrelated, so MSD is ballistic, \(\langle v^2\rangle t^2\). At intermediate times particles may be caged. A linear fit is justified only in a resolved diffusive regime.
4. A worked slope and units
Suppose an analytic teaching line has MSD slope \(0.12\ \mathrm{nm^2/ps}\) in 3D. Then \(D=0.02\ \mathrm{nm^2/ps}=2\times10^{-8}\ \mathrm{m^2/s}\) because \(1\ \mathrm{nm^2/ps}=10^{-6}\ \mathrm{m^2/s}\). In 2D the denominator would be four, not six. This is unit arithmetic, not a reported trajectory result. A finite intercept should not be forced to zero when fitting a long-time region that follows a ballistic transient.
5. Reliable analysis and limitations
Overlapping time origins improve use of data but are correlated. At long lag, few origins remain; error grows even if the plotted curve appears smooth. Report fit intervals, replica variation and finite-size tests. Remove imposed streaming flow from self-diffusion only according to the physical question. Thermostats can alter dynamical correlations. Periodic hydrodynamic finite-size effects can bias diffusion even when local structure is converged.
For viscosity and conductivity, Green–Kubo formulas involve stress or current correlations and often long noisy tails. A visually decayed correlation is not proof of a stable integral plateau. Distinguish self, collective and tracer diffusion; they are not interchangeable in interacting mixtures.
6. Exercises
Why is wrapped-coordinate MSD bounded in a fixed box? Wrapped coordinates lie in a finite interval; their differences cannot represent accumulated crossings. The physical displacement requires unwrapping.
If \(g(r)=1\), what is the coordination number inside \(r_*\)? Integrating gives \(4\pi\rho r_*^3/3\), the density times sphere volume. Peaks alter this ideal-gas baseline rather than removing its geometric factor.
7. References
Original analytic teaching illustration under the stated assumptions; no simulation data.
8. Related theory and practice
Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso