Einstein conductivity¶

An electrolyte conducts because ions move. So can you get the conductivity by measuring how fast the ions diffuse, using the MSD machinery you already have?

Almost — and the gap between "almost" and "exactly" is the physics of this page.

Why self-diffusion is not enough¶

Conductivity is a collective property. Current flows only when positive and negative charge move in opposite directions. If a cation and an anion drift along together as a neutral pair, both have a perfectly respectable self-diffusion coefficient and together they carry no current at all.

Estimating conductivity from self-diffusion alone gives the Nernst–Einstein result

\[ \sigma_\mathrm{NE} = \frac{e^{2}}{V k_B T}\sum_\alpha N_\alpha z_\alpha^{2} D_\alpha , \]

which assumes every ion moves independently of every other. Real electrolytes — concentrated ones and ionic liquids especially — have correlated motion, and the true conductivity usually lands 20–70 % below \(\sigma_\mathrm{NE}\). The ratio \(\sigma/\sigma_\mathrm{NE}\) is the ionicity, or degree of uncorrelated ion motion, and measuring it is often the entire point of the calculation.

To get \(\sigma\) itself you need a quantity that already contains the cancellation. That quantity is the collective charge displacement

\[ \mathbf{M}(t) = \sum_a q_a \mathbf{r}_a(t), \]

sometimes called the Helfand moment or the itinerant dipole. A cation–anion pair moving together contributes nothing to \(\mathbf{M}\): their charges are equal and opposite, so the pair cancels itself out, exactly as it should.

The Einstein relation, applied to charge¶

Now run the MSD argument again with \(\mathbf{M}\) in place of a particle position:

\[ \mathrm{MSD}_M(\tau) = \big\langle |\mathbf{M}(t+\tau)-\mathbf{M}(t)|^{2}\big\rangle_{t}, \]

\[ \boxed{\;\sigma = \lim_{\tau\to\infty}\frac{1}{6\,V k_B T}\, \frac{\mathrm{d}\,\mathrm{MSD}_M(\tau)}{\mathrm{d}\tau}\;} \]

Same shape as \(D = \lim \mathrm{MSD}/6\tau\), same factor of 6 from three dimensions, same requirement to find a genuinely linear region before fitting. Everything on the MSD page transfers — including that \(\mathbf{M}\) must be built from unwrapped coordinates. One ion teleporting across the box injects a spurious jump of \(qL\) into the collective sum.

Computing it¶

MolPy will not scan a trajectory for you here, because only you know which atoms carry which charge. Assemble \(\mathbf{M}(t)\) yourself, shape (n_frames, 3), then hand it over:

import numpy as np
from molpy.compute import EinsteinConductivity, LinearFit

rng = np.random.default_rng(1)
M = np.ascontiguousarray(np.cumsum(rng.normal(0, 0.02, size=(400, 3)), axis=0))

raw = EinsteinConductivity().compute(M, dt=10.0, max_correlation_time=100)
print(sorted(raw), raw["msd"].shape)     # -> ['lag_times', 'msd'] (101,)

In real work M comes from your own charges and unwrapped positions:

# docs: skip — needs your own trajectory and charge array
M = np.array([
    (charges[:, None] * unwrapped[i]).sum(axis=0)
    for i in range(len(unwrapped))
])

The compute returns the raw curve and stops. Fit the linear region yourself — the window is a fraction of the curve, so (0.1, 0.5) means from 10 % to 50 % of the way along it:

fit = LinearFit(0.1, 0.5).fit(raw["lag_times"], raw["msd"])
print(round(fit["r2"], 3))                # -> 0.989

Check r2 before believing the slope. This synthetic \(\mathbf{M}\) is a pure random walk, so it is linear by construction and scores 0.989 — the shortfall from 1 is finite-sample noise, not curvature. A real \(\mathrm{MSD}_M\) with a caging shoulder scores much worse if you include the shoulder in the window, which is precisely the warning you want.

Getting to S/m¶

fit["slope"] is in \(e^2\,\text{Å}^2\,\text{fs}^{-1}\) when charges are in elementary units, positions in Å, and dt in fs. Converting to SI is one constant:

\[ \sigma\ [\mathrm{S\,m^{-1}}] = 3.0988\times10^{9}\; \frac{\text{slope}\ [e^{2}\,\text{Å}^{2}\,\text{fs}^{-1}]} {V\ [\text{Å}^{3}]\; \times\; T\ [\mathrm{K}]} \]

def conductivity_si(slope, volume_a3, temperature_k):
    """Einstein conductivity in S/m from a fitted MSD_M slope."""
    return 3.0988e9 * slope / (volume_a3 * temperature_k)

Resist applying that to the fit above. The M in this example is a random walk with an arbitrary step size, not a physical charge displacement, so it would produce an arbitrary number that happens to look like a plausible conductivity. The helper is right; the input is a placeholder.

That constant is \(e^2\,\text{Å}^2 / (6\,\text{fs}\cdot\text{Å}^3 k_B)\) evaluated in SI. The factor of 6 from the Einstein relation is already inside it — do not divide by 6 again.

No figure on this page yet — TODO

A meaningful \(\mathrm{MSD}_M\) figure needs an electrolyte trajectory with charges: a molten salt, or a solvated salt with Ewald electrostatics. The reference system behind the other compute pages is neutral argon, for which \(\mathbf{M} \equiv 0\). Plotting a random walk here and labelling it "conductivity" would be decoration, not measurement. This page gets a figure when a charged reference trajectory exists under scripts/docs_data/.

When it goes wrong¶

\(\mathrm{MSD}_M\) has huge discrete jumps. Wrapped coordinates. One ion crossing the box adds \(qL\) to \(\mathbf{M}\) in a single step.

\(\sigma\) comes out far too large. Check that you summed signed \(q_a\) rather than \(|q_a|\). Check also that the system is neutral: unless \(\sum_a q_a = 0\), the value of \(\mathbf{M}\) depends on where you put the origin, and so does your answer.

\(\mathrm{MSD}_M\) is far noisier than a particle MSD. Expected, and it is the main practical difficulty. \(\mathbf{M}\) is one collective vector per frame, so each time origin gives one sample instead of \(N\). Collective transport needs much longer trajectories than self-diffusion — often tens of nanoseconds where \(D\) would be converged in one.

\(\sigma\) changes a lot with the fitting window. There is no linear region yet. Plot \(\mathrm{MSD}_M/\tau\) and look for a plateau before quoting anything.

\(\sigma \approx \sigma_\mathrm{NE}\) exactly. Suspicious for a concentrated electrolyte. Check that \(\mathbf{M}\) is a genuine collective sum and not a per-ion quantity averaged afterwards.

Check yourself¶

Compute \(\sum_a q_a\) for your system. If it is not zero to machine precision, stop — \(\mathbf{M}\) is not well defined and neither is your conductivity.
Build \(\mathbf{M}\) for a frozen configuration repeated many times. The \(\mathrm{MSD}_M\) must be identically zero.
Compare your \(\sigma\) with \(\sigma_\mathrm{NE}\) from ion self-diffusion. A ratio above 1 is a red flag; 0.3–0.8 is typical.

References¶

E. Helfand, Phys. Rev. 119, 1 (1960) — the moment whose mean squared displacement yields a transport coefficient.
H. K. Kashyap et al., J. Phys. Chem. B 115, 13212 (2011) — collective versus self transport in ionic liquids.
N. Molinari, J. P. Mailoa, B. Kozinsky, Chem. Mater. 31, 8748 (2019) — Nernst–Einstein deviations in concentrated electrolytes.