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PMSD

Textbook guide to the Einstein (charge-dipole MSD) route to ionic conductivity — collective polarization displacement of charge carriers.

Conventions

  • Build \(\mathbf{M}(t)=\sum_a q_a\mathbf{r}_a\) from unwrapped positions.
  • Time fs; compose raw curve → LinearFit → SI scale in your script.

1. Einstein conductivity

\[ \mathrm{MSD}_M(\tau) = \big\langle|\mathbf{M}(t+\tau)-\mathbf{M}(t)|^2\big\rangle_t, \qquad \sigma = \lim_{\tau\to\infty} \frac{1}{6\,V k_B T}\frac{\mathrm{d}{\mathrm{d}\tau}\mathrm{MSD}_M(\tau). \]

There is no trajectory-scanning recipe class: assemble \(\mathbf{M}(t)\) yourself, then EinsteinConductivity → LinearFit → SI prefactor.

Equivalent Green–Kubo current route: JACF. Frequency-dependent response: Dielectric.

2. Usage

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=(60, 3)), axis=0))
raw = EinsteinConductivity().compute(M, dt=10.0, max_correlation_time=20)
fit = LinearFit(start_frac=0.1, end_frac=0.5).fit(raw["lag_times"], raw["msd"])
slope = fit["slope"]  # then sigma = slope / (6 V k_B T) * SI_prefactor

3. Pitfalls

  1. Wrapped ion coordinates → nonsense \(\mathbf{M}\).
  2. Quoting \(\sigma\) at the last lag instead of the linear window.
  3. Mixing solvent total dipole into \(\mathbf{M}\) for electrolytes — see Dielectric decomposition.

See also