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Onsager

Textbook guide to Onsager phenomenological coefficients — collective displacement cross-correlations that encode multi-component transport coupling.

1. Collective coordinates and \(\Omega_{\alpha\beta}\)

\[ \mathbf{P}_\alpha(t) = \sum_{i\in\alpha}\mathbf{r}_i(t), \qquad L_{\alpha\beta}(\tau) = \big\langle\Delta\mathbf{P}_\alpha(\tau)\cdot\Delta\mathbf{P}_\beta(\tau)\big\rangle_t, \]
\[ \boxed{ \Omega_{\alpha\beta} = \lim_{\tau\to\infty} \frac{L_{\alpha\beta}(\tau)}{6\,k_B T\,V\,N_A\,\tau} } \]
  • Diagonal: collective MSD of species \(\alpha\).
  • Off-diagonal: cation–anion coupling (often negative under ion pairing).

Conductivity from the Onsager matrix:

\[ \sigma = \frac{e^2}{V k_B T}\sum_{\alpha\beta} z_\alpha z_\beta\,\Omega_{\alpha\beta}. \]

If off-diagonals vanish this is Nernst–Einstein from self-diffusion alone. The ratio \(\sigma/\sigma_\mathrm{NE}\) measures correlation suppression.

Onsager.correlation returns the raw \(L_{\alpha\beta}(\tau)\) only — fit the slope with LinearFit and apply the prefactor yourself.

2. Usage

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

rng = np.random.default_rng(0)
P_cat = np.ascontiguousarray(np.cumsum(rng.normal(0, 0.01, size=(40, 3)), axis=0))
P_an = np.ascontiguousarray(np.cumsum(rng.normal(0, 0.01, size=(40, 3)), axis=0))
L11 = Onsager.correlation(P_cat, P_cat, dt=10.0, max_correlation_time=10)
L12 = Onsager.correlation(P_cat, P_an, dt=10.0, max_correlation_time=10)
fit = LinearFit(0.1, 0.5).fit(L11["lag_times"], L11["correlation"])

3. Pitfalls

  1. Wrapped species COM sums.
  2. Fitting before the linear regime.
  3. Comparing \(\Omega\) across definitions of \(N_A\) / volume units.

See also