MSD of the Prony-GLE model.
prony_msd(t, lambda, nu = 1, ...)Vector of timepoints at which to calculate the MSD.
Vector of length K giving the inverse decorrelation times of the exponential decay terms.
Autocorrelation scaling factor.
Additional arguments to prony_coeff().
Vector of mean square displacements at the timepoints in t.
The Prony-GLE model satisfies the integro-differential equation $$ F_t - \int_{-\infty}^t \kappa(t-s) \dot X_s ds = 0, \qquad ACF_F(t) = k_B T \kappa(t), $$ where \(T\) is temperature, \(k_B\) is Boltzmann's constant, and the memory kernel is a sum of exponentials: $$ \kappa(t) = \nu sum_{k=1}^K exp(-\lambda_k t). $$ The solution process is of the form $$ X_t = C_0 B_t + \sum_{i=1}^{K-1} C_i W_{it}, $$ where \(B_t\) is Brownian motion and \(d W_{it} = -\rho_i W_{it} dt + d B_{it}\) are Ornstein-Uhlenbeck processes all independent of each other and of \(B_t\).