Frequency-Dependent Diffusion Constant of Quantum Fluids from Path Integral Monte Carlo and Tikhonov’s Regularizing Functional

Abstract

We present a novel implementation of the analytic continuation of the velocity autocorrelation function method that has been developed to study the transport properties of quantum liquids at finite temperatures. To invert the ill-posed linear Fredholm integral equation of the first kind, we combine Tikhonov’s first-order regularizing functional with several methods used for automatic selection of the regularization parameter. Taking into account our results, we recommend two methods for automatic selection of the regularization parameter, namely: L-curve and quasi-optimality criterion. We found that the frequency-dependent diffusion power spectrum of normal liquid 4He at T ) 4 K and F ) 0.01873 Å-3 (F ) 31.1 mmol cm-3) is characterized by a single asymmetric peak. The predicted self-diffusion coefficient of 4He at this state point of 0.57-0.58 Å2/ps is in excellent agreement with previous works. We demonstrate that, within proposed mathematical treatment of the quantum transport at finite temperatures, the entire real-time frequency-dependent diffusion power spectrum of liquid normal 4He, can be successfully reconstructed from the limited number of Trotter slices and without the knowledgeof covariance matrix. Moreover, the small values of regularization parameters (i.e., order of 10-7) indicate that the information about quantum dynamics of normal liquid 4He can be easily withdrawn from the high quality imaginary-time correlation function collected in the standard path integral Monte Carlo simulation.