Isothermal Molecular Dynamics in Classical and Quantum Mechanics

Abstract

The typical problem in statistical physics is the determination of ensemble averages given in terms of phase-space integrals (classical case) or traces of the density matrix (quantum case). Especially for strongly correlated, i.e. interacting many-body systems, the direct evaluation of these averages is usually impossible. Therefore, a large number of different approaches has been developed. In this contribution, we will focus on finite temperature or canonical ensemble properties.

A specific approach due to Nosé is based on classical molecular dynamics (MD) and time averages [1]. It permits one to calculate canonical ensemble averages by averaging over an ergodic deterministic isothermal time evolution. The key idea is to add an additional degree of freedom to the original system in order to mimic the influence of the heat bath on the dynamics. Nowadays, the most reliable (ergodic) techniques are the so-called Nosé-Hoover chains and the demon method.

Static finite temperature properties of quantum systems have been determined very successfully with path integral Monte Carlo (MC) techniques in the past, e.g., for liquid ⁴He [2]. A drawback of these methods is the fact that MC dynamics is highly artificial and non-physical. Therefore, the calculation of dynamical quantities is inhibited. Surprisingly, the calculation of path integrals using classical isothermal MD methods is also possible on the basis of the “classical isomorphism” due to Feynman [3], but this method does not shed light on the real-time quantum dynamics of the system either.

Hence, a more direct isothermal quantum molecular dynamics scheme is highly desirable in order to realistically describe the dynamics of a quantum system at finite temperature. As a first step, we are able to generalize the method of Nosé and Hoover to a genuine quantum system, the ubiquitous harmonic oscillator. The method is valid both for a single particle and for ideal Fermi and Bose gases. We discuss possibilities of further generalization.

In our contribution we review the various classical thermostat methods and sketch fields of applicability. We also discuss indirect (path integrals) and direct (our own work) applications of thermostat methods in quantum mechanical many-body problems.