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International Journal of Modern Physics B

Solving the Liouville equation to probe relaxation in strongly nonlinear systems

Tyler J Barrett, Surajit Sen

Published Mar 20, 2022• 0 citations
Publisher

Abstract

An established formal way to solve the Liouville equation for dynamical systems with hermitian Hamiltonians is to frame the problem via parametrization in a canonical ensemble. The goal is to solve for the dynamics in the canonical ensemble. This approach allows one to examine temperature-dependent relaxation functions as a part of the construction of the full solution to the Liouville equation. Of course, a full solution to the Liouville equation may not always be necessary as it is the relaxation function which often tends to be of interest. The formalism involves evaluation of infinite continued fractions and this can at times be an especially challenging problem. Here we explore the relaxation function in near-equilibrium systems in the presence of strong nonlinearity. Due to computational challenges, we consider small systems. However, even our small system considerations may be adequate to infer about the relaxation in large systems. We show that the nonlinear systems tend to lead to nonconvergent infinite continued fractions. These continued fractions cannot be approximated perturbatively. We close by presenting a new strategy to evaluate them, thereby demonstrating a successful way to examine relaxation processes in strongly nonlinear systems.

DOI
10.1142/s021797922240077x

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An established formal way to solve the Liouville equation for dynamical systems with hermitian Hamiltonians is to frame the problem via parametrization in a canonical ensemble. The goal is to solve for the dynamics in the canonical ensemble. This approach allows one to examine temperature-dependent relaxation functions as a part of the construction of the full solution to the Liouville equation. Of course, a full solution to the Liouville equation may not always be necessary as it is the relaxation function which often tends to be of interest. The formalism involves evaluation of infinite continued fractions and this can at times be an especially challenging problem. Here we explore the relaxation function in near-equilibrium systems in the presence of strong nonlinearity. Due to computational challenges, we consider small systems. However, even our small system considerations may be adequate to infer about the relaxation in large systems. We show that the nonlinear systems tend to lead to nonconvergent infinite continued fractions. These continued fractions cannot be approximated perturbatively. We close by presenting a new strategy to evaluate them, thereby demonstrating a successful way to examine relaxation processes in strongly nonlinear systems.
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The quantum-classical Liouville equation (QCL) describes most atoms of the molecular system under consideration by means of a classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the \\classical part" of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a rst order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of non-adia...
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Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.
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The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented.