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Entropy dynamics for a propeller-shaped quantum Brownian molecular rotator

J. Jeknić-Dugić, Igor Petrović, K. Kojic, M. Arsenijević, M. Dugić

Published Jan 1, 2023• 0 citations
Publisher

Abstract

We investigate and analyze the time dependence of the so-called differential entropy as a measure of the dynamical stability of a one-dimensional, propeller-shaped quantum Brownian molecular rotator. The larger the entropy change, the more profound the instability (lower control) of the rotator. The quantum molecular rotator is modelled by the quantum Caldeira-Leggett master equation while assuming the external harmonic field for the rotator. Rotational stability is found relatively high for the constructed Gaussian states, especially for molecules with a larger number of blades.

DOI
10.46793/iccbi23.082jd

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We investigate and analyze the time dependence of the so-called differential entropy as a measure of the dynamical stability of a one-dimensional, propeller-shaped quantum Brownian molecular rotator. The larger the entropy change, the more profound the instability (lower control) of the rotator. The quantum molecular rotator is modelled by the quantum Caldeira-Leggett master equation while assuming the external harmonic field for the rotator. Rotational stability is found relatively high for the constructed Gaussian states, especially for molecules with a larger number of blades.
We investigate dynamical stability of a single propeller-like shaped molecular cogwheel modelled as the fixed-axis rigid rotator. In the realistic situations, rotation of the finite-size cogwheel is subject to the environmentally-induced Brownian-motion effect that we describe by utilizing the quantum Caldeira-Leggett master equation. Assuming the initially narrow (classical-like) standard deviations for the angle and the angular momentum of the rotator, we investigate the dynamics of the first and second moments depending on the size, i.e. on the number of blades of both the free rotator as well as of the rotator in the external harmonic field. The larger the standard deviations, the less stable (i.e. less predictable) rotation. We detect the absence of the simple and straightforward rules for utilizing the rotator's stability. Instead, a number of the size-related criteria appear whose combinations may provide the optimal rules for the rotator dynamical stability and possibly control. In the realistic situations, the quantum-mechanical corrections, albeit individually small, may effectively prove non-negligible, and also revealing subtlety of the transition from the quantum to the classical dynamics of the rotator. As to the latter, we detect a strong size-dependence of the transition to the classical dynamics beyond the quantum decoherence process.
Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.
Shape and size are the basic characteristics of macroscopic, classical systems, and as such represent a special challenge within the fundamental problem of "transition from quantum to classical", but also the problem of quantum measurement. Propeller-shaped molecules are excellent candidates for analyzing the effects of size and shape because of the linear dependence of both the moment of inertia and the damping factor on the number of propeller blades. Also, such research can be useful for their practical application within molecular machines technology. The obtained results indicate a very complex dependence of dynamic stability on the size of the system, as well as some unexpected characteristics of the time evolution of standard deviations in some cases. The new method called "the quantum first passage time" also led to unexpected conclusions, just like the analysis of the role of decoherence in the transition from quantum to classical behavior.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
In this contribution, we discuss the statistical mechanical implications of the mean field Ehrenfest method of quantum/classical dynamics for a quantum harmonic oscillator in a classical heat bath using the Brownian motion Hamiltonian as a model. A mean field quantum/classical master equation is derived and compared to the corresponding Redfield master equation, and the deficiencies of the quantum/classical approach pointed out by analyzing the nature of energy/population relaxation and decoherence.