Dynamical stability of the one-dimensional rigid Brownian rotator: the role of the rotator’s spatial size and shape

Abstract

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 of the weakly nonharmonic propeller-shaped planar Brownian rotator

“…Figure 1 in Ref. [19]. This presents a limitation of our considerations: for the mutually dependent blades and/or mutually dependent local environments, the linear dependence may be expected to be lost.…”

“…in the scattering-model of interaction of the blades with the environmental molecules, the total strength of interaction with the environment can be modeled as a sum of the strengths of interaction for the individual blades. Thus linear dependence on the number N of the blades follows for both the moment of inertia as well as for the total damping factor [19]; cf. Figure 1 in Ref.…”

“…In order to compensate for this lack of the general case, we separately investigate the cases for the different combinations of the values for both ω and b. To this end, it is worth emphasizing that placing b = 0 for the calculations presented in Section V returns the results obtained for the pure harmonic model [19]. Thus, in order to facilitate the calculations, we formally consider the parameters ω and b as constants, whose values are independently varied.…”

“…For every N, 1.5ϕ (N ) max lies on the horizontal axis, while 1.6ϕ (N ) max lies below the horizontal stand for the moment of inertia and the damping factor that regard the molecules of the different chemical species and geometry-the only model assumption in considering the size of the propeller is the linear scaling, I = NI • and γ = Nγ • , for the molecule moment of inertia and the damping factor, respectively [19].…”

“…Dynamical stability has been investigated for the free rotator and rotator in the external harmonic potential [14]. The absence of simple rules or recipes for utilizing the rotator stability is acknowledged [19].…”