RESUMO
An operator formulation of centroid molecular dynamics (CMD) for rotational degrees of freedom is presented. The quasi-density operator concept was introduced by Jang and Voth [J. Chem. Phys 111, 2357 (1999)] and is used to obtain a phase-space mapping without the need for discretized path integrals. The approach allows the calculation of approximate Kubo-transformed time correlation functions. The particle on a ring is chosen as an illustrative example. Numerical results demonstrate that the proposed approach leads to accurate results when compared with exact diagonalization calculations for linear operators. At very low temperatures, it is found that rotational CMD yields results that are in exact agreement with the quantum dynamics of a spin-1 system.
RESUMO
A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature ß limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.
RESUMO
BACKGROUND: Pain sensitivity has been negatively associated with physical activity levels. Few studies have examined associations between experimentally induced pain sensitivity and physical activity in adults with chronic low back pain and pain-free controls. OBJECTIVE: The objective of this study was to examine associations between physical activity levels and how an individual processes pain using experimentally induced pain stimuli. METHODS: Seventy subjects (CLBP = 49; mean age = 46.8 ± 14.9; Pain-free = 21; mean age = 45.3 ± 18.2, n of females = 46) participated. A self-report questionnaire derived from the International Physical Activity Questionnaire (IPAQ) was used to calculate an activity level index. Pain sensitivity was assessed via quantitative sensory testing (QST) at the right lower extremity. RESULTS: Moderate (U = 688, p< 0.05) and vigorous (U = 649, p< 0.05) physical activity levels were higher in pain-free vs. individuals with CLBP. Activity level was not associated with pain sensitivity (Pain-free: R2 = 0.02, p> 0.05; CLBP: R2 = 0.01, p> 0.05). Both moderate (R2 = 0.49, p< 0.05) and vigorous (R2 = 0.68, p< 0.01) physical activity were associated with pain modulation amongst pain-free individuals. CONCLUSIONS: Findings suggest that physical activity influences pain modulation amongst pain-free individuals, however no relationship exists once CLBP is present. However, future investigation will elucidate the extent to which physical activity level either prevents CLBP or is effective in alleviating CLBP.
