RESUMO
We studied the Ehrenfest urn model in which particles in the same urn interact with each other. Depending on the nature of interaction, the system undergoes a first- or second-order phase transition. The relaxation time to the equilibrium state, the Poincaré cycles of the equilibrium state and the most far-from-equilibrium state, and the duration time of the states during first-order phase transition are calculated. It was shown that the scaling behavior the Poincaré cycles could serve as an indication to the nature of phase transition, and the behavior of the ratio of duration time of the states could be strong evidence of the metastability during first-order phase transition.
RESUMO
Paget's disease in the Far East is quite rare. Age at diagnosis is usually greater than 50 years, and the disease typically affects the spine. Most patients are usually diagnosed with Paget's disease following radiographic examination for other purposes. It usually occurs at multiple vertebral levels, with only 10-25% of vertebral Paget's disease being monostotic. The disease rarely causes neurologic complications resulting from compression of intraspinal nerve tissue. Here, we present 2 cases of monostotic vertebral Paget's disease of the third lumbar vertebra. The first patient, who may be the first documented case of Paget's disease in the lumbar spine with progressive neurologic deficiency in an Asian population, received decompressive laminectomy due to marked spinal stenosis with neurologic deficits. The symptoms were greatly relieved following surgery, and ambulatory ability was restored. The second patient was diagnosed with Paget's disease following surgical biopsy. He remained asymptomatic at the most recent follow-up.
