On the stochastic mechanics of the free relativistic particle

Given a positive energy solution of the Klein-Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with non-constant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation $(d\tau)^2=-\frac{1}{c^2}dX_{\nu}dX_{\nu}$. A random time-change transformation provides the bridge between the $t$ and the $\tau$ domain. In the $\tau$ domain, we obtain an \M^4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein-Gordon solution is an invariant, non integrable density for this Markov process. It satisfies a relativistically covariant continuity equation

Dissipation caused by a vorticity field and generation of singularities in Madelung fluid

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Diversidade de espécies e densidade de ninhos de abelhas sociais sem ferrão (Hymenoptera, Apidae, Meliponinae) em floresta de terra firme na Amazônia Central.

Stingless bees were collected between 1984 and 1990 in continuous forest, forest fragments and cleared areas 90 Km north of Manaus, Amazonas, Brazil. Several methods were employed. a total of 54 species of 21 genera were collected including two undescribed species of Plebeia Schwarz, 1938. The most abundant genera were Trigona Jurine, 1807; Melipona Illiger, 1806; Partamona Schwarz, 1939 and Tetragona Lepeletier, 1825. The most abundant species were Trigona crassipes (Fabricius, 1793) and T. fulviventris Guérin, 1835. Fruit fly traps baited with fragrances for euglossine bees showed to be an useful method for stingless bee collection too. The study area showed a great richness in relation to other regions of the world. However the density of nests found from a 100ha area of continuous forest were low (1 nest/6.67ha). The consequences of deforestation on stingless bees populations and of the decrease of these on the forest conservation are also discussed

Theory of the Relativistic Brownian Motion. The (1+1)-Dimensional Case

We construct a theory for the 1+1-dimensional Brownian motion in a viscous medium, which is (i) consistent with Einstein's theory of special relativity, and (ii) reduces to the standard Brownian motion in the Newtonian limit case. In the first part of this work the classical Langevin equations of motion, governing the nonrelativistic dynamics of a free Brownian particle in the presence of a heat bath (white noise), are generalized in the framework of special relativity. Subsequently, the corresponding relativistic Langevin equations are discussed in the context of the generalized Ito (pre-point discretization rule) vs. the Stratonovich (mid-point discretization rule) dilemma: It is found that the relativistic Langevin equation in the Haenggi-Klimontovich interpretation (with the post-point discretization rule) is the only one that yields agreement with the relativistic Maxwell distribution. Numerical results for the relativistic Langevin equation of a free Brownian particle are presented.Comment: see cond-mat/0607082 for an improved theor