This paper introduces the use of Markov stochastic process theory to study heliospheric modulation of cosmic rays. The basic cosmic-ray transport equation is reformulated with a set of stochastic di †eren-tial equations that describe the guiding center and momentum of individual charged particles randomly walking in the heliospheric magnetic Ðeld. The Fokker-Planck di †usion equation for the cosmic-ray isotropic distribution function can be derived from these stochastic di †erential equations of basic cosmicray transport. General exact solutions to the initial-boundary value problems of the Fokker-Planck di †usion equation are obtained in terms of time-backward Markov stochastic processes. It is demonstrated with a few examples that modulated cosmic-ray spectra can be calculated with Monte-Carlo simulation of stochastic processes, and the results from the stochastic process simulation are completely consistent with those from calculations that numerically solve the Fokker-Planck di †usion equation. In addition to the capability of modulation spectrum calculation, the stochastic process simulation reveals new information about the behavior of individual particles during their transit through the heliospheric magnetic Ðeld, which includes the trajectory of particles, momentum-loss history, source particle distribution as functions of location and momentum, and distribution of transport time or path length, all of which are normally not available by just solving the Fokker-Planck di †usion equation. This method will allow us to tackle very complicated time-dependent cosmic-ray modulation problems that are impossible by numerically solving the Fokker-Planck di †usion equation. Di †usive particle acceleration and interstellar propagation of Galactic cosmic rays may also be studied with this stochastic process method. Subject headings : cosmic rays È di †usion È interplanetary medium

This paper reported a simple template-free route to prepare hierarchical SnO2 nanostructures with hollow interiors. These high porous architectures, with the diameter about 200 nm, were assembled by nanosheets. Time-dependent experiments illustrated that these hierarchical and hollow nanostructures were transformed from solid SnO2 spheres composed of nanoparticles. Furthermore, the hierarchical SnO2 manifested high capacities and excellent cycle performances as the anode materials for lithium ion batteries, which can deliver a reversible capacity of 545 mA h g−1 up to 50 cycles. Even at high rates, the electrode exhibited excellent properties; the reversible capacities were about 590 and 460 mA h g−1 at the rate of 1.6 and 10 C. The improved electrochemical properties could be ascribed to the large surface area, enhanced structure stability, and short diffusion length for both lithium ion and electron.

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