The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if $S \to \infty$ and the inverse temperature $\beta \to 0$ in such a way that $\beta S$ stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.Comment: 11 pages, pdfLate

Variability of nitrogen compoud concentrations in undergraound sources of drinking water

Concentrations of nitrogen compounds ( NO3,NO2 and NH4 ) have been monitored for 21 years, and the data sets obtained within this time span were subject to statistical analysis. Samples were collected in the water-works of Wielkopolska.Analyzed were a number of variables describing the water sources and thier location. Smoothed time profiles were established for the parameters characterizing the pollution leves in the various from which water for drinking purposes is drawn.In the investigated period,the overall pollution load did not undergo noticeable variations.Most of the variability depended on the depth of the water source