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Theory of Cold Atoms: Basics of Quantum Statistics

The aim of this Tutorial is to present the basic mathematical techniques required for an accurate description of cold trapped atoms, both Bose and Fermi. The term {\it cold} implies that considered temperatures are low, such that quantum theory is necessary, even if temperatures are finite. And the term {\it atoms} means that the considered particles are structureless, being defined by their masses and mutual interactions. Atoms are {\it trapped} in the sense that they form a finite quantum system, though their number can be very large allowing for the use of the methods of statistical mechanics. This Tutorial is the first part of several tutorials, giving general mathematical techniques for both types of particle statistics. The following tutorials will be devoted separately to Bose atoms and Fermi atoms. The necessity of carefully explaining basic techniques is important for avoiding numerous misconceptions often propagating in literature.