The purpose of this monograph is twofold: The one is to present a comprehensive account of the so-called Renormalization-Group (RG) method and its extension, which we named the doublet scheme, in a geometrical point of view with various examples. The second is the application of the doublet scheme in the RG method to the derivation of the so-called second-order (causal) fluid dynamics from the relativistic as well as non-relativistic Boltzmann equations with quantum statistics. All the contents of the monograph are virtually based on the authors original work except for some review parts.
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