Linear Algebra of Reduced Units and Discussion of Temperature Parameter

A formal linear vector field representation for scientific equations is developed to rationalize the intuitive methods that are constantly employed. It is shown that unlike mechanical units that appear in the basis of the space, the reduced temperature and Boltzmann parameter cannot be described by the basis set individually and can only be described as a product. Further, the definition and determination of temperature is dependent on theory and not on standard mechanical units. It is shown that there is no reason to reduce the number of degrees of freedom in temperature determination via equipartition since stochastic variables are involved, and this observation is significant in that the temperature variable reported in simulation studies would have a discrepancy to the extent of using the decreased number of freedom, which is most cases is not large nor significant. The standard assignments used in reduced units do not lead to errors because operationally the resulting reduced temperature parameter represents the reduced product of the temperature and Boltzmann parameters. The non-independence of these quantities explains why entropy and other associated functions cannot be calculated directly, but are always scaled in dimensionless increments of the Boltzmann parameter