Deformed density matrix and generalized uncertainty relation in thermodynamics

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. In our opinion the approach proposed may lead to the proofs of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Planck scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Planck's. The associated deformation of a canonical Gibbs distribution is given explicitly.