Homogeneous functions of degree one and heat phenomena in potential fields

The variational argument is presented to establish the attainability of homogeneity of degree one in the number of particles for any functional $F[n, f]$ that depends on both the state variable $f$ and the particle count $n$. Euler's integration of homogeneous functions applies to any such functional. This argument is employed to examine the heat equilibrium of a system containing an undefined and unconserved number of indistinguishable particles within each cell $h^3$ in the quantized phase-space of particle coordinates and momenta, with $h$ representing the Planck constant, including for the case of the smallest system of a single elementary volume. The system does not exchange particles with a reservoir, and the uncertainty in particle count is intrinsic to the system itself. The system is maintained at a constant temperature $T$ with a chemical potential denoted by $\mu$. The definition of chemical potential is based on variational principles related to homogeneous functions of degree one. The equilibrium particle density is analyzed in the presence of gravitational and electric fields characterized by a central $\frac{1}{r}$ reciprocally decaying potential, where the local density of potential field sources is provided for partition functions defined by the nature of the particles. A star is a point source of gravitation embedded in a rarefied ambient gas, where heat phenomena create a 'dark' illusion of additional mass presence. For an atomic core as a point source of an electric field in an electron gas, the study explores the temperature-dependent potential barrier in the electric field in atoms, where electron states correspond to particle states moving in the potential field with a barrier situated between a well and a valley. The homogeneous functionals for particle density energy and particle energy itself are discussed.