[AntonioLao]

At thermal equilibrium, the population (N) for each energy state (E) of an atomic system is inversely proportional to the natural exponential function of energy divided by the product of Boltzmann’s constant (k) and the absolute temperature (T) where constant of proportionality is arbitrarily set to unity: N=1/exp{E/kT}. An equivalent expression is that the product of N and exp{E/kT} is unity: Nexp{E/kT}=1. At different energy states, their populations hold the same unit proportion. Therefore, N(1)exp{E(1)/kT}= N(2)exp{E(2)/kT}= N(3)exp{E(3)/kT}= N(4)exp{E(4)/kT}= N(5)exp{E(5)/kT}= N(6)exp{E(6)/kT}=…= N(∞)exp{E(∞)/kT}. These imply that E(1)< E(2)< E(3)< E(4)< E(5)< E(6)<…< E(∞) while N(1)>N(2)>N(3)>N(4)>N(5)>N(6)>…>N(∞). At both extreme, if E is exactly zero then N is exactly infinite and if E is infinite then N is exactly 1. Since infinite energy is not physically realizable (equivalently, infinite shift of the Hamiltonian cannot be measured), the population can never be exactly 1. At the least, N=2.

The philosophical implication of this dualism is that in any domain of reality there always exist two independent fundamental principles, for examples: field (wave) and particle, yin and yang, mind and brain, form and content, potential and kinetic, passive and active, left and right, top and bottom, front and back, life and death, good and evil. Since these dualistic principles are perfectly and absolutely symmetrical in the infinitesimal domain of space-time, understanding them requires a principle of directional invariance for all local infinitesimal motions, uniform or non-uniform; equilibrium or non-equilibrium. Nevertheless, the minimum population number is always 2.