Bernstein Polynomials: A Bridge Between the Weierstrass Theorem and a Landau Temperature Fluctuation Problem

We address a Landau temperature fluctuation problem by applying Bernstein polynomials and the Weierstrass approximation theorem.We consider the problem detailed in [X.Wang and Q.H. Liu, Temperature fluctuations for a finite system of classical spin-1/2 particles, Annals of Physics 322 (2007), 2168-2178] in which temperature fluctuations diverge as the temperature of a particle system tends to 0. An application of Bernstein polynomials allows us to solve this issue by proving that the m'th moment of the configurational temperature of a finite particle system tends to the temperature of the surroundings, implying the fluctuations in the temperature must tend to 0.