By means of a stochastic model suggested in this paper for the systems with local non-equilibrium excited thermal fluctuations, the famous Shannon entropy is extended to include the heat conduction processes controlled externally by boundary constraints of constant temperature gradients at two sides.Meanwhile,using the description of master equation for the continuous Markov processes a balance equation of stochastic entropy production valid for one dimension gaseous heat conduction systems with high values of Prandtl number has been also established.Based on it,a general expression for both the stochastic entropy production and the entropy production of fluctuations have been further deduced by theΩ-expansions.In this formalism,all kinds of stochastic contributions to the dissipation from the non-equilibrium thermal fluctuation and internal noise turn explicit.
