The expression of ensemble average internal energy in long-range interaction complex system and its statistical physical properties

In this paper, we attempt to derive the expression of ensemble average internal energy in long-range interaction complex system. Further, the Shannon entropy hypothesis is used to derive the probability distribution function of energy. It is worth mentioning that the probability distribution function of energy can be equivalent to the q-Gaussian distribution given by Tsallis based on nonextensive entropy. In order to verify the practical significance of this model, it is applied to the older subject of income system. The classic income distribution is two-stage, the most recognized low-income distribution is the exponential form, and the high-income distribution is the recognized Pareto power law distribution. The probability distribution can explain the entire distribution of United States income data. In addition, the internal energy, entropy and temperature of the United States income system can be calculated, and the economic crisis in the United States in recent years can be presented. It is believed that the model will be further improved and extended to other areas.