Abstract: Thermal phenomena are the results of the statistical behavior of microscopic particles. Traditionally, one is accustomed to deriving the laws of thermodynamics from statistical physics. However, the fundamental assumptions on which statistical physics relies, such as the principle of equal a priori probability and the ergodic (or traversing) hypothesis, are difficult to verify directly by experiment. In contrast, thermodynamics is founded on solid experimental facts such as heat engines and thermal cycles. Therefore, it is of great significance to deduce the equilibrium statistical distribution from the fundamental principles of thermodynamics. This paper based on our series of studies on quantum thermodynamics over the past two decades, by defining internal energy through an (undetermined) statistical distribution over quantum states and establishing the concept of heat self-consistently through the mechanical definition of work, we derive the fundamental element of statistical physics—the probability distribution, the inverse temperature β naturally emerges as the integrating factor of heat. The key method is to extend Carathéodory’s 1909 idea of integrability to the microscopic level, introducing the concept of quantum thermodynamic integrability, and to obtain the entropy integrability equation satisfied by the statistical distribution and the temperature. The special solution of this equation under detailed balance is precisely the canonical equilibrium distribution, while more general noncanonical solutions describe finite systems away from the thermodynamic limit, capable of characterizing the conversion of black-hole information into correlations of emitted radiation, thus addressing the information-loss paradox.