Schottky Anomaly of five-Dimensional de Sitter Hairy Spacetime based on effective thermodynamic quantities

Abstract

Taking the mass, charge, hair parameter, and cosmological constant of the 5-dimensional de Sitter hairy spacetime as the state variables of a thermodynamic system, and based on the satisfaction of the universal first law of thermodynamics, we obtain the effective thermodynamic quantities of the spacetime. The thermodynamic properties of the system in the coexistence region of the black hole and cosmological horizons are discussed. We find that under certain conditions, the heat capacity of the effective thermodynamic system in the two-horizon coexistence region of de Sitter hairy spacetime, as a function of either temperature or the ratio of the horizon positions, exhibits a peak-like behavior similar to that observed in paramagnetic systems. Further analysis reveals that, under specific conditions, the heat capacity of this effective thermodynamic system in dS spacetime resembles that of a two-level system composed of two horizons with different radiation temperatures. By comparing these results, we derive the expressions for the number of microscopic particles on the two horizons within the coexistence region. This outcome reflects the quantum nature of dS spacetime and provides a new pathway for further in-depth investigation into the thermodynamic and quantum properties of the two-horizon coexistence region in dS spacetime.