Continuous phase transition of the de Sitter spacetime with charged black holes and cloud of strings and quintessence

Recently,some meaningful results have been obtained by studying the phase transition,critical exponents,and other thermodynamical properties of different black holes.Especially for the Anti-de Sitter（AdS） black holes,their thermodynamical properties nearby the critical point have attracted considerable attention.However,there exists little work on the thermodynamic properties of the de Sitter（dS） spacetime with black holes.In this paper,based on the effective thermodynamical quantities and the method of the Maxwell's equal-area law,we explore the phase equilibrium for the de Sitter spacetime with the charged black holes and the cloud of string and quintessence（i.e.,C-dSSQ spacetime）.The boundaries of the two-phase coexistence region in both Peff-Teff and Teff=S diagrams are obtained.The coexistent curve and the latent heat of phase transition for this system are also investigated.Furthermore,we analyze the effect of parameters（the state parameter ω and the ratio of two horizon radii x=r+/rc） on the two-phase coexistence region boundary.The results indicate that the phase transition in CdSSQ spacetime is analogous to that in a van der Waals fluid（vdw） system,which is determined by the electrical potential at the horizon.These results are helpful for understanding the basic properties of black holes and are also of great value for the establishment of quantum gravity.