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Based on Reissner-Nordstrom-anti-de Sitter(RN-AdS) black hole surrounded by perfect fluid dark matter, we study the thermodynamics and phase transition by extending the phase space defined by the charge square

Black hole thermodynamics are one of the most important topics in modern physics research and have been widely studied in recent years. The laws of black hole dynamics and thermodynamics were analyzed by Bekenstein and Hawking [

From recent observations, we know that our universe is dominated by Dark Energy and Dark Matter [

This paper focuses on the study of the influence of perfect fluid dark matters on thermodynamics and phase transition for Reissner-Nordstrom-anti-de Sitter black hole. In Section

We consider the dark matter field minimally coupled to gravity, electromagnetic field, and cosmological constant [

For the Reissner-Nordstrom-AdS spacetime metric in perfect fluid DM, the black hole solution is as follows [

In Section

Now we study the thermodynamical properties of the black hole with perfect fluid dark matter by extending to new phase space. This phase space is constructed by the entropy

Now we study the critical behaviour near the phase transition with perfect fluid dark matter in new phase space. We first define

From the above analysis, we find that these critical exponents resemble those in Van der Waals system, implying that the critical phenomenon can be explained by mean field theory [

The perfect fluid dark matters around black holes could explain the formation of supermassive black holes in approximate stationary situation. In the past decades, some high redshift quasars have been discovered and the centre supermassive black hole with the mass beyond 10 billion of solar masses. Usually such black hole is difficult to form in the Universe less than one billion years old. Because perfect fluid dark matter black holes satisfy the first law of thermodynamics, the perfect fluid dark matter could accelerate the formation of supermassive black holes.

In the black hole thermodynamics, the thermodynamic geometry method is a usual tool to understand the property of black hole thermodynamics system. In the work, we use the Ruppeiner metric to study the thermodynamical effect of the perfect fluid dark matter on the microscopical structure of Reissner-Nordstrom-AdS black hole [

From (

From geometry, the Ricci scalar can be calculated for perfect fluid dark matter around Reissner-Nordstrom-AdS black holes by

We know that the sign of the Ricci scalar

In the paper, we study the perfect fluid dark matter influence on thermodynamics and phase transition of Reissner-Nordstrom-AdS black hole by extending phase space defined by the charge square

The Reissner-Nordstrom-AdS black hole surrounded by dark matters could appear in the Universe. In the future work we plan to study the observed effects of perfect fluid dark matter on black holes and the influence of perfect fluid dark matter on gravitational lensing and the evolution of dark matter in the universe.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The authors acknowledge the financial support from the National Natural Science Foundation of China, 11573060 and 11661161010.