巨正则系综中$f(R)$ AdS黑洞的相变和热力学几何
Phase transition and thermodynamic geometry of $f(R)$ AdS black holes in the grand canonical ensemble
摘要：
在巨正则系综下研究了常曲率$R+f(R)$引力场中四维带电AdS黑洞解的相变，发现了不同于正则系综的新特征.当$f(R)$黑洞的电荷保持不变时，$T-S$曲线不存在临界点，而以往的研究发现$T-S$曲线和$T-r_+$曲线都存在临界点。此外，我们还导出了$f(R)$ AdS黑洞在电势一定时的比热、体积膨胀系数和等温压缩系数的近似表达式.比热$C_Φ$在$0<Φ<b$时出现发散，而在$Φ>b$情况下没有发散。这一发现也不同于正则系综中的结果，正则系综中的比热$C_Q$可能有两个、一个或没有发散点。为了研究巨正则系综中新发现的相结构，我们借助热力学几何工具，导出了Weinhold标量曲率和Ruppeiner标量曲率的解析表达式.它表明，他们分歧正是在比热$C_Φ$分歧。
英文摘要：
The phase transition of four-dimensional charged AdS black hole solution in the $R+f(R)$ gravity with constant curvature is investigated in the grand canonical ensemble, where we find novel characteristics quite different from that in canonical ensemble. There exists no critical point for $T-S$ curve while in former research critical point was found for both the $T-S$ curve and $T-r_+$ curve when the electric charge of $f(R)$ black holes is kept fixed. Moreover, we derive the explicit expression for the specific heat, the analog of volume expansion coefficient and isothermal compressibility coefficient when the electric potential of $f(R)$ AdS black hole is fixed. The specific heat $C_Φ$ encounters a divergence when $0<Φ<b$ while there is no divergence for the case $Φ>b$. This finding also differs from the result in the canonical ensemble, where there may be two, one or no divergence points for the specific heat $C_Q$. To examine the phase structure newly found in the grand canonical ensemble, we appeal to the well-known thermodynamic geometry tools and derive the analytic expressions for both the Weinhold scalar curvature and Ruppeiner scalar curvature. It is shown that they diverge exactly where the specific heat $C_Φ$ diverges.
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