Complexity growth rate, grand potential and partition function

Abstract: We examine the complexity/volume conjecture and further investigate the possible connections between complexity and partition function. The complexity/volume 2.0 states that the complexity growth rate $\mathcal{\dot{C}}\sim PV$. In the standard statistics, there is a fundamental relation among $PV$, the grand potential $\Omega$ and the partition function $\mathcal{Z}$. By using this relation, we are able to construct an ansatz between complexity and partition function. The complexity/partition function relation is then utilized to study the complexity of the thermofield double state of extended SYK models for various conditions. The relation between complexity growth rate and black hole phase transition is also discussed.