Thermal Casimir effect in a classical liquid in a quasi-periodically identified conical spacetime

In this paper, we study the finite-temperature quantum fluctuation of a classical liquid induced by the topology of an effective conical spacetime, as well as by a quasi-periodic boundary condition. The conical spacetime could be either a disclination or a cosmic string. In this context, we consider a phonon field representing quantum excitations of the liquid density, which obeys an effective Klein- Gordon equation with the sound velocity replaced by the light velocity. We obtain closed analytic expressions for the thermal Hadamard function, and consequently, the renormalized mean square density fluctuation of the liquid along with thermodynamics quantities such as internal energy, free energy, total energy, and entropy densities. We also discuss the limiting cases, including low and high-temperature regimes, and the situations in which there are only either the conical spacetime or quasi-periodicity.