Fractal thermodynamics under imaginary rotation: the characteristics of free massless boson gas as function of the statistical angle χ related to imaginary angular frequency ΩI in Eq. (3). We show only the energy density as it is connected to other thermodynamic quantities via Eq. (16). The normalization to the non-rotating gas is implied (15). The plots show various zoom scales from 10 0 to 10 −3 .

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... Fig. 2, we show the energy density of a massless boson gas as a function of the statistical angle χ, which is related to the imaginary angular velocity Ω I in Eq. (3). According to Eq. (16), this thermodynamic quantity is normalized to the non-rotating boson gas value at a fixed temperature (15). Thermodynamics exhibits an explicit fractal ...

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... have also found the fractal behavior of free-fermion thermodynamics, similar to the bosonic fractal fountains shown in Fig. ...

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... factor 2 in Eq. (18) corresponds to a spin degeneracy while the factor q gives us the thermal degeneracy similarly to the bosonic case (13). As a result, one fermion produces 2q ghosts (18). The thermodynamics of fermions under the imaginary rotation also has a fractal fountain structure similar to the one of Fig. 2. The fractal fountains, however, acquire a two-color structure due to even/odd partitioning of the free energy (18). The relations (18) can be understood along the same analysis that we performed in the case of bosons, taking into account the anti-periodicity of fermions in the τ direction. One should also mention that the rotation ...