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It is often stated that heat baths with finite degrees of freedom i.e.,finite baths, are sources of Tsallis distributions for the classicalHamiltonian systems. By using well-known fundamental statistical mechanicalexpressions, we rigorously show that Tsallis distributions with fat tails arepossible \textit{only} for finite baths with constant negative heat capacitywhile constant positive heat capacity finite baths yield decays with sharpcut-off with no fat tails. However, the correspondence between Tsallisdistributions and finite baths holds at the expense of violating equipartitiontheorem for finite classical systems at equilibrium. Finally, we comment on theimplications of the finite bath for the recent attempts towards a$q$-generalized central limit theorem.

Do Tsallis distributions really originate from the finite baths?