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We analyze in the context of geometrothermodynamics the behavior of idealquantum gases which satisfy either the Fermi statistics or the Bose statistics.Although the corresponding Hamiltonian does not contain a potential, indicatingthe lack of classical thermodynamic interaction, we show that the curvature ofthe equilibrium space is non-zero, and can be interpreted as a measure of theeffective quantum interaction between the gas particles. In the limiting caseof a classical Boltzmann gas, we show that the equilibrium space becomes flat,as expected from the physical viewpoint. In addition, we derive a thermodynamicfundamental equation for the Bose-Einstein condensation and, using theEhrenfest scheme, we show that it can be considered as a first order phasetransition which in the equilibrium space corresponds to a curvaturesingularity. This result indicates that the curvature of the equilibrium spacecan be used to measure the thermodynamic interaction in classical and quantumsystems.

A geometrothermodynamic approach to ideal quantum gases and Bose-Einstein condensates