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A particularly simple relation of proportionality between internal energy andpressure holds for scale invariant thermodynamic systems, including classicaland quantum Bose and Fermi ideal gases. One can quantify the deviation fromsuch a relation by introducing the internal energy shift as the differencebetween the internal energy of the system and the corresponding value for scaleinvariant gases. We discuss general thermodynamic properties associated to thescale invariance, provide criteria for which the internal energy shift densityis a bounded function of temperature. We then study the internal energy shiftand deviations from the energy-pressure proportionality in low dimensionalmodels of gases interpolating between the ideal Bose and the ideal Fermi gases,focusing on the Lieb-Liniger model in 1d and on the anyonic gas in 2d. In 1dthe internal energy shift is determined from the thermodynamic Bethe ansatzintegral equations and an explicit relation for it is given at hightemperature. Our results show that the internal energy shift is positive, itvanishes in the two limits of zero and infinite coupling (respectively theideal Bose and the Tonks Girardeau gas) and it has a maximum at a finite,temperature depending, value of the coupling. Remarkably, at fixed coupling theenergy shift density saturates to a finite value for infinite temperature. In2d we consider systems of Abelian anyons and non Abelian Chern-Simonsparticles: as it can be seen also directly from a study of the virialcoefficients, in the usually considered hard-core limit the internal energyshift vanishes and the energy is just proportional to the pressure, with theproportionality constant being simply the area of the system. Soft-coreboundary conditions at coincident points for the two-body wavefunctionintroduce a length scale, and induce a non-vanishing internal energy shift.

Energy-pressure relation for low-dimensional gases