The Sommerfeld expansion is a mathematical technique used in statistical mechanics to evaluate the thermodynamic properties of quantum gases, especially at low temperatures. Named after the physicist Arnold Sommerfeld, this method is particularly useful for calculating integrals that arise in the context of Fermi-Dirac statistics for fermions (like electrons in metals) and Bose-Einstein statistics for bosons (like photons or helium-4 at low temperatures).
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The Sommerfeld expansion approximates integrals weighted by the Fermi-Dirac distribution when and is smooth near . Applying it to the number integral at fixed particle number gives . The sign depends on the slope of the density of states. For an unbounded parabolic band, , yielding opposite quadratic shifts in one and three dimensions. In two dimensions the leading algebraic shift vanishes, but the finite band-bottom cutoff still gives an exponentially small negative shift.