  In a thermal field theory, the cumulants of the momentum distribution can be
extracted from the dependence of the Euclidean path integral on a shift in the
fields built into the temporal boundary condition. When combined with the Ward
identities associated with the invariance of the theory under the Poincare'
group, thermodynamic potentials such as the entropy or the pressure can be
directly inferred from the response of the system to the shift. Crucially the
argument holds, up to harmless finite-size and discretization effects, even if
translational and rotational invariance are broken to a discrete subgroup of
finite shifts and rotations such as in a lattice box. The formulas are thus
applicable at finite lattice spacing and volume provided the derivatives are
replaced by their discrete counterpart, and no additive or multiplicative
ultraviolet-divergent renormalizations are needed to take the continuum limit.
In this paper we present a complete derivation of the relevant formulas in the
scalar field theory, where several technical complications are avoided with
respect to gauge theories. As a by-product we obtain a recursion relation among
the cumulants of the momentum distribution, and formulae for finite-volume
corrections to several well-known thermodynamic identities.
