  Using the Bethe ansatz solution, we analytically study expansionary, magnetic
and interacting Gr\"uneisen parameters (GPs) for one-dimensional (1D)
Lieb-Liniger and Yang-Gaudin models. These different GPs elegantly quantify the
dependences of characteristic energy scales of these quantum gases on the
volume, the magnetic field and the interaction strength, revealing the caloric
effects resulted from the variations of these potentials. The obtained GPs
further confirm an identity which is incurred by the symmetry of the thermal
potential. We also present universal scaling behavior of these GPs in the
vicinities of the quantum critical points driven by different potentials. The
divergence of the GPs not only provides an experimental identification of
non-Fermi liquid nature at quantum criticality but also elegantly determine low
temperature phases of the quantum gases. Moreover, the pairing and depairing
features in the 1D attractive Fermi gases can be captured by the magnetic and
interacting GPs, facilitating experimental observation of quantum phase
transitions. Our results open to further study the interaction- and
magnetic-field-driven quantum refrigeration and quantum heat engine in quantum
gases of ultracold atoms.
