Good knowledge of the far-infrared and millimeter emission from dust in the interstellar medium is important to get reliable estimates of the dust mass, to trace and understand the evolution of pre-stellar structures, and to accurately subtract the foreground emission in the cosmological background anisotropy measurements. Up to now the modeled dust emission profile in FIR and millimeter wavelength range is deduced from the wings of some mid-infrared fundamental lattice-resonances inside the silicate material, which is known to be the dominant constituent of this dust component. However recent astronomical observations have shown that the dust emission profile could be significantly more complicated than expected. In addition, spectroscopic studies in the laboratory on analogues of amorphous interstellar grains have revealed that additional processes can occur in that spectral range, which are strongly temperature-dependent. We propose a new model for far-infrared and millimeter dust emission which takes into account results from the solid state physics, used to interpret these laboratory data. This model explicitly incorporates the effect of the disorder in the internal structure of the dust grain. We show that this model can give a satisfactory interpretation for the astronomical observations. It opens new perspectives to derive some new dust characteristics from the shape of the dust emission spectrum.