  Submillimeter galaxies (SMGs) at $z\gtrsim1$ are luminous in the far-infrared
and have star-formation rates, SFR, of hundreds to thousands of solar masses
per year. However, it is unclear whether they are true analogs of local ULIRGs
or whether the mode of their star formation is more similar to that in local
disk galaxies. We target these questions by using Herschel-PACS to examine the
conditions in the interstellar medium (ISM) in far-infrared luminous SMGs at
z~1-4. We present 70-160 micron photometry and spectroscopy of the [OIV]26
micron, [FeII]26 micron, [SIII]33 micron, [SiII]34 micron, [OIII]52 micron,
[NIII]57 micron, and [OI]63 micron fine-structure lines and the S(0) and S(1)
hydrogen rotational lines in 13 lensed SMGs identified by their brightness in
early Herschel data. Most of the 13 targets are not individually
spectroscopically detected and we instead focus on stacking these spectra with
observations of an additional 32 SMGs from the \herschel\ archive --
representing a complete compilation of PACS spectroscopy of SMGs. We detect
[OI]63 micron, [SiII]34 micron, and [NIII]57 micron at >3sigma in the stacked
spectra, determining that the average strengths of these lines relative to the
far-IR continuum are $(0.36\pm0.12)\times10^{-3}$,
$(0.84\pm0.17)\times10^{-3}$, and $(0.27\pm0.10)\times10^{-3}$, respectively.
Using the [OIII]52/[NIII]57 emission line ratio we show that SMGs have average
gas-phase metallicities $\gtrsim Z_{\rm sun}$. By using PDR modelling and
combining the new spectral measurements with integrated far-infrared fluxes and
existing [CII]158 micron data we show that SMGs have average gas densities, n,
of $\sim10^{1-3}{\rm cm^{-3}}$ and FUV field strengths, $G_0\sim10^{2.2-4.5}$
(in Habing units: $1.6\times10^{-3}{\rm erg~cm^{-2}~s^{-1}}$), consistent with
both local ULIRGs and lower luminosity star-forming galaxies.
