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We study ultracold Bose gases in periodic potentials as described by theBose-Hubbard model. In 1D and at finite temperature, we simulate ultracold Bosegases in imaginary time with the gauge $P$ representation. We study variousquantities including the Luttinger parameter $K$, which is important forlocating the boundaries of the Mott insulator lobes, and find a simple relationfor the kinetic energy part of the Bose-Hubbard Hamiltonian.  We show that for J=0, the stepwise pattern of the average number of particlesper lattice site versus the chemical potential vanishes at temperatures above$T \approx 0.1 U$. Also, at chemical potential $\mu=0.5 U$ and temperature$T=0.5 U$ by increasing $J$, the relative value of the number fluctuationdecreases and approaches that of a coherent state.

Finite temperature correlations in the Bose-Hubbard model: application of the Gauge $P$ representation