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In order to understand thermodynamical properties of N D-branes with chemicalpotentials associated with R-symmetry charges, we study a one dimensional largeN gauge theory (bosonic BFSS type model) as a first step. This model isobtained through a dimensional reduction of a 1+D dimensional SU(N) Yang-Millstheory and we use a 1/D expansion to investigate the phase structure. We findthree phases in the \mu-T plane. We also show that all the adjoint scalarscondense at large D and obtain a mass dynamically. This dynamical mass protectsour model from the usual perturbative instability of massless scalars in anon-zero chemical potential. We find that the system is at least meta-stablefor arbitrary large values of the chemical potentials in D \to \infty limit. Wealso explore the existence of similar condensation in higher dimensional gaugetheories in a high temperature limit. In 2 and 3 dimensions, the condensationalways happens as in one dimensional case. On the other hand, if the dimensionis higher than 4, there is a critical chemical potential and the condensationhappens only if the chemical potentials are below it.

Thermodynamics of Large N Gauge Theories with Chemical Potentials in a 1/D Expansion