We theoretically predict the local density of nucleosomes on DNA brushes in a solution of molecules, which are necessary for transcription and the assembly of nucleosomes. Our theory predicts that in a confined space, DNA brushes show phase separation, where a region of relatively large nucleosomal occupancy coexists with a region of smaller nucleosomal occupancy. This phase separation is driven by an instability arising from the fact that the rate of transcription increases as the nucleosomal occupancy decreases due to the excluded volume interactions between nucleosomes and RNA polymerase during thermal diffusion and, in turn, nucleosomes are (in some cases) desorbed from DNA when RNA polymerase collides with nucleosomes during transcription. The miscibility phase diagram shows critical points, which are sensitive to the rate constants involved in transcription, the changes of interactions of DNA chain segments by assembling nucleosomes, and pressures that are applied to the brushes.