A model violating the Boltzmann distribution
  We study a model of an ionic conductor having interstitial ions that jump from site to site. The conductor is subject to an external electric field. According to classical mechanics, the ion number density follows the Boltzmann distribution. But according to quantum mechanics, we show that the Boltzmann distribution is violated by a factor of 2. And the violation provides a mechanism to explain the Haven ratio observed in some experiments. 
Yong-Jun Zhang
cond-mat.stat-mech quant-ph