A Symmetry Property of Momentum Distribution Functions in the   Nonequilibrium Steady State of Lattice Thermal Conduction
  We study a symmetry property of momentum distribution functions in the steady state of heat conduction. When the equation of motion is symmetric under change of signs for all dynamical variables, the distribution function is also symmetric. This symmetry can be broken by introduction of an asymmetric term in the interaction potential or the on-site potential, or employing the thermal walls as heat reservoirs. We numerically find differences of behavior of the models with and without the on-site potential. 
Akira Ueda and Shinji Takesue