A hard disk analysis of momentum deficit due to dissipation
  When a Brownian object is in a nonequilibrium steady state, actual force exerted on it is different from one in a thermal equilibrium. In our previous paper [Phys. Rev. Lett. 108 (2012), 160601] we discovered a general principle which relates the missing force to dissipation rates through a concept of momentum deficit due to dissipation (MDD). In this article, we examine the principle using various models based on hard disk gases and Brownian pistons. Explicit expressions of the forces are obtained analytically and the results are compared with molecular dynamics simulations. The good agreement demonstrates the validity of MDD. 
Ryoichi Kawai, Antoine Fruleux and Ken Sekimoto
cond-mat.stat-mech