  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.
