The Burgers equation is a nonlinear hydrodynamic model describing the evolution of the velocity field of sticky dust. The particles in this kind of medium interact only when they hit and stick to each other forming clumps. Some ergodic properties of this system with white-noise forcing and mostly in compact domains are known, but there are several interesting unanswered questions, especially for unbounded domains. In this talk a new simpler model for forcing based on Poissonian point field is proposed. The advantage of this model is that although it preserves many characteristic features of the white-noise model, it is easier to work with and visualize the resulting behavior. In fact, the model can be studied by looking at optimal paths through the Poissonian environment. In the unbounded domain case, if the spatial component of the measure driving the Poisson process has finite first moment, we obtain ergodic results for this model: one force-one solution principle; existence, uniqueness and some properties of a global skew-invariant solution including its behavior at infinity and a description of its basin of forward and pullback attraction; existence and uniqueness of a stationary distribution. Even for the Burgers equation on the circle this model provides a new insight into the behavior of the global minimizer.