  We study the dissipation mechanism of a stochastic particle system for the
Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes
equations can be expressed as an expected value of a stochastic process based
on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3
(2008) 330-345]. In this paper we study a particle system for the viscous
Burgers equations using a Monte-Carlo version of the above; we consider N
copies of the above stochastic flow, each driven by independent Wiener
processes, and replace the expected value with $\frac{1}{N}$ times the sum over
these copies. A similar construction for the Navier-Stokes equations was
studied by Mattingly and the first author of this paper [Iyer and Mattingly
Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle
system for the Burgers equations shocks almost surely in finite time. In
contrast to the full expected value, the empirical mean $\frac{1}{N}\sum_1^N$
does not regularize the system enough to ensure a time global solution. To
avoid these shocks, we consider a resetting procedure, which at first sight
should have no regularizing effect at all. However, we prove that this
procedure prevents the formation of shocks for any $N\geq2$, and consequently
as $N\to\infty$ we get convergence to the solution of the viscous Burgers
equation on long time intervals.
