In Monte Carlo integration for Molecular dynamics simulation, why is a Boltzmann distribution assumed? In statistical physics, the calculation of partition function for an ensemble takes a Boltzmann's distribution of the Hamiltonian. Similarly, in Monte-Carlo integration of Molecular Dynamics simulation, the Boltzmann distribution is used to update the configuration of each particle. The essence of my question is that why do we assume that the Hamiltonian follows the Bolzmann's distribution and not any other probability distribution for that matter?
 A: 
Similarly, in Monte-Carlo integration of Molecular Dynamics simulation, the Boltzmann distribution is used to update the configuration of each particle.

It is the other way round: in physical problems one is interested in thermodynamic properties of a system, which are described by Boltzmann distribution. We then use molecular dynamics to find this distribution (not the distribution to solve the molecular dynamics).
There is an extra level to it: Hamiltonian Monte Carlo is a rather general method of sampling from probability distributions, used well beyond physics. This method is based on reducing the problem of sampling from a distribution to studying particles dynamics in a fictitious potential, equal to the negative logarithm of the probability density of interest - see my answer to Hamiltonian Monte Carlo: Kinetic and Potential energies for more details and references.

[...] why do we assume that the Hamiltonian follows the Bolzmann's distribution and not any other probability distribution for that matter?

Hamiltonian does not follow Boltzmann distribution - Hamiltonian dynamics does not contain any statistical information. On the other hand, Boltzmann distribution is determined in terms of Hamiltonian (and temperature)
$$
w\propto e^{-\frac{H}{k_BT}}.
$$
