In many places I see it mentioned that the Maxwell-Boltzmann distribution
$$ n_i \propto \exp(-\beta E_i) $$
(number $n_i$ of particles in state $i$ decreases exponentially with the energy $E_i$ of the state)
holds only if the particles do not interact or the interactions are negligible. For example in the "MB statistics" Wiki article it is mentioned several times:
In statistical mechanics, Maxwell–Boltzmann statistics describes the average distribution of non-interacting material particles over various energy states
(Limits of applicability) Note, however, that all of these statistics [MB, FD, BE] assume that the particles are non-interacting and have static energy states.
(Derivation) Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles: (...) In each case it is necessary to assume that the particles are non-interacting
However I've been following Tolman's text and it seems that the MB distribution holds for classical particles even if they interact with one another, so long as the entire dynamical system follows canonical equations of motion (actually I'm not sure what the exact requirement is, but it seems to be pretty liberal).
I've skimmed back portions of chapters 3 and 4 (ideally I would reread Tolman's entire derivation of the MB distribution, but it's so so long-winded) and this seems to indeed be the case, but I'm not 100% sure yet, and so I come here to ask.
Sketch of derivation of MB distribution for gas of centrally interacting particles
Consider a gas consisting of a large number $n$ of classical point particles, distinguishable but otherwise identical, interacting with one another through central forces (say, each particle is bound to every other particle by an ideal spring):
$$ H = \frac1{2m}\sum_i p_i^2 + \sum_{i,j} V(|\mathbf q_i-\mathbf q_j|) $$
First note that the complete dynamical (micro)state of the gas is specified by the coordinates $x_i, y_i, z_i$ and momenta $p_{xi}, p_{yi}, p_{zi}$ of the particles. Or, the phase space of the gas is a product of $n$ single-particle phase spaces. This parallels Tolman's development; we have this footnote p. 71:
This implies that we can regard the coordinatos and momenta of the molecules themselves as giving a sufficiently precise specification of the intermolecular fiold, and can neglect tho circumstance that, in general, a knowledge of additional variables would really be neeessary to give an exact spocification to tho complieated electromagnetic ffeld existing betweon tho molecules. This approximation is valid when tho velocities of the electric charges involved are small compared with that of light.
Anyway, suppose now that the only thing we know about the gas is that its total energy is $E$. By the principle of equal a priori probabilities of statistical mechanics, it is equally likely to be in any one of the dynamical microstates compatible with this constraint on the energy.
This is the microcanonical distribution $\rho$. Notice that since the particles are interacting, the dynamical microstate of the system is changing over time as particles exchange energy. However, from the canonical equations of motion we have Liouville's theorem. And so $\rho$, which depends only on $E$, a constant of the motion, is constant over time. This means that the system is appropriately represented by the microcanonical ensemble every instant in time.
And so we may apply the usual argument. "What is the probability $n_i/n$ that a given gas particle is in the (single-particle) state $i$?" (we have partitioned the phase space of the particles into discrete units etc…). With the usual argument we arrive at
$$ n_i \propto \exp(-\beta E_i) $$
where $E_i$ is the energy of the $i$th state.
So, is that right?
This answer and some of its comments seem to agree roughly with this (MB distribution works for interacting classical particles) (though it's unclear exactly what the restrictions on the interaction should be), but they don't go too deep on the subject.
As far as I know, the Boltzmann distribution is valid even for interacting gases, provided the particle interaction is short-range
It seems that within classical statistical mechanics the velocity distribution in thermal equilibrium is always the Maxwell-Boltzmann distribution, no matter what the interactions are (as long as they don't depend on the velocities).
Given how much I see it stated that MB only works for non-interacting particles, I would like more details on the matter. Who's wrong? Where did this idea that it had to be non-interacting come from?