# Do particle velocities in liquid follow the Maxwell-Boltzmann velocity distribution?

Generally speaking, it depends on the nature of the liquid. The assumptions behind the Maxwell-Boltzmann distribution are fairly simple: molecules can be approximated as point-masses and their only interactions are through collisions that exchange momentum and energy.

So if you have liquids where this is true, then yes, the distribution will be correct. However, if you have liquids (or gases for that matter) that have large molecules (so that the point-mass assumption is invalid) or that have long-range interaction forces (like water for example), then the distribution will not be Maxwell-Boltzmann.

Classical particles must follow the Maxwell-Boltzmann velocity distribution, and this is a consequence of separability of momenta and position in the partition function, and that the Boltzmann factor weighing the probability of each state is $\propto \exp(-\beta\mathcal{H})$. This is not, however, to say that this is the distribution one would always obtain experimentally. I am not aware of an experimental procedure where one can directly sample velocities, but rather you observe displacements between times $0$ and $t$, say, and then you divide the size of the displacement by the time $t$ to get an effective velocity.

Now, remember that the diffusion constant is defined as (in 3D) $$D = \lim_{t\to\infty}\frac{1}{6t}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$

Note that here we assume that the displacements are taken an infinite time apart. Because the collisions cause random motion, and in stochastic Ito calculus the time is proportional to the square of displacements, this indeed evaluates as a finite constant in most cases. And in most cases the distribution of displacements (and therefore of effective velocities) at infinity is Gaussian.

Now there are important exceptions to this rule. Fractional Brownian motion, for example, does not yield a regular diffusion constant, but undergoes something called anomalous diffusion (a hot topic in current physics research, anything dealing with anomalous diffusion, or diffusion seemingly anomalous often gets published in top venues). Anomalous diffusion looks like the following: $$D = \lim_{t\to\infty}\frac{1}{6t^\alpha}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$

where the limit makes sense only for some $\alpha$. If $\alpha$ is larger than $1$, one calls the motion superdiffusive, and when it is smaller, subdiffusive.

How is this related to velocity distributions? Well, to get an idea through the diffusion constant of the underlying velocity distribution, one wants the displacements to be a very short time apart. $$D = \lim_{t\to0}\frac{1}{6t^\alpha}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$

Note the change of limit. Now what is $\alpha$ going to be? Obviously $2$, because there are no collisions in very short time scales and one is then dealing with ballistic motion, where displacements scale linearly with time. Shortly after (if the limit in $t$ were to be in the picosecond scale) one enters the subdiffusive regime caused by viscoelastic behaviour of the material, and finally, when the displacements are measured long enough apart, to the normal, diffusive regime (in most liquids, that is, but there are exceptions, of course, like fractional Brownian motion).

Very few experimental methods can access the femtosecond, ballistic, regime, and it is only here that the displacement distribution should follow Maxwell-Boltzmann. For longer time scales, as you typically observe and are interested in, a Gaussian distribution might be a better approximation, but this does depend on the type of liquid.

I think Maxwell-Boltzmann distribution should be valid for molecules in liquid too, at least according to classical statistical physics, because the factor $e ^{−\beta p^2/2m}$ in the Gibbs-Boltzmann probability density does not depend on potential energy and is the same whether the molecule is in gas, or a liquid. I do not know if there is a measurement supporting this theoretical result.

are collisions in liquid still (on average) elastic?

Elastic collision means that appreciable change in the kinetic and potential energy of two bodies happens to them only during short time interval and the energy long after that is the same as the energy long before that - the interaction of the two molecules is thought of as a scattering process. In liquids the interaction of the molecules may not be idealizable in this way, as the molecules are believed to be in incessant complicated motion constantly influencing each other (Brownian motion...) This does not seem to be a reason to abandon the Boltzmann statistics, however.