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

The Maxwell-Boltzmann velocity distribution arises from non-reactive elastic collisions of particles and is usually discussed in the context of the kinetic theory (for gases). There are various offhand remarks, for example here (slide 5), that state without reference that particles observe a similar velocity distribution in liquid. Is that true? References?

The main reason I'm curious is that it seems as though the mean free path would be extremely short in liquid vs. gas. I'm actually most curious about the nature of the collisions in liquid vs. gas, i.e., are collisions in liquid still (on average) elastic?

EDIT: The linked post on particle velocity in liquids is definitely interesting and weighs in on this question, and I appreciate the distinction made between local position fluctuation vs. long range movement. Still, let me frame this question in a few different ways.

1. Gillespie proposed a stochastic framework for simulating chemical reactions which is formulated on the idea that non-reactive elastic collisions serve to 'uniformize' particle position so that the assumption of well-mixedness is always satisfied (see page 409 in the linked version). This is formulated from kinetic theory. A corollary to this is that a non-reactive collision between two molecules that are able to react does not induce local correlation, i.e., two particles able to react with each other that just collided, but didn't react are no more likely to react with each other in the next dt than any other particle pair in the volume. Gillespie's algorithm is commonly used in biology where biochemical species are modeled in the aqueous environment of cells. Is this valid, and if so why?

2. On a microscopic scale, suppose we are interested in two 'A' particles diffusing in one dimension in an aqueous environment. The two A particles collide but don't react. What is their behavior immediately after the collision? Is it a 'reflection' which conserves velocity and might correspond to a Neumann BC? In a gas that approach seems natural, but in liquid the short mean-free path makes me think that diffusive forces would rapidly dissipate any momentum from the A-A collision, which might imply the A particles collide and 'stop'. How should I be thinking about this?

Just to bring it back to the original question, I think both (1) and (2) depend on the statistical velocity behavior of particles in liquid.

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I'm curious about it too. This question arised recently in my work, and I haven't found a satisfatory answer yet. – Hydro Guy May 20 '14 at 21:46
I think your edits 1 and 2 would be better posted as a new question and possibly as two separate questions. – John Rennie May 21 '14 at 15:59
What @JohnRennie said. – Kyle Oman May 21 '14 at 18:20
@vector07, I do not think that your question was well answered here or in the "duplicate" question suggested by the people below. 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. – Ján Lalinský May 21 '14 at 19:34
@JánLalinský I agree that this isn't a duplicate. Discussion continued here – vector07 May 27 '14 at 14:48

In terms of the Hamiltonian formalism the MB distribution can be obtained for a set of particles without potential energy between them (it means free particles), under the assumption that in equilibrium they satisfy the MB statistics. It is worth noting that this is not an obvious fact, the MB statistics can be applied to any kind of object, in particular applied to an ideal gas gives the MB distribution for velocities.

The free particle approximation is not valid for the liquid state, the mean free path in liquids is very short due to the intermolecular interactions and to derive the distribution of velocities is needed to take into account the potential term.

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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.

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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.

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Another very important example of fluids with long-range interaction forces are plasmas. – Robin Ekman May 21 '14 at 15:52

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.

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