# Justification behind using probability density in kinetic gas theory

Let's consider an ideal gas, of some huge number of particles. The Maxwell-Boltzmann distribution describes the probability of measuring a particle speed in a range of speeds, through integration.

Why are we justified in assuming that there is a continuum of speeds to integrate over? Certainly we have a huge number of particles in the gas, but there are still only finitely many - 10^30 or 10^40 or whatever particles is huge, but still doesn't comprise a continuum. No matter how many particles there are, surely the speed distribution should remain discrete?

I understand that this is some approximation, but then how many particles is 'big enough' for this approximation to hold? 10^20? 10^30? These sort of numbers are huge by everyday experience, but don't compare to some sort of uncountable continuum. I'd be interested to know what people think!

• Even a single particle can access a continuum of momenta... – Nephente Jul 29 at 13:06
• Ohh I see, I think I've misunderstood what people mean when they say "the integral over whatever range gives the probability of finding a particle with a speed in that range". I interpreted that as meaning the probability of measuring the speed of a given particle in a certain range has some sort of dependence on what the collection of particles is doing, rather than "here is a particle. Measure it's speed, and it's described by this distribution". Is that what you're saying here? – M. Whyte Jul 29 at 13:23
• Are you aware that Kinetic Theory itself is mathematically incorrect due to Poincare recurrences? But, we accept entropy as a truth because the Poincare recurrences would take too long for us to care, at least compared to the current age of the universe. If you're willing to accept this, then you should be willing to approximate the distribution function as being continuous. Once again, this is based purely on physical arguments, not mathematical ones. – Feel My Black Hole Jul 29 at 13:29
• Kinetic theory and/or statistical mechanics are approximations, like any other theory, and they have their limitations. In the limit of large N, the evolution of the particle velocity distribution can be approximated by a continuous model function. This is done for mathematical convenience and so far as we can tell, it works quite well (e.g., fluid equations of motion, for example the Navier-Stokes equation, are just velocity moments of the Boltzmann equation). – honeste_vivere Aug 13 at 13:18