A new definition of pressure based on kinetic theory derivations – article published on overleaf

Typical derivations of kinetic theory equations often exchange the contact time of the particle on a wall with the period of the particle’s motion between walls. In this paper we redefine pressure as time-dependent in order to solve this issue and show that this definition makes much more intuitive and theoretical sense than our old definition of pressure.

About a year ago, I decided to revisit this derivation. I began by considering a single particle bouncing off the wall of a rectangular prism and applying Newton’s second law as anyone might, but I hit a road block when trying to derive the contact time of the particle with the wall. Online research showed that physicists typically solve this problem by simply plugging in the period of the particle’s journey to and from the wall.

This clearly does not equal the contact time and results in a totally different impulse applied to the wall, but nevertheless the derivation gives correct results. It turns out that the problem does not lie in Newton’s equations (heaven forbid), but rather in our definition of pressure.

Consider, for instance, a cubic box with a single particle in it at temperature T. Empirical evidence confirms the equation we just derived, so we know for a fact that this temperature determines the velocity of our single particle in the box. This velocity in turn sets the impact force of our particle on the wall. Now, if we double the length of our box along an isotherm, the ideal gas equation tells us that the pressure must halve as the volume doubles and $NkT$ stays constant.

In addition, consider an infinite wall sitting in open space with a single particle moving towards it. The particle hits the wall once, rebounds, and proceeds in the opposite direction, never to return. According to our old definition of pressure, the particle exerts a small force on this wall because it came in contact with it. However, we can call this wall a box of infinite length, area, and volume; as such the pressure must be zero according to the ideal gas law.

Let’s start the new derivation by examining how the force on our one wall changes over time. For the purpose of simplicity we’ll assume that the force is constant for the entire interaction, growing to some value immediately and falling back to zero after impact. We can therefore represent our force function as a sort of square wave (see Figure \ref{fig:wave}). Each "blip" represents the particle hitting the wall with force $F$ and duration $\Delta t$, and each event repeats after a period of $\tau$.

We have one last test for our new definition of pressure: on large scales, it must tend towards the regular definition of pressure, $P=\frac{F}{A}$, similar to Bohr’s correspondence principle in quantum mechanics. If we imagine a total force function $F(t)$ for a large number of particles, this is clearly the case; statistically speaking, this graph would be approximately flat, since particles constantly and randomly hit all walls of the box. Thus, the average value of the function over time would equal the value of the flat function itself; $P$ would really equal $\frac{F}{A}$.

Our new definition of pressure represents little more than a curiosity; it leads to the same macroscopic predictions and laws and assumes a classical universe. This new definition may help modern physics in laying the framework for quantum thermodynamics and operator counterparts, but it mainly serves to clear up confusion for those first working through the derivation and learning kinetic theory.