I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from this knowledge and assuming that there are no collisions between particles, I tried to show that an ideal gas maximises the enclosed volume. 


More formally for any compact Euclidean manifold $M$ in $\mathbb{R}^3$ with elastic boundary $\partial M$ containing a macroscopic number of gas particles, a gradual increase in temperature ultimately results in a sphere. 


My method of proof is to take for granted that $\partial M$ allows a maximum pressure $P^\*$ and that if we gradually increase the temperature $T$ of the particles inside $M$, 


$$ P^\* \text{ everywhere on $\partial M$} \implies M \text{ is spherical} \tag{\*}$$


I think this method of proof would work but I realised that it's not easy to fill in the details considering that I assume that this balloon $M$ is contained in a gas with a different temperature and pressure. For this reason, I'd like to know whether this result has already been proven rigorously for a macroscopic number of particles in order to compare my work. 


I think this might be as hard as proving the double bubble conjecture so it might be that this result has not yet been rigorously demonstrated. 


**Note 1:** I am interested in the special case of Hookean elasticity as I think that Hookean elasticity would be a sufficiently good approximation. I also assume that the boundary $\partial M$ has negligible thickness. 


**Note 2:** So far I have the following handwavy argument that points to a proof. Assuming that $P^\*$ has been attained uniformly on $\partial M$, the magnitude of the force on $\partial M$ must be approximately constant all over $\partial M$. In consequence, by the Maxwell-Boltzmann distribution and momentum conservation, the distance between opposite extremities of the balloon must be approximately constant. 



rigorous derivation of isoperimetric inequality from ideal gas equation
For the $n$-dimensional phase space of $n$ particles on a line the isoperimetric inequality of an ideal gas was derived in: [Phase space measure concentration for an ideal gas](https://arxiv.org/abs/0811.1719) (2009).



> 
> We point out that a special case of an ideal gas exhibits
>  concentration of the volume of its phase space, which is a sphere,
>  around its equator in the thermodynamic limit. The rate of approach to
>  the thermodynamic limit is determined. Our argument relies on the
>  spherical isoperimetric inequality of Lévy and Gromov.
> 
> 
> 


