If the heat produced and the volume are each proportional to the number of qubits (R proportional to cube root of n), and the surface area bounding the qubits is as small as it could be while bounding that much volume (as a pessimistic assumption) (and therefore a sphere) ...
hm, the rate that heat passes through a surface by conductance is proportional to the surface area multiplied by the gradient of temperature across the surface, right?
If the heat production is uniform within the ball, then... well, the core of the ball would be the hottest... supposing that the surface of the ball is held at a constant temperature (with the system in a steady state, as far as temperature goes)
Let u(x) the temperature at location x. Let \alpha be the thermal diffusivity (assume to be constant throughout the material and over time). Assume that within the ball of radius R, \alpha \nabla^2 u = k for k the heat production density divided by the specific heat capacity (assumed to be constant over the range of temperatures involved) .
For spherically symmetric u(x), a function of just distance from the center..
ok, so, need solutions of Laplace's equation, \nabla^2 u = f , where f is some constant times the indicator function of the ball of radius r? Uh, I was thinking to have a boundary condition at the surface of the ball, fixing a particular temperature there, and seeing what temperature enforced there is enough to produce a small enough temperature at the center of the ball...
(In that case I guess f can just be a constant, rather than the indicator function of the ball)
uhh.. does this have an analytic solution? This is ending up as a more difficult computation than I anticipated...
edit: oh, for it to be steady state, the rate of heat going through any sphere centered at the origin, must be equal to the rate of heat produced within the ball that it bounds, so for r so, g'(r) ~ r ,
so g(r) - g(0) ~ r^2 .
So... if I haven't messed up too badly, I would think that, the difference in temperature of the center, and the temperature of the surface, should be proportional to (heat production per qubit) * ((radius of ball)^2) ~ (heat production per qubit) * ((number of qubits)^{2/3})
which... given a particular upper bound on working temperatures for the core of the ball, would put an upper bound on the number of qubits if packed in a ball like that. Though, I would imagine that if you instead have the inner (some number) fraction of the ball not have qubits, and not produce heat, then that wouldn't apply. Though this would require the surface area grow faster than (number of qubits)^{2/3} .