Hi, I didn't see this reply before, but I think that's a wonderfully simple way of looking at it. Thanks for the intuition and step-by-step construction. That

Hi, I didn't see this reply before, but I think that's a wonderfully simple way of looking at it. Thanks for the intuition and step-by-step construction. That makes me think of the normal distribution and the heat kernel, which I'd be very interested to hear your thoughts on. The heat kernel is the Green's function solution of the heat equation (governing heat or other diffusive transport in the absence of material movement transporting the quantity along with it): dT/dt = dT/dx^2 (pretend those are partial derivatives) If we have an initial condition of T=1 at the origin and zero everywhere else (e.g., a spiked Dirac delta), the solution at time t>0 is the same as a normal distribution after appropriate normalization. The variance simply gets spatially bigger as time goes on and the thermal energy continues to diffuse. Thinking about your intuition, the first derivative at the origin is zero as well (because the heat should diffuse the same in every direction absent any conductivity anisotropy). The second derivative is also near zero around the origin, for sufficiently small distances and sufficiently large T>0. Because the heat kernel is flat to second (not first) order near the origin, heat flux vanishes to second order. At sufficiently large distances/small times, the flux vanishes to any order. The sweet spot is in the middle, at the steep part of the heat kernel/Gaussian. There, difference of heat and exiting a point is infinitesimally different, to the first order of the temperature gradient. But that doesn't mean that heat transport in an out of a point is proportional to the temperature gradient! One point can only transport heat to infinitesimally nearby points. The difference in temperature between infinitesimally close points, where the temperature gradient is infinitesimal, is doubly infinitesimal.