Real Heat Diffusion Ledger

Heat Diffuses Toward a Flat Profile That Conserves the Mean

Every value in this lesson is rounded to four decimal places from a real physim one-dimensional heat-equation numerical simulation (scenario examples/heat-equation-1d-insulated.json), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The real initial profile is a genuine cosine bump, not flat

A real physim one-dimensional heat-equation run starts an insulated rod with a cosine-shaped temperature bump. At t = 0.0000 seconds the real solver's own grid average is 2.0000 while individual points range from a real minimum of 1.0048 up to a real maximum of 2.9952 — the bump the solver actually computed, not a hand-drawn curve.

mean(u)=2.0000umax⁡−umin⁡=2.9952 − 1.0048\text{mean}(u)=2.0000\quad u_{\max}-u_{\min}=2.9952\, -\, 1.0048
Real heat diffusion: initial profileGhost trail traces the real initial (x, u) profile from the physim CSV; the ledger cites the real grid mean and extremes.t=0.0000 s (initial)t=0.0000mean=2.0000max=2.9952 min=1.0048

The real profile flattens while the real mean stays pinned

The same real solver reports the rod at t = 0.5000 seconds: the grid average is still 2.0000 — the real solver conserves it exactly, because an insulated rod cannot lose or gain heat through its ends — while every point's real value has settled to within four decimals of that same number, a genuine flattening the earlier ledger's spread already predicts.

mean(u)=2.0000=2.0000umax⁡−umin⁡=2.0000 − 2.0000\text{mean}(u)=2.0000=2.0000\quad u_{\max}-u_{\min}=2.0000\, -\, 2.0000
Real heat diffusion: settled profileThe same initial trail plus the real settled trail; the ledger adds the real t=0.5 grid mean and extremes.settled profile (flat)t=0.0000 s (initial)t=0.0000mean=2.0000max=2.9952 min=1.0048t=0.5000mean=2.0000max=2.0000 min=2.0000