Mathematical models for thermal diffusion and heat content in disordered geomedia, derived from first principles in Mathematical Geoenergy (GEM), Chapter 14 (Thermal Energy: Diffusion and Heat Content).
Chapter 14 applies Maximum Entropy (MaxEnt) uncertainty quantification to thermal diffusion problems — heat conduction in heterogeneous soils, geothermal heat exchangers, and the ocean heat content (OHC) response to greenhouse forcing.
The central observation is that real geological media are never perfectly
uniform: rocks, soil layers, pockets of different composition, and varying
moisture content all modulate the local thermal diffusivity. Without detailed
knowledge of every spatial variation, the best we can assume is a mean
diffusion coefficient D₀. Applying MaxEnt to D then yields an exponential
distribution p(D) = (1/D₀)·exp(−D/D₀), which is integrated over the
standard Gaussian heat kernel to give a closed-form dispersive thermal
kernel.
Two levels of MaxEnt disorder are considered:
Disorder in D only — integrating the Gaussian heat kernel over the
MaxEnt prior for D replaces the erfc cumulative response with a pure
exponential exp(−x/√(D₀t)) (Eq. 14-1) and eliminates the singular
initial transient of the standard solution.
Disorder in D and interface position — a second MaxEnt prior for the
spatial uncertainty x (mean x₀) gives the doubly-dispersive response
(1/2)/(x₀ + √(D₀t)) (Eq. 14-2), a Lomax-type decay in √t.
The same dispersive kernel reappears in:
Eq. 14-3 — Standard Gaussian heat kernel (ordered medium)
\[\Delta T(x,t) = \frac{C}{\sqrt{Dt}}\cdot e^{-x^2/(Dt)}\]Impulse response of the heat equation in a uniform medium. The Gaussian
widens with time as √(Dt).
Eq. 14-1 — Dispersive thermal cumulative response (MaxEnt D)
\[T(x,t) = T_1\cdot e^{-x/\sqrt{D_0 t}} + T_0\]Obtained by integrating the Gaussian Green’s function over the MaxEnt prior
p(D) = (1/D₀)·exp(−D/D₀) using the Laplace-Bessel identity (Eq. 20-6 in
GEM Ch. 20). The erfc is replaced by a pure exponential — no singularity
at t = 0, and the tail is fatter than the Gaussian prediction.
Eq. 14-4 — Dispersive impulse response (MaxEnt D)
\[\Delta T(x,t) = \frac{C}{\sqrt{D_0 t}}\cdot e^{-x/\sqrt{D_0 t}} \left(1 + \frac{x}{\sqrt{D_0 t}}\right)\]The disordered counterpart of Eq. 14-3. The (1 + u)·e^{−u} envelope
(with u = x/√(D₀t)) broadens the peak and increases the late-time tail
compared with the Gaussian, consistent with observations in disordered
heterogeneous media.
Eq. 14-2 — Doubly-dispersive thermal response
\[T(x,t) = \frac{1}{2}\cdot\frac{1}{x_0 + \sqrt{D_0 t}}\]Derived by averaging the dispersive impulse response n(x',t) = exp(−x'/√(D₀t))/(2√(D₀t))
over a MaxEnt distribution of source positions p(x') = (1/x₀)·exp(−x'/x₀):
The result is a Lomax-type decay in the variable √t — slower than exponential and consistent with long thermal tails measured in heat-exchanger experiments.
Eq. 14-6 — Smeared diffusive step response
\[\Delta T(t) = \frac{1}{1 + \sqrt{t/\tau}}\]The fraction of heat remaining at the source after time t, where
τ = L²/D₀ and L is the characteristic depth scale. Derived by averaging
the dispersive cumulative exp(−x/√(D₀t)) over a MaxEnt depth distribution
p(x) = (1/L)·exp(−x/L) and taking the complement:
Matches impulse-response data from earthen heat-exchanger tests (Witte 2006).
Eq. 14-5 — Convolution for thermal box model
\[\text{Response}(t) = \text{Input}(t) \otimes \text{Transfer}(t) = \int_0^t \text{Input}(\tau)\cdot\text{Transfer}(t-\tau)\\,d\tau\]Models a thermal input coupled through a dispersive transfer function. Used to construct the modulated CPU and geothermal transient responses (Eqs. 14-7, 14-8).
Eq. 14-9 — Depth-integrated heat content
\[I(t) = \int_0^\infty \Delta T(t\mid x)\cdot e^{-x/L}\\,dx\]The dispersive impulse response ΔT(t|x) (Eq. 14-4) is convolved with the
exponential depth weight exp(−x/L), modelling the total heat absorbed
by an ocean layer of characteristic depth L.
Eq. 14-10 — Ocean heat content response
\[I(t) = \frac{\sqrt{D_0 t}/L + 2}{\bigl(\sqrt{D_0 t}/L + 1\bigr)^2}\]Closed-form result of integrating the dispersive impulse response
ΔT(t|x) = (1/σ)·exp(−x/σ)·(1+x/σ) (Eq. 14-4, σ = √(D₀t)) over
the exponential depth weight exp(−x/L). Setting s = √(D₀t)/L:
As s → 0 (early time): I → 2 (maximum, pulse at the surface).
As s → ∞ (late time): I ~ 1/s → 0 (heat diffuses to great depths).
Eq. 14-12 / 14-13 — Convolution with linear forcing
\[F(t) = k\cdot(t - t_0)\] \[\frac{R(t)}{k} = \tfrac{2L}{3}(D_0 t)^{3/2} - L^2 D_0 t + 2L^3\sqrt{D_0 t} - 2L^4\ln\Bigl(\tfrac{\sqrt{D_0 t}+L}{L}\Bigr)\]The convolution R(t) = F(t) ⊗ I(t) gives the accumulated OHC under a
linearly growing greenhouse forcing, with t₀ ≈ 1960 as the onset year.
| File | Purpose |
|---|---|
thermal_dispersion_symbolic.py |
Symbolic derivation of all Chapter 14 equations using SymPy |
thermal_dispersion_numerical.py |
Numerical implementation, validation, and composite figure |
thermal_dispersion_model_output.png |
Output figure (6 panels) generated by thermal_dispersion_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python thermal_dispersion_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python thermal_dispersion_numerical.py
Disorder transforms erfc → exp: introducing MaxEnt uncertainty in the
diffusion coefficient D (Eq. 14-1) replaces the complementary error
function profile with a pure exponential exp(−x/√(D₀t)). The
initial singularity of the standard kernel vanishes and the late-time
tail is heavier.
Double MaxEnt gives Lomax-in-√t: adding MaxEnt uncertainty in the
interface position x gives a Lomax-type 1/(x₀ + √(D₀t)) response
(Eq. 14-2) — the same functional form that appears in lake sizes (Ch. 15)
and contamination spreading (Ch. 20), demonstrating the universality of
dispersive MaxEnt.
Geothermal heat-exchanger evidence: the smeared response
1/(1 + √(t/τ)) (Eq. 14-6) matches borehole heat-exchanger
measurements with a single parameter τ, far more naturally than the
standard erfc formula because geological soil is inherently disordered.
Ocean heat content as a single-parameter model: Eq. 14-10 reproduces the observed OHC increase at multiple ocean depths with a single effective diffusion coefficient D₀ ≈ 2–3 cm²/s. No ocean-circulation model or parameter tuning beyond D₀ and the layer thickness L is needed.
Same dispersive kernel across scales: exp(−x/√(D₀t)) / (2√(D₀t))
appears in geothermal heat diffusion (Ch. 14), battery discharge (Ch. 18),
and contaminant transport (Ch. 20), confirming that MaxEnt disorder in D
is a universal mechanism across geophysical and engineering systems.