Geometric Heat Flow, Entropy and Free Energy

Entropy · Free Energy · Heat Flow

Heat flow as entropy descent on a geometric space

Start with a concentrated heat density and let it diffuse. On a flat torus, a dumbbell domain, a hidden metric, or a curved surface, the visible picture may change slowly or quickly depending on geometry. The simulation tracks the mathematical entropy, physical entropy, free energy, and Dirichlet energy while the heat distribution evolves.

Entropy viewpoint

A concentrated density has large mathematical entropy. Pure heat flow spreads the density and decreases this quantity; equivalently, physical entropy increases.

Mathematical entropy: \( \mathcal E(\rho)=\int_M \rho\log\rho\,dV_g \) Physical entropy: \( S(\rho)=-k_B\int_M \rho\log\rho\,dV_g \)

Free-energy viewpoint

In the pure heat examples below, the potential term is zero, so the free energy is the entropy term. Geometry still matters because it changes the Laplacian, boundary behavior, and transport paths.

Free energy: \( \mathcal G_\beta(\rho)=\int_M V\rho\,dV_g+\beta^{-1}\int_M\rho\log\rho\,dV_g \) Pure heat mode: \( V=0,\quad \partial_t\rho=\Delta_g\rho \)

geometry: –

boundary: –

entropy trend: –

time: –

Interactive heat-flow panels

Heat densitydrag 3D surfaces to rotate

Geometry / metric viewdomain

Entropy and free energylive diagnostics

Diagnostics

scenario

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mathematical entropy \(\mathcal E\)

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physical entropy \(S\)

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free energy \(\mathcal G\)

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Dirichlet energy

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heat mass / max heat

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Legend

Heat density, low to high

Boundary / surface mesh

Metric barrier / slow diffusion

Hot concentrated region

Parameter notes

Control	Meaning	Mathematical interpretation	What to observe
`geometry`	Chooses the domain or surface where heat diffuses.	Changes the graph Laplacian or the surface-neighbor operator.	Same initial heat can spread very differently when topology, bottlenecks, or metric weights change.
`initial width σ`	Controls how concentrated the initial Gaussian is.	Smaller σ usually starts with larger \(\mathcal E=\sum p\log p\).	Sharp peaks have more entropy to dissipate.
`neck / barrier`	For dumbbell it controls the neck width; for hidden metric it controls the strength of the slow-diffusion wall.	Bottlenecks reduce transport and delay equilibration.	Entropy decreases more slowly when heat cannot easily cross the middle.
`temperature β⁻¹`	Scales the entropy term in free energy.	With \(V=0\), \(\mathcal G_\beta=\beta^{-1}\mathcal E\).	It rescales the plotted free energy but does not change the pure heat equation.
`color scale`	Renormalized color imitates the Julia GLMakie animations; absolute color shows decay relative to the initial peak.	Renormalization reveals shape; absolute scaling reveals amplitude decay.	Switch modes to see either transport geometry or smoothing strength.