= Ledoux criterion
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The <Ledoux criterion> tests local adiabatic <stellar convective stability> in the presence of a composition gradient. Define
$$
\nabla=\frac{d\log T}{d\log P},\qquad
\nabla_\mu=\frac{d\log\mu}{d\log P},\qquad
\delta=-\left(\frac{\partial\log\rho}{\partial\log T}\right)_{P,\mu},\qquad
\phi=\left(\frac{\partial\log\rho}{\partial\log\mu}\right)_{P,T},
$$
where $\mu$ is the <mean molecular weight> and the partial derivatives describe the <equation of state>. For positive $\delta$, the stratification is stable to an infinitesimal composition-preserving <adiabatic process> displacement when
$$
\nabla<\nabla_{\rm ad}+\frac{\phi}{\delta}\nabla_\mu.
$$
The parcel adjusts its <pressure> to its surroundings, retains its composition and changes <temperature> along the <adiabatic temperature gradient>. Comparing its resulting <mass density> with the ambient value determines the restoring <buoyancy>. For positive $\phi/\delta$, increasing mean molecular weight inward gives $\nabla_\mu>0$ and an additional stabilizing contribution. With uniform composition the condition reduces to the <Schwarzschild criterion>. Thermal or compositional diffusion introduces further instability questions beyond this adiabatic parcel test.
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