Microcanonical shell

Let $\Gamma$ be a classical phase space with Liouville volume element $d\Gamma$, and let $H(x)$ be the Hamiltonian . Fix an energy value $E$ and an energy resolution (window width) $\Delta E>0$.

The microcanonical shell at energy $E$ with width $\Delta E$ is the subset

A common symmetric convention is $\{x:\, |H(x)-E|\le \Delta E/2\}$; the physics is unchanged as long as $\Delta E$ is small on macroscopic scales but large compared to microscopic level spacing (in quantum settings).

The “ideal” energy surface $\Sigma_E=\{x:\,H(x)=E\}$ has codimension one in $\Gamma$ and therefore has zero Liouville volume, which is why the shell (a thickened surface) is typically used in defining probabilities.

Phase-space volume of the shell

The Liouville volume of $\Sigma_{E,\Delta E}$ is

In terms of the density of states $g(E)$, for small $\Delta E$ one has the approximation

Physical interpretation

A classical isolated system evolves at constant energy (up to experimental resolution), so its accessible microstates lie (approximately) in $\Sigma_{E,\Delta E}$. The microcanonical shell is the geometric object on which the microcanonical measure places equal a priori weight.

In practice one may further restrict the shell by other conserved quantities (e.g., total momentum), but the defining idea remains: the shell encodes the accessible region in phase space given macroscopic constraints.