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A grand canonical ensemble describes a system exchanging both energy and particles with a reservoir at fixed $T$, volume, and chemical potential $\mu$. For a noninteracting fermion level $r$, occupation is $0$ or $1$, soFactorization over levels gives $\mathcal{Z}={\prod }_r{\mathcal{Z}}_r$, and differentiation with respect to $\beta \mu$ yields the Fermi-Dirac distribution

For $X=(x,p{)}^T$,Solving $A\sigma +\sigma A^T=b{b}^T$ givesThe canonical ensemble density proportional to $e^{-H/(k_{B}T)}$ for $H=p^2/(2m)+\lambda x^2/2$ factorizes into independent centered Gaussians. Its equipartition variances are exactly $⟨x^2⟩=k_{B}T/\lambda$ and $⟨p^2⟩=m{k}_{B}T$, while the absence of an $xp$ term gives ${\sigma }_{12}=0$.

A statistical Hamiltonian assigns an energy to a microscopic configuration. Its Boltzmann weight defines the canonical ensemble, and summing or integrating that weight gives a partition function. A reduced Hamiltonian absorbs inverse temperature into its coefficients, so the weight is $e^{-H_{\mathrm{reduced}}}$. An Ising model uses discrete Ising spin configurations, while Landau-Ginzburg theory uses a coarse scalar field.

For a positive quadratic bending operator $K$ with a real orthonormal basis of eigenfunctions, the equipartition theorem gives $C(x,y)=k_{B}T{\sum }_n{W}_n(x)W_n(y)/{\mu }_n$, where $K{W}_n={\mu }_n{W}_n$. This is the inverse-operator Green function multiplied by Boltzmann constant and temperature. Unconstrained zero-energy filament modes prevent a normalizable canonical ensemble, and negative modes signal an unstable quadratic model.

For both ends clamped and $K=A{\partial }_x^4$, the diagonal is $C(x,x)=k_{B}T{x}^3(L-x{)}^3/(3A{L}^3)$. This differs from a clamped-free tip variance. The center variance is $k_{B}T{L}^3/(192A)$.

A zero-energy mode lies in the kernel of the quadratic elastic filament operator. For pure bending, zero curvature makes the displacement affine. Free-free endpoints therefore leave translation and tilt modes, while slope-constrained, force-free endpoints leave only translation. Both clamped-clamped and hinged-hinged endpoints remove these modes. An unrestricted modal amplitude has constant Boltzmann distribution weight, so no normalizable canonical ensemble or finite displacement variance exists. Fixing the rigid degrees of freedom allows the equipartition theorem on the positive complement.