Integrating and rescaling a statistical model ($e^{-{\mathcal{H}}^{\prime }(S)}={\sum }_{s}B(S\mid s)e^{-\mathcal{H}(s)}$)

Use a dimensionless statistical Hamiltonian $\mathcal{H}=H/(k_{B}T)$, so that the partition function is $Z={\sum }_s{e}^{-\mathcal{H}(s)}$. Choose a normalized blocking kernel $B(S\mid s)$ with ${\sum }_{S}B(S\mid s)=1$. For the majority example it is one when every block has its prescribed majority, and zero otherwise. Define the coarse statistical Hamiltonian by the displayed sum. Then an elementary interchange of finite sums provesThe probability of a retained pattern is therefore its original marginal probability. For any observable $A(S)$ of the retained variables,This is the sense in which elimination preserves predictions for retained information. It does not reconstruct every microscopic observable from a block sign.

Next express positions in units of the enlarged spacing: $x^{\prime }=x/b$, where $b>1$. In a continuum version, rescale the retained scalar field as well to restore a chosen gradient energy coefficient; any change in the integration measure contributes to the field-independent term. The result is a new set of dimensionless couplings. Repeating the procedure defines a renormalization-group flow. Additional interactions and the additive constant generally appear, even if the starting statistical Hamiltonian was simple. Keeping only a few terms is an approximation; the equality of partition functions above applies to the complete coarse weight. See also the Wilsonian coarse-grained statistical Hamiltonian for integration over fast Fourier modes.