Coarse-grained variable

A coarse-grained variable retains selected collective information while averaging over microscopic details. Dividing a magnetic sample into small cells and assigning each cell its mean magnetization gives a field that ignores individual spins but still describes variations over many lattice sites. The cell size should be specified relative to both microscopic spacing and the spatial changes of interest.

Different microstates can give the same coarse-grained variable. At thermal equilibrium, their statistical weights contribute an entropic term to the effective free energy; the effective cost is therefore not simply the microscopic energy evaluated at a smoothed configuration. In statistical field theory, a local order parameter is a common retained variable. Changing the averaging scale can change its effective parameters while leaving long-distance predictions consistent. Coarse graining is useful precisely because the retained field answers a chosen macroscopic question without reconstructing every microscopic coordinate.