The memory equation of a characteristic function in surface diffusion
The decay of a density wave, the Fourier component of the adparticle density seen by a coherent probe, obeys in an interacting adlayer an exact memory equation of Mori's form, determined by the local rate and the memory function. Because the density is conserved, the correlation function is a characteristic function, and the kernel is built from that of a single jump; positivity and normalization then make the local rate and the memory function equilibrium averages, whatever the interaction. The local rate obeys an exact sum rule, which makes de Gennes narrowing a theorem on the lattice and gathers the interaction into one effective hop rate. Detailed balance makes the decay a superposition of relaxation modes, approached from below by a convergent hierarchy of closed forms. The theory fixes the local rate, the memory function and the relaxation spectrum. The experiment provides the static structure factor and the decay, recorded from the instant of scattering through a ballistic regime, a jump regime in which the memory acts, and a diffusive regime. A single exponential fitted, as experiments do, over a time interval in the diffusive regime gives the slowest relaxation rate, the constant rate of that regime, and not the local rate. The correlation time, defined as the area beneath the normalized decay, needs no time interval and is the mean inverse relaxation rate. Under a standard condition on the jump rates, met in one-dimensional diffusion when the rate of leaving a site is independent of the site reached, the correlation time, the static structure factor and the lattice geometry give the effective hop rate at long wavelength, with no model or fit and with a parameter-free test. There the sum rule becomes a Darken-type relation.