Irreversibility of the renormalization group flow and information theory

Monotonic entropy growth for a nonlinear model of random exchanges

“…The main idea behind this proof is that any such evolution may be viewed as some kind of coarse-graining since it includes integration over initial data (cf. [14]). Note that for nonlinear evolution, when P(x, u; p) depends on the distribution function p(u), as in (2), and hence changes on each step, this approach does not work, because now p 0 is not an instantaneous equilibrium andν = p 0 (x).…”

“…(1). Thus, on each step the nonlinear evolution of the gaslike model may be obtained by a kind of 'projection' procedure from the linear transformation (4) by choosing the special initial conditions (13) and by the subsequent elimination of one variable from the resulting two-particle distribution function (14).…”

“…But elimination of exactly half of the variables, as in Eq. (14), is also related to some monotonicity property. For example, for the real space decimation renormalization transformation in spin systems, when on each step of renormalization we divide the lattice into two identical sublattices and half of all spins are summed away [15], the entropy per lattice site was shown to grow monotonically [13].…”