β-Peptide foldamer helices with tailored diameters

Ageing without detailed balance in the bosonic contact and pair-contact processes: exact results

“…In the three cases (i) α < α C and d > 2, (ii) α = α C and 2 < d < 4 and (iii) α = α C and d > 4 this leads by eqs. (34,35), respectively, to the upper bounds |C 1 | s ζ−d/2 , s (ζ−1)d/2 and s 2ζ−d/2 which vanish for s large more rapidly than G(0; t, s) ∼ s 1−d/2 , s 0 and s 2−d/2 , respectively and which are derived below. Hence C 1 (t, s, t p ) is irrelevant for the determination of b and the scaling functions and will be dropped in what follows.…”

“…A lattice version (with the 'fermionic' constraint of not more than one particle per site) of the model contains the reactions 2A → ∅ and 2A → 3A together with single-particle diffusion A∅ ↔ ∅A and was first studied numerically in [31]. An intense debate on the universality class of this model followed, see [28], and several mutually exclusive conclusions on the critical behaviour continue to be drawn, see [30,32,33,34,35,36] for recent work. The bosonic pair-contact process has a dynamic exponent z = 2 [26] and is hence distinct from the PCPD where z < 2.…”

“…The most interesting cases are d > 2 and α ≤ α C , which we will treat first. The asymptotic expression for F (0, t) is of the form F age (0, t) = Aρ 2 0 t ξ , where ξ and the prefactor A can be read of from equations (34)- (35).…”

Cluster mean-field approximations with the coherent-anomaly-method analysis for the driven pair contact process with diffusion

“…However, the accurate measurement of critical indices is only limited to rather simple DP systems. More complex critical behaviors like in the PC and the PCPD classes could not have been probed with a reasonable accuracy as yet by the CAM analysis [21,22].…”