Dynamics and statistical mechanics of non-linear systems

The exact partition function of the sine-Gordon field is calculated. This provides a complete thermal description of the system; as an example we have calculated the thermal densities of the soliton excitations. In the overlayers of adsorbed atoms the coupling of the displacement field of the atoms to the strain imposed on it is shown to preserve the continuous nature of the commensurate-incommensurate transition, with a shift in the transition temperature. The coupling is shown to produce a very inhomogeneous strain. The inclusion of thermal fluctuations in the transition is shown to produce a square-root dependence of the observed misfit parameters on the temperature (pressure). It is also shown that a crystalline order is established in the commensurate phase, but there is no true long-range order in the incommensurate phase. We present a theoretical model which predicts an insulator-to-metal transition in doped trans-polyacetylene. An inhomogeneous charge distribution in the polyacetylene fibrils with a localisation depth of 5 Å gives the experimentally observed critical dopant concentration of y⁎ ≈ 1 %. We present a thorough non-linear analysis of the one-dimensional magnetic materials CsNiF3 and (CH3)4NMnCJ3 (TMMC). The neutron scattering data on CsNiF3 are interpreted as the first observation of a breather soliton. We give an argument which indicates that CsNiF3 is a quantum sine-Gordon system with a substantial quantum correction to the kink mass. The results are extended to TMMC.