Disorder Fields, Phase Transitions, pp. 1–742, Singapore: World
Scientific, 1989.
Most physical systems follow complicated nonlinear equations. For a small number of degrees of freedom, numerical methods may lead to a satisfactory theoretical understanding. In macroscopic many-body systems, however, this number is immensely large and such an approach is hopeless. The best we can achieve is an approximate understanding of the statistical behavior, averaged either with respect to thermal or to quantum mechanical fluctuations. The chance of gaining such an approximate understanding depends crucially on a successful separation of linear and nonlinear excitations. A good example to illustrate this is crystalline material: For zero absolute temperature, the equilibrium configuration consists of a perfect regular array of atoms/ This array is called the ground state of the system. If the crystal is perturbed weakly, the atoms are displaced slightly from their equilibrium position. If released, they begin to vibrate. For small enough displacements these vibrations are goveed by the expansion of the energy up to quadratic terms in the displacements. The resulting equations of motion are all linear and can be diagonalized by a Fourier analysis. The eigenmodes correspond to a decoupled set of independent harmonic oscillators.
Most physical systems follow complicated nonlinear equations. For a small number of degrees of freedom, numerical methods may lead to a satisfactory theoretical understanding. In macroscopic many-body systems, however, this number is immensely large and such an approach is hopeless. The best we can achieve is an approximate understanding of the statistical behavior, averaged either with respect to thermal or to quantum mechanical fluctuations. The chance of gaining such an approximate understanding depends crucially on a successful separation of linear and nonlinear excitations. A good example to illustrate this is crystalline material: For zero absolute temperature, the equilibrium configuration consists of a perfect regular array of atoms/ This array is called the ground state of the system. If the crystal is perturbed weakly, the atoms are displaced slightly from their equilibrium position. If released, they begin to vibrate. For small enough displacements these vibrations are goveed by the expansion of the energy up to quadratic terms in the displacements. The resulting equations of motion are all linear and can be diagonalized by a Fourier analysis. The eigenmodes correspond to a decoupled set of independent harmonic oscillators.