Reaction-Diffusion Processes, Critical Dynamics, and Quantum Chains

Abstract

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and qdeformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.

Monte Carlo simulations of surface reactions

“…(32). From this we see that for (---~0 the avalanche size diverges, which leads to global oscillations.…”

“…That this is indeed the case can be seen with the help of Eq. (32). Mean field predicts that 0BOX(, SO that according to Eq.…”

“…According to some authors this is indeed the case [47,61 ]. However, there are indications that even in the limit of infinitely fast diffusion the mean-field approach is not quantitatively correct [32]. Monte Carlo results show that the first-order transition should shift to higher values of y and the second-order transition to lower values of y when diffusion is included.…”

“…Unimolecular reactions, like monatomic desorption and adsorption, can often be treated exactly using simple analytical methods [25,311. Exact results can also be obtained for some bimolecular reactions, e.g., 1D models of annihilation and coagulation [32]. Here we will discuss the ZiffGulari-Barshad model and the Lotka model.…”

Lattice kinetics of diffusion-limited coalescence and annihilation with sources

“…The solution of such equations usually poses great mathematical difficulties; however, the simple topology of the one-dimensional lattice frequently allows derivation of exact solutions [1][2][3][4][5][6]. These results may be used to test the validity of various approximations.…”