Reaction-diffusion systems

Diffusion is usually caused by thermal or other stationary fluctuations of the system in which the diffusive particle is embedded. However, also non-stationary, relaxational processes may lead to diffusion processes by way of time-dependent momentum transfer or by catalysing the random motion of the particle. In such a case the dynamics will generically be non-Markovian and lead to anomalous diffusive behavior. This is indeed the case for random hopping catalyzed or triggered by the presence of other mobile particles (e.g. excitons) which themselves undergo reaction-diffusion dynamics [49]. Currently we are investigating the possibility of such processes in the glass transition.

Because of the underlying diffusive dynamics, density correlations build up by the particle reactions are quickly washed out in sufficiently high-dimensional systems. Hence, above a certain critical dimension (where fortunately quite often ) mean-field calculations are a perfectly adequate tool for the study of reaction-diffusion dynamics. By the same token, however, such a rate equation approach fails in low-dimensional systems where diffusive mixing is inefficient and long-range correlations persist for all times. The result of these correlations are anomalous critical exponents which are well-known e.g. in the reaction-diffusion dynamics of laser-induced excitons on the one-dimensional backbone of polymer chains. An exactly solvable toy model for this dynamics [59] allows for a verification and a very detailed understanding of the old phenomenological theories of Smoluchowski who predicted the corresponding anomalous dynamics almost 100 years ago.

Whenever random motion results from thermally activated hopping it is natural to expect anomalous subdiffusive behaviour caused by an environment with a random energy landscape to lead to anomalous critical exponents in a reaction-diffusion system. This can be proved rigorously [34] and studied in detail in diffusion-limited annihilation in one space dimension, where both inefficient diffusive mixing and quenched random hopping rates lead to anomalies [36]. However, also if single-particle hopping is ordered, our work on partially solvable reaction- diffusion models (in any space dimension) has shown that quenched disorder in the reaction rates may lead to disordered collective dynamics [21,59]. To which extent anomalous dynamics may ensue is an open question at this point.

Among the most multifaceted problems in the study of nonequilibrium relaxation of many-body systems are the extent to which the time evolution depends on the initial conditions (i.e., the question of ergodicity) and the closely related phenomenon of aging. In most general terms aging simply refers to the process of slow relaxation, where `slow' is somewhat vaguely understood with reference to typical observation times. It typically occurs in systems which are dominated by quenched disorder as amorphous polymer systems and glasses, spin glasses and random magnets, but also ordered structures with stochastic dynamics may exhibit aging. To sharpen the notion we adopt the more restrictive point of view where aging refers to the phenomenon of increasing relaxation times, i.e., the longer one waits (waiting time from the initial state), the slower the relaxation becomes (measured at time after ). This definition includes the special case where the autocorrelation function of some measurable quantity depends the ratio of the two times which is in contrast to stationary processes for which the autocorrelation function would depend only on the time difference . Physically, the dependence of correlations on the ratio of the times implies a scaling property: a young system and an old system evolve in the same way if their evolution times are measured in units of their respective ages. Hence one would expect aging to be a generic feature of dynamically scale-invariant systems. Indeed, one might suspect that already algebraic relaxation to stationarity of order parameter is sufficient for aging to occur. Both simple reaction-diffusion models with fast diffusive dynamics (i.e., lacking spatial correlations) [35] and strongly correlated quantum spin chains [43] have been shown to confirm this conjecture, thus strengthening the viewpoint that aging and power law relaxation are intimately linked both in classical and genuinely quantum mechanical systems.