On the physical relevance of extremal dynamics

– Taking the Bak-Sneppen evolution model as an example (Phys. Rev. Lett., 71 (1993) 4083), we study extremal dynamics where M > 1 smallest barriers are simultaneously updated as opposed to models in the limit where only the smallest barrier is updated. We investigate the scaling properties of the nearest-neighbour and the random-neighbour model. We demonstrate that the behaviour of models with extremal dynamics in the limit of single update per time step is irrelevant to physical observations. Introduction. – Extremal dynamics is used to model the temporal evolution of many different systems. Invasion percolation is one of the most prominent examples [1] but attempts have been also made to catch some of the salient features of biological evolution in models involving extremal dynamics [2, 3] as well as in some other models related to self-organised criticality [4, 5]. In extremal models, the dynamics consist of a global search for the site in the system with the smallest (or largest) value of the dynamical variable. This site and its neighbours are then updated according to the specific algorithm of the model considered, whereupon the procedure is repeated. To be specific, let us consider the particular case of invasion percolation. As pointed out by Wilkinson and Willemsen, the procedure of updating only the link with smallest resistance corresponds to the limit of zero flux [1]. From this perspective it appears immediately relevant to study the behaviour as one approaches the limit of slow but finite drive, that is, study the system when the sites with the M smallest values of the dynamical variable are updated simultaneously. It has previously been tacitly assumed, that the smallest and the second smallest dynamical variables are well separated, in which case the single update limit is appropriate. We find, however, that the probability density of the separation between the smallest and second smallest dynamical variables has its maximum at zero separation. Thus one will inevitably (∗) E-mail: [email protected] (∗∗) E-mail: [email protected] (∗∗∗) E-mail: [email protected]