Finite size scaling in rotational sandpile models

Singularities of thermodynamic quantities in equilibrium critical phenomena are characterized by well defined critical exponents. The set of critical exponents determine the universality class of a system in the thermodynamic limit. In numerical simulations, the results are very often limited by the finite system size L. There is rounding and shifting of critical singularities in finite systems depending on the ratio of correlation length to the linear dimension L of a system. In order to obtain the behaviour of a macroscopic system in the thermodynamic limit, results of finite systems are often extrapolated using finite-size scaling (FSS) ansatz. Self-organized criticality (SOC) [1] is the non existence of any single characteristic size or time in a non-equilibrium steady state of a class of slowly driven systems without fine tuning of any parameter of the system. There are many physical systems found to exhibit SOC where intermittent bursts of activities occur during non-linear transport of certain physical quantities like energy, stress, mass, etc. These intermittent bursts of activities are called avalanches. In this process, a system evolves into a non equilibrium steady state characterized by power law scaling behavior as observed in equilibrium critical phenomena. The physical properties of such systems at the non equilibrium steady state is then expected to follow the usual FSS ansatz as observed in equilibrium critical phenomena. Surprisingly, the scaling behavior of the Bak, Tang and Wiesenfeld (BTW) sandpile model [2] is found to obey a peculiar multiscaling [3] rather than FSS. This observation then posed a question that what would be the condition to have FSS in a system at out of equilibrium situation. The lattice statistical models under rotational constraint in equilibrium critical phenomena exhibit nontrivial critical behavior and obey usual FSS