Both the deterministic and stochastic sandpile models are studied on the percolation backbone, a random fractal, generated on a square lattice in $2$-dimensions. In spite of the underline random structure of the backbone, the deterministic Bak Tang W
iesenfeld (BTW) model preserves its positive time auto-correlation and multifractal behaviour due to its complete toppling balance, whereas the critical properties of the stochastic sandpile model (SSM) still exhibits finite size scaling (FSS) as it exhibits on the regular lattices. Analysing the topography of the avalanches, various scaling relations are developed. While for the SSM, the extended set of critical exponents obtained is found to obey various the scaling relation in terms of the fractal dimension $d_f^B$ of the backbone, whereas the deterministic BTW model, on the other hand, does not. As the critical exponents of the SSM defined on the backbone are related to $d_f^B$, the backbone fractal dimension, they are found to be entirely different from those of the SSM defined on the regular lattice as well as on other deterministic fractals. The SSM on the percolation backbone is found to obey FSS but belongs to a new stochastic universality class.
A dissipative stochastic sandpile model is constructed on one and two dimensional small-world networks with different shortcut densities $phi$ where $phi=0$ and $1$ represent a regular lattice and a random network respectively. In the small-world reg
ime ($2^{-12} le phi le 0.1$), the critical behaviour of the model is explored studying different geometrical properties of the avalanches as a function of avalanche size $s$. For both the dimensions, three regions of $s$, separated by two crossover sizes $s_1$ and $s_2$ ($s_1<s_2$), are identified analyzing the scaling behaviour of average height and area of the toppling surface associated with an avalanche. It is found that avalanches of size $s<s_1$ are compact and follow Manna scaling on the regular lattice whereas the avalanches with size $s>s_1$ are sparse as they are on network and follow mean-field scaling. Coexistence of different scaling forms in the small-world regime leads to violation of usual finite-size scaling, in contrary to the fact that the model follows the same on the regular lattice as well as on the random network independently. Simultaneous appearance of multiple scaling forms are characterized by developing a coexistence scaling theory. As SWN evolves from regular lattice to random network, a crossover from diffusive to super-diffusive nature of sand transport is observed and scaling forms of such crossover is developed and verified.
A two state sandpile model with preferential sand distribution is developed and studied numerically on scale free networks with power-law degree ($k$) distribution, {em i.e.}: $P_ksim k^{-alpha}$. In this model, upon toppling of a critical node sand
grains are given one to each of the neighbouring nodes with highest and lowest degrees instead of two randomly selected neighbouring nodes as in a stochastic sandpile model. The critical behaviour of the model is determined by characterizing various avalanche properties at the steady state varying the network structure from scale free to random, tuning $alpha$ from $2$ to $5$. The model exhibits mean field scaling on the random networks, $alpha>4$. However, in the scale free regime, $2<alpha<4$, the scaling behaviour of the model not only deviates from the mean-field scaling but also the exponents describing the scaling behaviour are found to decrease continuously as $alpha$ decreases. In this regime, the critical exponents of the present model are found to be different from those of the two state stochastic sandpile model on similar networks. The preferential sand distribution thus has non-trivial effects on the sandpile dynamics which leads the model to a new universality class.
A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and the clusters
are grown from these nucleation centers with a tunable growth probability g. As the growth probability g is varied from 0 to 1 two distinct regimes are found to occur. For gle 0.5, the model exhibits continuous percolation transitions as ordinary percolation whereas for gge 0.8 the model exhibits discontinuous percolation transitions. The discontinuous transition is characterized by discontinuous jump in the order parameter, compact spanning cluster and absence of power law scaling of cluster size distribution. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously till the crossover from continuous to discontinuous transition is completed.
A dissipative stochastic sandpile model is constructed and studied on small world networks in one and two dimensions with different shortcut densities $phi$, where $phi=0$ represents regular lattice and $phi=1$ represents random network. The effect o
f dimension, network topology and specific dissipation mode (bulk or boundary) on the the steady state critical properties of non-dissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and non-dissipative avalanches display stochastic scaling at $phi=0$ and mean-field scaling at $phi=1$, the dissipative avalanches display non trivial critical properties at $phi=0$ and $1$ in both one and two dimensions. In the small world regime ($2^{-12} le phi le 0.1$), the size distributions of different types of avalanches are found to exhibit more than one power law scaling with different scaling exponents around a crossover toppling size $s_c$. Stochastic scaling is found to occur for $s<s_c$ and the mean-field scaling is found to occur for $s>s_c$. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.
A two parameter percolation model with nucleation and growth of finite clusters is developed taking the initial seed concentration rho and a growth parameter g as two tunable parameters. Percolation transition is determined by the final static config
uration of spanning clusters. A finite size scaling theory for such transition is developed and numerically verified. The scaling functions are found to depend on both g and rho. The singularities at the critical growth probability gc of a given rho are described by appropriate critical exponents. The values of the critical exponents are found to be same as that of the original percolation at all values of rho at the respective gc . The model then belongs to the same universality class of percolation for the whole range of rho.
Discontinuous transition is observed in the equilibrium cluster properties of a percolation model with suppressed cluster growth as the growth parameter g0 is tuned to the critical threshold at sufficiently low initial seed concentration rho in contr
ast to the previously reported results on non- equilibrium growth models. In the present model, the growth process follows all the criteria of the original percolation model except continuously updated occupation probability of the lattice sites that suppresses the growth of a cluster according to its size. As rho varied from higher values to smaller values, a line of continuous transition points encounters a coexistence region of spanning and non- spanning large clusters. At sufficiently small values of rho (less equal 0.05), the growth parameter g0 exceeds the usual percolation threshold and generates compact spanning clusters leading to discontinuous transitions.
In the rotational sandpile model, either the clockwise or the anti-clockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class
. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anti-clockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behaviour of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anti-clockwise (or clockwise) rotational toppling rule. As the anti-clockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behaviour of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the non-existence of the Manna class.
A dissipative sandpile model (DSM) is constructed and studied on small world networks (SWN). SWNs are generated adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities $phi$. Three differen
t regimes are identified as regular lattice (RL) for $philesssim 2^{-12}$, SWN for $2^{-12}<phi< 0.1$ and random network (RN) for $phige 0.1$. In the RL regime, the sandpile dynamics is characterized by usual Bak, Tang, Weisenfeld (BTW) type correlated scaling whereas in the RN regime it is characterized by the mean field (MF) scaling. On SWN, both the scaling behaviors are found to coexist. Small compact avalanches below certain characteristic size $s_c$ are found to belong to the BTW universality class whereas large, sparse avalanches above $s_c$ are found to belong to the MF universality class. A scaling theory for the coexistence of two scaling forms on SWN is developed and numerically verified. Though finite size scaling (FSS) is not valid for DSM on RL as well as on SWN, it is found to be valid on RN for the same model. FSS on RN is appeared to be an outcome of super diffusive sand transport and uncorrelated toppling waves.
Self-organized criticality is characterized by power law correlations in the non-equilibrium steady state of externally driven systems. A dynamical system proposed here self-organizes itself to a critical state with no characteristic size at ``dynami
cal equilibrium. The system is a random solid in contact with an aqueous solution and the dynamics is the chemical reaction of corrosion or dissolution of the solid in the solution. The initial difference in chemical potential at the solid-liquid interface provides the driving force. During time evolution, the system undergoes two transitions, roughening and anti-percolation. Finally, the system evolves to a dynamical equilibrium state characterized by constant chemical potential and average cluster size. The cluster size distribution exhibits power law at the final equilibrium state.
