%0 Book Section %1 statphys23_0675 %A Costa, N.V. Da %A Fulco, U.L. %A Lyra, M.L. %A Gléria, I.M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K critical dynamic irreversible nonequilibrium phenomena point statphys23 thermodynamics topic-2 %T Criticality of a contact process with coupled diffusive and non-diffusive fields %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=675 %X We consider a model of interacting species. One of the species corresponds to the individuals of a population that can be in either the inactive (non-infected) state $P_i$ or in the active (infected) state $P_a$. They are free to diffuse in the lattice. The sites on the lattice also assume two possible states: active or inactive. The presence of at least one infected individual in a given site turns it infective. There is no direct contamination from individual to individual or from site to site. Although, an inactive individual becomes immediately active whenever it occupies the same site of another active individual, in most of the cases it is activated when visiting a site that was contaminated at a previous time. We consider a finite lifetime for the active state for both the individuals and the sites. Therefore they become inactive with a constant rate. The reaction-rate equations that capture the essence of the model are given by: eqnarray eq1 & & P_a+V_i P_a+V_a ~~at rate ~~ k_4 \\ & & P_i+V_a P_a+V_a ~~at rate ~~ k_2 \\ & & P_a P_i ~~ at rate ~~ k_3 \\ & & V_a V_i ~~ at rate ~~ k_1 eqnarray where $V_i$ ($V_a$) stands for inactive (active) vector sites. The constants $k_1$ and $k_3$ are the recovering rates, while $k_2$ and $k_4$ represent the interaction between the populations and promote the possible survival of the density of active particles. We will concentrate in the critical behavior of the present model in $1D$ where fluctuations are predominant. In our simulations the initial state of all sites is the inactive state. We randomly distribute a given number of individuals in the lattice. Their states are also randomly chosen. The dynamics of the model takes place in the following way: the sites with at least one infected individual become itself infective. Then all non-infected individuals in an active site become themselves infected. The activity removal step takes place by allowing all infected individuals and infective sites to become inactive with their given decay rates. Afterwards, all the individuals diffuse in the lattice with equal probability in both directions. One lattice sweep is taken as the unit of time. We measure the average fraction of active individuals in the stationary state as a function of the total population density. Whenever the system becomes trapped in the vacuum state we replace a randomly chosen non-infected individual by an infected one. In such a way the vacuum state is replaced by a reflecting boundary. Figure 1 presents the results for various lattice sizes. The observed growth at the vicinity of $0$ reflects the fact that the very few particles present in this limit are constantly changing from the active to the inactive state and therefore $\Psi(0)1/2$. At finite but low densities the state of the system frequently visits the reflective vacuum state. As $Lınfty$ we clearly note a transition from the vacuum to the active state as the density $\rho$ increases. To determine the critical density, we measured the relative fluctuation on the number of infected individuals, which is expected to be independent of the lattice size at the critical state. We shows $U_L()$ for various lattice sizes. Notice that the crossing points of all curves depict almost no spread which indicates that corrections to scaling are small for the chain sizes we simulated. From these crossing points we can estimate $\rho_c$ to lie approximately in the range $0.81,0.82$. This procedure does not allow for a finer tuning of the critical density due to its discrete nature in finite chains and to the inherent difficult in estimating precisely any correction to scaling. The critical relative fluctuation was found to be $U_c=0.15$ which is below the reported value for the one-dimensional contact process but within the error bar of the recently estimated value for the conserved directed-percolation universality class. Our data at the border of the critical region were fitted to the above power laws and presented to estimate the exponents $\beta/\nu$ and $1/\nu$, respectively. eqnarray eq4 & & \beta\nu=0.28,\; \nu=1.83\; for\; \rho_c=0.81, \\ & & \beta\nu=0.23,\; \nu=1.80\; for\; \rho_c=0.82. eqnarray We observe that, although the value for $\beta\nu$ is in agreement with the directed percolation universality class ($\beta\nu=0.252$), the same is not true for $\nu$ ($\nu=1.097$ for directed percolation). The exponents $\beta$ and $\nu$ were used to collapse the data for different lattice sizes and for both values of $\rho_c$.

@incollection{statphys23_0675, abstract = {We consider a model of interacting species. One of the species corresponds to the individuals of a population that can be in either the inactive (non-infected) state $P_i$ or in the active (infected) state $P_a$. They are free to diffuse in the lattice. The sites on the lattice also assume two possible states: active or inactive. The presence of at least one infected individual in a given site turns it infective. There is no direct contamination from individual to individual or from site to site. Although, an inactive individual becomes immediately active whenever it occupies the same site of another active individual, in most of the cases it is activated when visiting a site that was contaminated at a previous time. We consider a finite lifetime for the active state for both the individuals and the sites. Therefore they become inactive with a constant rate. The reaction-rate equations that capture the essence of the model are given by: \begin{eqnarray} \label{eq1} & & P_a+V_i \rightarrow P_a+V_a ~~\mbox{at rate} ~~ k_4 \nonumber \\ & & P_i+V_a \rightarrow P_a+V_a ~~\mbox{at rate} ~~ k_2 \nonumber \\ & & P_a \rightarrow P_i ~~ \mbox{at rate} ~~ k_3 \nonumber \\ & & V_a \rightarrow V_i ~~ \mbox{at rate} ~~ k_1 \end{eqnarray} where $V_i$ ($V_a$) stands for inactive (active) vector sites. The constants $k_1$ and $k_3$ are the recovering rates, while $k_2$ and $k_4$ represent the interaction between the populations and promote the possible survival of the density of active particles. We will concentrate in the critical behavior of the present model in $1D$ where fluctuations are predominant. In our simulations the initial state of all sites is the inactive state. We randomly distribute a given number of individuals in the lattice. Their states are also randomly chosen. The dynamics of the model takes place in the following way: the sites with at least one infected individual become itself infective. Then all non-infected individuals in an active site become themselves infected. The activity removal step takes place by allowing all infected individuals and infective sites to become inactive with their given decay rates. Afterwards, all the individuals diffuse in the lattice with equal probability in both directions. One lattice sweep is taken as the unit of time. We measure the average fraction of active individuals in the stationary state as a function of the total population density. Whenever the system becomes trapped in the vacuum state we replace a randomly chosen non-infected individual by an infected one. In such a way the vacuum state is replaced by a reflecting boundary. Figure 1 presents the results for various lattice sizes. The observed growth at the vicinity of $\rho \rightarrow 0$ reflects the fact that the very few particles present in this limit are constantly changing from the active to the inactive state and therefore $\Psi(\rho \rightarrow 0)\rightarrow 1/2$. At finite but low densities the state of the system frequently visits the reflective vacuum state. As $L\rightarrow \infty$ we clearly note a transition from the vacuum to the active state as the density $\rho$ increases. To determine the critical density, we measured the relative fluctuation on the number of infected individuals, which is expected to be independent of the lattice size at the critical state. We shows $U_L(\rho )$ for various lattice sizes. Notice that the crossing points of all curves depict almost no spread which indicates that corrections to scaling are small for the chain sizes we simulated. From these crossing points we can estimate $\rho_c$ to lie approximately in the range $[0.81,0.82]$. This procedure does not allow for a finer tuning of the critical density due to its discrete nature in finite chains and to the inherent difficult in estimating precisely any correction to scaling. The critical relative fluctuation was found to be $U_c=0.15$ which is below the reported value for the one-dimensional contact process but within the error bar of the recently estimated value for the conserved directed-percolation universality class. Our data at the border of the critical region were fitted to the above power laws and presented to estimate the exponents $\beta/\nu$ and $1/\nu$, respectively. \begin{eqnarray} \label{eq4} & & \frac{\beta}{\nu}=0.28,\; \nu=1.83\; \mbox{for}\; \rho_c=0.81, \nonumber \\ & & \frac{\beta}{\nu}=0.23,\; \nu=1.80\; \mbox{for}\; \rho_c=0.82. \end{eqnarray} We observe that, although the value for $\frac{\beta}{\nu}$ is in agreement with the directed percolation universality class ($\frac{\beta}{\nu}=0.252$), the same is not true for $\nu$ ($\nu=1.097$ for directed percolation). The exponents $\beta$ and $\nu$ were used to collapse the data for different lattice sizes and for both values of $\rho_c$.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Costa, N.V. Da and Fulco, U.L. and Lyra, M.L. and Gléria, I.M.}, biburl = {https://www.bibsonomy.org/bibtex/202130652085b35a8e5a7f50bd9fc6024/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {d41ce888e1cc4f78d8fd9150ff7e3a00}, intrahash = {02130652085b35a8e5a7f50bd9fc6024}, keywords = {critical dynamic irreversible nonequilibrium phenomena point statphys23 thermodynamics topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:26.000+0200}, title = {Criticality of a contact process with coupled diffusive and non-diffusive fields}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=675}, year = 2007 }