Monte Carlo simulations of bosonic reaction-diffusion systems and comparison to Langevin equation description

Using the Monte Carlo simulation method for bosonic reaction-diffusion systems introduced recently [S.-C. Park, Phys. Rev. E {\bf 72}, 036111 (2005)], one dimensional bosonic models are studied and compared to the corresponding Langevin equations derived from the coherent state path integral formalism. For the single species annihilation model, the exact asymptotic form of the correlation functions is conjectured and the full equivalence of the (discrete variable) master equation and the (continuous variable) Langevin equation is confirmed numerically. We also investigate the cyclically coupled model of bosons which is related to the pair contact process with diffusion (PCPD). From the path integral formalism, Langevin equations which are expected to describe the critical behavior of the PCPD are derived and compared to the Monte Carlo simulations of the discrete model.Comment: Proceedings of the 3rd International Conference NEXT-SigmaPh

δ\delta-exceedance records and random adaptive walks

We study a modified record process where the $k$'th record in a series of independent and identically distributed random variables is defined recursively through the condition $Y_k > Y_{k-1} - \delta_{k-1}$ with a deterministic sequence $\delta_k > 0$ called the handicap. For constant $\delta_k \equiv \delta$ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of $\delta$ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park \textit{et al} 2015 {\it Phys. Rev.} E \textbf{91} 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences $\delta_k$, focusing in particular on the case when $\delta_k$ matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when $\delta_k$ is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where $\delta_k$ corresponds to the change of the deterministic fitness component after $k$ mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.Comment: minor changes. Publishe

Crossover from the pair contact process with diffusion to directed percolation

Crossover behaviors from the pair contact process with diffusion (PCPD) and the driven PCPD (DPCPD) to the directed percolation (DP) are studied in one dimension by introducing a single particle annihilation/branching dynamics. The crossover exponents $\phi$ are estimated numerically as $1/\phi \simeq 0.58\pm0.03$ for the PCPD and $1/\phi \simeq 0.49 \pm 0.02$ for the DPCPD. Nontriviality of the PCPD crossover exponent strongly supports non-DP nature of the PCPD critical scaling, which is further evidenced by the anomalous critical amplitude scaling near the PCPD point. In addition, we find that the DPCPD crossover is consistent with the mean field prediction of the tricritical DP class as expected