Percolation theory is of interest in problems of phase transitions in condensed matter physics, and in biology and chemistry. More recently, concepts of percolation theory have been invoked in studies of color deconfinement at high temperatures in Quantum Chromodynamics. In the present paper we briefly review the basic concept of percolation theory, exemplify its application to the Ising model, and present the arguments for a possible relevance of percolation theory to the problem of color deconfinement.

Many food products are sensitive to oxygen which can promote rancidity, aerobic microorganism growth, browning, vitamin degradation, flavor loss, and other undesirable reactions. Active packaging concepts have been successfully applied to delay these deteriorative phenomena. In particular, oxygen scavengers help limit O2 ingress into packaging thus extending product shelf-life and thereby reducing food waste. Besides oxygen barrier polymers, other composite materials have been developed to reduce the overall O2 permeability into the package by adding oxygen scavengers or barrier fillers into the packaging structures. Typical thermoplastic food packaging barrier properties are summarized, including commercially available ones, and reported in the patent literature. Furthermore, various mathematical models have been developed to predict the oxygen transport behavior in active packaging to simulate oxygen transport in polymeric and composite materials. Such models are helpful to provide a better understanding of the mass transport fundamentals involved and shorten product development cycle time and cost. This review provides an overview of various oxygen absorber systems used in active food packaging. It also summarizes the mathematical models that simulate oxygen transport and absorption in different polymer films. Finally, these models can be adapted for other polymers and O2 scavengers to simulate O2 transfer on active polymeric materials. Simulation tools help the fundamental understanding of the O2 transport behavior and facilitate to developing novel active packaging systems that extend food shelf-life and enhance their safety.

We consider the critical short-time evolution of magnetic and droplet-percolation order parameters for the Ising model in two and three dimensions, through Monte Carlo simulations with the ͑local͒ heat-bath method. We find qualitatively different dynamic behaviors for the two types of order parameters. More precisely, we find that the percolation order parameter does not have a power-law behavior as encountered for the magnetization, but develops a scale ͑related to the relaxation time to equilibrium͒ in the Monte Carlo time. We argue that this difference is due to the difficulty in forming large clusters at the early stages of the evolution. Our results show that, although the descriptions in terms of magnetic and percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interpret percolation observables at short times. In particular, this concerns the attempts to describe the dynamics of the deconfinement phase transition in QCD using cluster observables.

We present results of our numerical study of the critical dynamics of percolation observables for the twodimensional Ising model. We consider the (Monte Carlo) short-time evolution of the system with small initial magnetization and heat-bath dynamics. We find qualitatively different dynamic behaviors for the magnetization M and for Ω, the so-called strength of the percolating cluster, which is the order parameter of the percolation transition. More precisely, we obtain a (leading) exponential form for Ω as a function of the Monte Carlo time t, to be compared with the power-law increase encountered for M at short times. Our results suggest that, although the descriptions in terms of magnetic or percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interpret percolation observables at short times.

We present preliminary results of our numerical study of the critical dynamics of percolation observables for the two-dimensional Ising model. We consider the (Monte-Carlo) short-time evolution of the system obtained with a local heat-bath method and with the global Swendsen-Wang algorithm. In both cases, we find qualitatively different dynamic behaviors for the magnetization and Ω, the order parameter of the percolation transition. This may have implications for the recent attempts to describe the dynamics of the QCD phase transition using cluster observables.