We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Its effects on the dynamics of complex networks have hardly been studied. Aimed at filling this gap, we analyze the contact process, i.e., the simplest propagation model, with quenched disorder on complex networks. We find Griffiths phases and other rare-region effects, leading rather generically to anomalously slow (algebraic, logarithmic, …) relaxation, on Erdos-Rényi networks. Similar effects are predicted to exist for other topologies with a finite percolation threshold. More surprisingly, we find that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of topological heterogeneity in networks with finite topological dimension. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks.

PACS 75.10.Jm -Quantized spin models PACS 75.10.Pq -Spin chain models PACS 03.65.Ud -Entanglement and quantum nonlocality Abstract. -We consider two prototypical quantum models, the spin-1/2 XY chain and the quantum Ising chain and study their entanglement entropy, S(ℓ, L), of blocks of ℓ spins in homogeneous or inhomogeneous systems of length L. By using two different approaches, free-fermion techniques and perturbational expansion, an exact relationship between the entropies is revealed. Using this relation we translate known results between the two models and obtain, among others, the additive constant of the entropy of the critical homogeneous quantum Ising chain and the effective central charge of the random XY chain.

We consider the one-dimensional partially asymmetric exclusion process with random hopping rates, in which a fraction of particles (or sites) have a preferential jumping direction against the global drift. In this case, the accumulated distance traveled by the particles, x, scales with the time, t, as x approximately t(1/z), with a dynamical exponent z>0. Using extreme value statistics and an asymptotically exact strong disorder renormalization group method, we exactly calculate z(PW) for particlewise disorder, which is argued to be related as z(SW)=z(PW)/2 for sitewise disorder. In the symmetric case with zero mean drift, the particle diffusion is ultraslow, logarithmic in time.

Networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems. Studying the role of quenched disorder, both intrinsic to nodes and topological, is a key challenge. With this in mind, here we analyze the contact process (i.e., the simplest model for propagation phenomena) with node-dependent infection rates (i.e., intrinsic quenched disorder) on complex networks. We find Griffiths phases and other rare-region effects, leading rather generically to anomalously slow (algebraic, logarithmic, etc.) relaxation, on Erdős-Rényi networks. We predict similar effects to exist for other topologies as long as a nonvanishing percolation threshold exists. More strikingly, we find that Griffiths phases can also emerge--even with constant epidemic rates--as a consequence of mere topological heterogeneity. In particular, we find Griffiths phases in finite-dimensional networks as, for instance, a family of generalized small-world networks. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks, and are relevant for the analysis of both models and empirical data.