Network Evolution Induced by the Dynamical Rules of Two Populations

We study the dynamical properties of a finite dynamical network composed of two interacting populations, namely; extrovert ($a$) and introvert ($b$). In our model, each group is characterized by its size ($N_a$ and $N_b$) and preferred degree ($\kappa_a$ and $\kappa_b\ll\kappa_a$). The network dynamics is governed by the competing microscopic rules of each population that consist of the creation and destruction of links. Starting from an unconnected network, we give a detailed analysis of the mean field approach which is compared to Monte Carlo simulation data. The time evolution of the restricted degrees \moyenne{k_{bb}} and \moyenne{k_{ab}} presents three time regimes and a non monotonic behavior well captured by our theory. Surprisingly, when the population size are equal $N_a=N_b$, the ratio of the restricted degree \theta_0=\moyenne{k_{ab}}/\moyenne{k_{bb}} appears to be an integer in the asymptotic limits of the three time regimes. For early times (defined by $t<t_1=\kappa_b$) the total number of links presents a linear evolution, where the two populations are indistinguishable and where $\theta_0=1$. Interestingly, in the intermediate time regime (defined for $t_1<t<t_2\propto\kappa_a$ and for which $\theta_0=5$), the system reaches a transient stationary state, where the number of contacts among introverts remains constant while the number of connections is increasing linearly in the extrovert population. Finally, due to the competing dynamics, the network presents a frustrated stationary state characterized by a ratio $\theta_0=3$.Comment: 21 pages, 6 figure

Fourier's law on a one-dimensional optical random lattice

We study the transport properties of a one-dimensional hard-core bosonic lattice gas coupled to two particle reservoirs at different chemical potentials which generate a current flow through the system. In particular, the influence of random fluctuations of the underlying lattice on the stationary-state properties is investigated. We show analytically that the steady-state density presents a linear profile. The local steady-state current obeys the Fourier law $j=-\kappa(\tau)\nabla n$ where $\tau$ is a typical timescale of the lattice fluctuations and $\nabla n$ the density gradient imposed by the reservoirs.Comment: 9 pages, 2 figure

Relaxation in the XX quantum chain

We present the results obtained on the magnetisation relaxation properties of an XX quantum chain in a transverse magnetic field. We first consider an initial thermal kink-like state where half of the chain is initially thermalized at a very high temperature $T_b$ while the remaining half, called the system, is put at a lower temperature $T_s$. From this initial state, we derive analytically the Green function associated to the dynamical behaviour of the transverse magnetisation. Depending on the strength of the magnetic field and on the temperature of the system, different regimes are obtained for the magnetic relaxation. In particular, with an initial droplet-like state, that is a cold subsystem of finite size in contact at both ends with an infinite temperature environnement, we derive analytically the behaviour of the time-dependent system magnetisation

Dynamical phase transition of a 1D transport process including death

Motivated by biological aspects related to fungus growth, we consider the competition of growth and corrosion. We study a modification of the totally asymmetric exclusion process, including the probabilities of injection $\alpha$ and death of the last particle $\delta$. The system presents a phase transition at $\delta_c(\alpha)$, where the average position of the last particle $$ grows as $\sqrt{t}$. For $\delta>\delta_c$, a non equilibrium stationary state exists while for $\delta<\delta_c$ the asymptotic state presents a low density and max current phases. We discuss the scaling of the density and current profiles for parallel and sequential updates.Comment: 4 pages, 5 figure

Entanglement evolution after connecting finite to infinite quantum chains

We study zero-temperature XX chains and transverse Ising chains and join an initially separate finite piece on one or on both sides to an infinite remainder. In both critical and non-critical systems we find a typical increase of the entanglement entropy after the quench, followed by a slow decay towards the value of the homogeneous chain. In the critical case, the predictions of conformal field theory are verified for the first phase of the evolution, while at late times a step structure can be observed.Comment: 15 pages, 11 figure

Quantum repeated interactions and the chaos game

Inspired by the algorithm of Barnsley's chaos game, we construct an open quantum system model based on the repeated interaction process. We shown that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system's evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time $t$) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.Comment: 21 pages, 8 figue

A matrix product solution for a nonequilibrium steady state of an XX chain

A one dimensional XX spin chain of finite length coupled to reservoirs at both ends is solved exactly in terms of a matrix product state ansatz. An explicit representation of matrices of fixed dimension 4 independent of the chain length is found. Expectations of all observables are evaluated, showing that all connected correlations, apart from nearest neighbor z-z, are zero.Comment: 11 page

Critical dynamics in trapped particle systems

We discuss the effects of a trapping space-dependent potential on the critical dynamics of lattice gas models. Scaling arguments provide a dynamic trap-size scaling framework to describe how critical dynamics develops in the large trap-size limit. We present numerical results for the relaxational dynamics of a two-dimensional lattice gas (Ising) model in the presence of a harmonic trap, which support the dynamic trap-size scaling scenario.Comment: 7 page