The κ-Minkoswki space-time provides a quantum noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional κ-Minkoswki space-time using the more general framework of Lorentzian noncommutative geometry. We show that this notion allows specific causal relations, but limited by a general constraint which is a quantum generalization of the traditional speed of light limit.
We consider a system of aggregated clusters of particles, subjected to coagulation and fragmentation processes with mass dependent rates. Each monomer particle can aggregate with larger clusters, and each cluster can fragment into individual monomers with a rate directly proportional to the aggregation rate. The dynamics of the cluster densities is governed by a set of Smoluchowski equations, and we consider the addition of a source of monomers at constant rate. The whole dynamics can be reduced to solving a unique non-linear differential equation which displays self-oscillations in a specific range of parameters, and for a number of distinct clusters in the system large enough. This collective phenomenon is due to the presence of a fluctuating damping coefficient and is closely related to the Liénard self-oscillation mechanism observed in a more general class of physical systems such as the van der Pol oscillator.
We consider a Brownian particle with diffusion coefficient D in a d -dimensional ball of radius R with reflecting boundaries. We study the maximum M x ( t ) of the trajectory of the particle along the x -direction at time t . In the long time limit, the maximum converges to the radius of the ball M x ( t ) → R for t → ∞. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations Δ( t ) = [ R − M x ( t )]/ R in the limit of large t in all dimensions. We find that the distribution of Δ( t ) exhibits a rich variety of behaviors depending on the dimension d . These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in d = 2 to study the convex hull of the trajectory of the particle in a disk of radius R with reflecting boundaries. We find that the mean perimeter ⟨ L ( t )⟩ of the convex hull exhibits a slow convergence towards the perimeter of the circle 2 πR with a stretched exponential decay 2 π R − ⟨ L ( t ) ⟩ ∝ R ( D t ) 1 / 4 e − 2 2 D t / R . Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations.
We study the entanglement dynamics of thermofield double (TFD) states in integrable spin chains and quantum field theories. We show that, for a natural choice of the Hamiltonian eigenbasis, the TFD evolution may be interpreted as a quantum quench from an initial state which is low-entangled in the real-space representation and displays a simple quasiparticle structure. Based on a semiclassical picture analogous to the one developed for standard quantum quenches, we conjecture a formula for the entanglement dynamics, which is valid for both discrete and continuous integrable field theories, and expected to be exact in the scaling limit of large space and time scales. We test our conjecture in two prototypical examples of integrable spin chains, where numerical tests are possible. First, in the XY-model, we compare our predictions with exact results obtained by mapping the system to free fermions, finding excellent agreement. Second, we test our conjecture in the interacting XXZ Heisenberg model, against numerical iTEBD calculations. For the latter, we generally find good agreement, although, for some range of the system parameters and within the accessible simulation times, some small discrepancies are visible, which we attribute to finite-time effects.
It is shown that the partial amplitudes of the pure point part of the diffraction spectrum of an aperiodic Delone point pattern of finite local complexity are linked by a set of linear constraints. These relations can be explicitly derived from the geometry of the prototile space of the underlying tiling.
