Criticality and Strong Intermittency in the Lorentz Channel A. K. Karlis, 1, * F. K. Diakonos, 1,† C. Petri, 2,‡ and P. Schmelcher 2,§ 1 Department of Physics, University of Athens, GR-15771 Athens, Greece 2 Zentrum fu ¨r Optische Quantentechnologien, Universita ¨t Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany (Received 30 December 2011; revised manuscript received 11 July 2012; published 11 September 2012) We demonstrate the emergence of criticality due to power-law cross correlations in an ensemble of noninteracting particles propagating in an infinite Lorentz channel. The origin of these interparticle long- range correlations is the intermittent dynamics associated with the ballistic corridors in the single particle phase space. This behavior persists dynamically, even in the presence of external driving, provided that the billiard’s horizon becomes infinite at certain times. For the driven system, we show that Fermi acceleration permits the synchronization of the particle motion with the periodic appearance of the ballistic corridors. The particle ensemble then acquires characteristics of self-organization as the weight of the phase space regions leading to critical behavior increases with time. DOI: 10.1103/PhysRevLett.109.110601 PACS numbers: 05.60.Cd, 05.45.Ac, 05.45.Pq, 05.65.+b The Lorentz gas (LG) [1] acts in the theory of dynamical systems as a paradigm allowing us to address fundamental issues of statistical mechanics, for instance, ergodicity and mixing [2–4], as well as transport processes, such as diffu- sion in the configuration space [5–9]. The static periodic LG comprises a regular lattice of circular fixed scatterers and an ensemble of noninteracting particles traveling freely be- tween collisions and scattering elastically off the circular obstacles. The transport properties of such a system are determined by the billiard’s geometry, that is the specific lattice symmetry and the lattice constant. If the maximum free path length is not bounded from above, then the setup possesses a so-called infinite horizon (IH) and the diffusion in configuration space is anomalous [10,11]. For a more compact packing of the scatterers, i.e., finite horizon (FH), arbitrarily long flights are not possible and the system exhibits normal diffusion [6]. From a dynamical point of view, if the system has FH, then it is fully hyperbolic. However, in the case of the IH geometry, the chaoticity weakens and an ordered portion of phase space emerges, due to the existence of ballistic corridors, leading to an intermittent behavior [12]. Alternatively, chaoticity can be reduced also by using different scatterer geometry as in Ref. [13] where algebraically decaying autocorrelations have been observed, although the corresponding horizon is finite. Time-dependent generalizations of the original periodic Lorentz gas model have been introduced, in which the scatterers are allowed to oscillate [14–16], rendering the study of diffusion in momentum space possible. This pro- cess is intimately linked to Fermi acceleration [17], which is considered a fundamental acceleration mechanism in many areas of physics [18]. The mechanism consists in the indefinite increase of the mean energy of particles as a result of random collisions with moving scatterers. In this Letter, we show the emergence of power-law (critical) cross correlations between noninteracting particles propagating in the LG with IH in a channel geometry. To reveal these cross correlations, a spatially coarse grained description of the dynamics is employed. The dynamically infinite horizon (DIH) is introduced as a property of driven extended billiards for which ballistic corridors open up and close periodically in time, i.e., exist only for certain time intervals. The development of Fermi acceleration then en- ables the particles to synchronize their motion with the periodic appearance of the ballistic corridors, such that they can perform free flights of arbitrary length, which, in turn, gives rise to intermittent dynamics and the appearance of critical correlations. In this sense, it is shown that Fermi acceleration can act as an effective driving force to steer an ensemble of propagating particles towards a critical state, imparting to the system’s dynamics characteristics of self- organized criticality. This work adds to the recent intensive efforts to clarify the interrelation between intermittency, criticality, and self-organization [19–22]. The device investigated is an infinite strip of a periodic LG, known as Lorentz channel (LC) [23–25], consisting in hard circular scatterers placed on a semi-infinite triangular lattice as shown in Fig. 1. The central disks of the device with radii b can oscillate with amplitude A and angular frequency !. Without loss of generality, provided that the flat segments between the static discs are small enough to prohibit bouncing orbits in the y direction [26], we choose the radii of the static semi-circles a=w ¼ 0:48, where w is the lattice constant. In the following, distance and time is measured in units of w and 1=!, respectively. Statistical aspects of the transport properties of the static Lorentz gas, have been modeled by a suitable hopping process, either as a two-dimensional completely uncorre- lated random walk [5] or by taking into account correla- tions between jumps [8,27,28]. Inspired by these studies, we introduce here, a coarse grained description of the dynamics in the driven LC by integrating out the motion of the particle inside a unit cell [marked by dashed blue PRL 109, 110601 (2012) PHYSICAL REVIEW LETTERS week ending 14 SEPTEMBER 2012 0031-9007= 12=109(11)=110601(5) 110601-1 Ó 2012 American Physical Society