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Coherence Properties of Coupled Chaotic Map Lattices Vol. 120 (2011) ACTA PHYSICA POLONICA A No. 6-A Proceedings of the 5th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, Poland, May 2022, 2011 Coherence Properties of Coupled Chaotic Map Lattices M. Janowicza;b;∗ and A. Orªowskia;b aChair of Computer Science, Warsaw University of Life Sciences, Nowoursynowska 159, 02-766 Warsaw, Poland bInstitute of Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland Strong global correlations in the systems of coupled chaotic map lattices based on a modied logistic map are investigated. It is shown that, in the parameter range close to the edge of chaos as dened for an individual map, the systems exhibit o-diagonal long-range order and single-particle reduced density matrices dened in a natural way possess one strongly dominant eigenvalue. In addition, pattern formation [13] in the above systems has been investigated. PACS: 05.45.Ra, 45.70.Qj, 03.75.Hh, 03.75.Nt 1. Introduction the condensate-like behavior (or, more generally, strong global correlations and coherence) of some other, similar Coupled map lattices (CMLs) [1, 2], that is systems systems. It is the purpose of this work to present some of coupled maps which simulate spatially extended non- results for CMLs based on modied logistic map such -linear systems, have long become a useful tool to investi- that every individual map can take negative values. In gate spatiotemporal chaos and other non-linear phenom- addition, we provide some results concerning pattern for- ena [36]. Several CMLs have found interesting appli- mation in the above CMLs for we believe that the subject cations in physical modeling. One should mention here is very far from being exhausted in spite of the existence CMLs developed to describe the RayleighBenard con- of already classic papers on the subject by Kapral and vection [7], dynamics of boiling [8, 9], formation and dy- Kaneko [5, 6, 11]. namics of clouds [10], crystal growth processes and hy- drodynamics of two-dimensional ows [11]. Our analysis is in the spirit of classical eld theory, es- In our previous work [12] we have argued that CMLs pecially the GrossPitaevskii equation which is very ex- based on the ubiquitous logistic map exhibit properties tensively used in the theory of BoseEinstein condensa- which are characteristic for the BoseEinstein conden- tion [14, 15]. Applications of the classical eld-theoretical sate. We have done this by describing CMLs with the methods in the physics of condensates have been de- help of a variable interpreted as a classical eld dened scribed, e.g., in [1618]. on discrete space-time. This has allowed us to dene the The main body of this work is organized as follows. single-particle reduced density matrix in a natural way. The mathematical model as well as the basic denitions That latter quantity enables one to give precise quantita- of reduced density matrix and reduced wave function are tive meaning to the terms correlations, coherence and introduced in Sect. 2. Section 3 provides a justication long-range order which are often loosely attributed to of our claim that the coupled map lattices based on mod- the spatially extended classical systems. It has turned ied logistic map exhibit properties which are analogous out that CMLs based on the logistic map exhibit, for to those of the BoseEinstein condensates (BEC). The a broad range of parameters, o-diagonal long-range or- description of numerical results concerning pattern for- der. What is more, there exists one dominant eigenvalue mation are contained in Sect. 4, while Sect. 5 comprises of the reduced density matrix as well as a single domi- a few concluding remarks. nant mode in the Fourier transform of the eld describing CML. It is to be noted that the CMLs cannot of course be called condensates, rst of all because they are merely 2. The model somewhat remote models of reality, and not physical sys- tems. In addition, in those models there are no natural rst integrals like the energy or the number of particles. We consider a classical eld (x; y; t) dened on a two- Therefore, we say that CMLs exhibit condensate-like -dimensional spatial lattice. Its evolution in (dimension- behavior. less, discrete) time t is given by the following equation: Needless to say, the subject requires further inves- (x; y; t + 1) = (1 − 4d)f( (x; y; t)) tigations. One natural path to follow is checking for [ + d f( (x + 1; y; t)) + f( (x − 1; y; t)) ] + f( (x; y + 1; t)) + f( (x; y − 1; t)) ; (1) ∗ corresponding author; e-mail: [email protected] where the function f is given by (A-114) Coherence Properties of Coupled Chaotic Map Lattices A-115 f( ) = 1 − a 2; (2) of that density matrix is not obvious. We can use, how- and the parameters and are constant. The set of val- ever, the classical-eld approach to the theory of Bose a d Einstein condensation [17, 21] and dene the quantities: ues taken by is the interval [−1; 1], so that negative values of are allowed unlike in the work [12]. NX−1 0 h 0 i (4) A single map of the form (t + 1) = f( (t)) exhibits ρ¯(x; x ) = (x; y) (x ; y) t ; the accumulation of period-doubling at a = 1:40155 ::: y=0 and and the band merging from period-2 band to a single / X band state at a = 1:542 ::: ρ(x; x0) =ρ ¯(x; x0) ρ¯(x; x) : (5) In the following the coecient a will be called the non- x 0 -linear parameter while the coecient d will be called We shall call the quantity ρ(x; x ) the reduced density the diusion constant. It is assumed that satises matrix of CML. The above denition in terms of an av- the periodic boundary conditions on the borders of sim- eraged quadratic form made of seems quite natural, ulation box. The size of that box is N × N. All our especially because ρ is a real symmetric, positive-denite simulations have been performed with N = 256. matrix with the trace equal to 1. The sharp brackets Let ~ be the two-dimensional discrete Fourier trans- h:::it denote the time averaging form of , 1 XT NX−1 NX−1 h(:::)it = (:::) ; 2π i mx=N 2π i ny=N Ts ~(m; n) = e e (x; y) : (3) t=T −Ts x=0 y=0 where T is the total simulation time and Ts is the averag- Thus, ~ may be interpreted as the momentum repre- ing time. In our numerical experiments T has been equal sentation of the eld . to 3000, and Ts has been chosen to be equal to 1000. Below we investigate the relation between a CML Let W be the largest eigenvalue of ρ. We will say described by Eq. (1) and a BoseEinstein condensate. that CML is in a condensed state if W is signicantly Therefore, let us invoke the basic characteristics of the larger than all other eigenvalues of ρ. If this is the case, latter which are so important that they actually form a the system possesses property (1) of the BoseEinstein part of its modern denition. These are [15, 19, 20]: condensates. Further, we can provide the quantitative meaning to 1. The presence of one eigenvalue of the one-particle the concept of ODLRO by saying that it is present in the reduced density matrix which is much larger than system if all other eigenvalues. ρ(x1 + x; x1 − x) does not go to zero with increasing [20] for any . If 2. The presence of o-diagonal long-range order x x1 (ODLRO). this is the case, the system possesses the basic property (2) of the BoseEinstein condensates. For technical convenience, namely, to avoid dealing The property (1) corresponds to the well-known intu- with too large matrices, the above denition of the re- itive denition of the BoseEinstein condensate. Taking duced density matrix involves not only temporal, but into account that the following decomposition of the one- also spatial averaging over y. Let us notice that we might -particle reduced density matrix ρ(1) has the following equally well consider averaging over x without any qual- decomposition eigenvalues λj and eigenvectors jϕji: X itative change in the results. (1) All the above denitions are modeled after the corre- ρ = λjjϕjihϕjj ; sponding denitions in the non-relativistic classical eld j theory. we can realize that if one of the eigenvalues is much larger than the rest, then the majority or at least a substantial 3. Condensate-like features fraction of particles is in the same single-particle quan- tum state. We have performed our numerical experiment with In addition, for an idealized system of the Bose parti- six values of the non-linear parameter a (1:5 + 0:1i, cles with periodic boundary conditions and without ex- i = 0; 1;:::; 5), ve values of the diusion constant d ternal potential, the following signature of condensation (0:05j, j = 1; 2;:::; 5), two dierent initial conditions, is also to be noticed: and periodic boundary conditions. The following initial conditions have been investigated. The rst type A 3. The population of the zero-momentum mode is initial conditions are such that (x; y; t) is excited much larger than population of all other modes. only at a single point at t = 0: (N=2; N=2; 0) = 0:5, and (x; y; 0) is equal to zero at all other (x; y). By type B The properties (1) and (2) acquire quantitative mean- initial conditions we mean those with (x; y; 0) being ing only if the one-particle reduced density matrix is de- a Gaussian function, (x; y; 0) = 0:5 exp(−0:01((x − ned.
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