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Open graph forming a chain composed of N = 6 identical unit cells. The five lengths that compose the unit cell take different values. (In this regard, no specific length can be associated with the periodicity of the chain.) For the graph of Fig. 4, the transition probabilities from bond to bond P bb ′ are given by P bb ′ =
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We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator that generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-or...
Citations
... As a matter of fact, the classical dynamics of particles on graphs can be described by simple maps. Trajectories of a particle on a graph, undergoing scattering at its vertices, are in one-to-one correspondence with the ones of one-dimensional piecewise chaotic maps [19][20][21]. ...
... where the transition matrix is given by W 1 in (19) and G β ≡ g β 0 0 l β . The region in the (g, l) with large fluctuations |β| < 2 is bounded between the curves defined by (13) with β = ±2. ...
We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval [S. Lepri, Chaos Solitons & Fractals, 139,110003 (2020)]. We determine the conditions for having fat-tailed invariant measures by considering approximate solution of the Perron-Frobenius equation for generic graphs. An analogy with the statistical mechanics of a directed polymer is presented that allows for a physically appealing interpretation of the statistical regimes. The connection between non-Gaussian statistics and the generalized Lyapunov exponents $L(q)$ is illustrated. Finally, some results concerning large graphs are reported.
... To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2] (see also [3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs whenh (the reduced Planck constant) goes to zero. ...
... One way to overcome this difficulty is to assign a probability to every possible path on the graph. Typically the probability of a certain path is postulated, and given in terms of the square modulus of the quantum transition (or stability) amplitudes (see, e.g., [ [17] and in several other works, see, e.g., [3], the review [14], and the monograph [4]. We have already noted that, up to a sign, the coefficients 2 n − δ ℓ,ℓ ′ coincide with the elements of the matrix S identifying both the classical and quantum scattering operators. ...
We consider the dynamics of a quantum particle of mass m on a n -edges star-graph with Hamiltonian $$H_K=-(2m)^{-1}\hbar ^2 \Delta $$ H K = - ( 2 m ) - 1 ħ 2 Δ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kreĭn’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $$(H_K,H_{D}^{\oplus })$$ ( H K , H D ⊕ ) , where $$H_{D}^{\oplus }$$ H D ⊕ is the Hamiltonian with Dirichlet conditions in the vertex.
... The idea is that light rays can be treated as particles undergoing chaotic diffusion and amplification. Indeed, the classical dynamics of particles on graphs is a chaotic type of diffusive process [17]. Trajectories of a particle on a graph, undergoing scattering at its vertices, are in one-to-one correspondence with the ones of one-dimensional piecewise chaotic maps [17,18]. ...
... Indeed, the classical dynamics of particles on graphs is a chaotic type of diffusive process [17]. Trajectories of a particle on a graph, undergoing scattering at its vertices, are in one-to-one correspondence with the ones of one-dimensional piecewise chaotic maps [17,18]. ...
... see Fig.1. If we consider the motion of a particles on the graph there drawn, f can be derived exactly as a suitable Poincaré section as described in [17]. 1 The map is everywhere expanding and is invariant for x → 1 − x, but it is straightforward to generalize to an asymmetric case and/or more complex graphs. ...
We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval. We investigate the possibility of having power-law tails in the invariant measure by approximate solution of the Perron-Frobenius equation and discuss the connection with the generalized Lyapunov exponents $L(q)$. We then consider the case of open maps where trajectories escape and demonstrate that stationary power-law distributions occur when $L(q)=r$, with $r$ being the escape rate. The proposed system is a toy model for coupled active chaotic cavities or lasing networks and allows to elucidate in a simple mathematical framework the conditions for observing L\'evy statistical regimes and chaotic intermittency in such systems.
... To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2] (see also [3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs whenh (the reduced Planck constant) goes to zero. ...
... One way to overcome this difficulty is to assign a probability to every possible path on the graph. Typically the probability of a certain path is postulated, and given in terms of the square modulus of the quantum transition (or stability) amplitudes (see, e.g., [ [17] and in several other works, see, e.g., [3], the review [14], and the monograph [4]. We have already noted that, up to a sign, the coefficients 2 n − δ ℓ,ℓ ′ coincide with the elements of the matrix S identifying both the classical and quantum scattering operators. ...
We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\u{\i}n's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\oplus})$, where $H_{D}^{\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.
... We consider closed graphs that are chaotic in the classical limit [20,22,23]. In that limit, amplitudes are replaced by probabilities, and interest centers on the B (2 )-dimensional vector r of occupation propabilities r 0 bd ⩾ for the B 2 directed bonds bd ( ). ...
... Equations (20) and (22) then show that the P Q ( , ) correlation functions in equations (17) and (19) can both be written as ...
For completely connected simple graphs with incommensurate bond lengths and
with unitary symmetry we prove the Bohigas-Giannoni-Schmit conjecture in its
most general form. For graphs that are classically mixing (i.e., graphs for
which the spectrum of the classical Perron-Frobenius operator possesses a
finite gap), we show that the generating functions for all $(P, Q)$ correlation
functions for both closed and open graphs coincide (in the limit of infinite
graph size) with the corresponding expressions of random-matrix theory.
... If G = (V, E) is a finite graph, we introduce, following ideas in [BG00,BG01], an algebraic sub-manifold Z G , which we call the "determinant manifold", of the torus T E := (R/2πZ) E which allows to compute the eigenvalues thanks to the so-called "secular equation". The Gauß map Γ associates to any point of the smooth part Z reg G of Z G the half-line of the cone P E := [0, +∞[ E \{0} which is orthogonal to Z G . ...
... In the case of a connected quantum graph, the classical dynamics is ergodic as shown from the study of the associated Perron-Frobenius operator done in [BG01]. The phase space Z of a Quantum graph G = (V, E, l) (the unit cotangent bundle) can be identified with the set of oriented edges: to a point x of an oriented edge, we associate the unit co-vector pointing in the direction given from the orientation. ...
In this paper, I describe the weak limits of the measures associated to the
eigenfunctions of the Laplacian on a Quantum graph for a generic metric in
terms of the Gauss map of the determinant manifold. I describe also all the
limits with minimal support (the "scars").
... This connection is described by the famous Selberg trace formula[4], that was first discovered as a number theoretic and functional analysis theorem. Apart from the deterministic chaotic systems, the expansion (3) can also describe the spectra of many classically stochastic systems, e.g. the so-called quantum graphs[5,6]and 2D ray splitting billiards[8]: as long as the stochastic dynamics generates trajectory patterns similar to the deterministic chaotic trajectories, they manifest themselves in the in the same physical context in quantum regime. ...
... In[17][18][19][20][21][22][23]it was shown that this approach can be applied to the case of the quantum graphs – simple quasi one dimensional, scaling, classically stochastic models (Fig. 6), that are often used to model deterministic chaotic behavior in low dimensional dynamical systems[5,6]. In[17][18][19]it was shown that there exist quantum graphs with different degrees of spectral irregularity defined in the sense of the bootstrapping hierarchy. For the simplest case of the regular quantum graphs[20][21][22][23][24], the spectrum obtained from ...
It has been long recognized that the task of semiclassical evaluation of quantum spectra for the classically nonintegrable systems is fundamentally more complex than for the classically integrable ones. Below it is argued that the quantum spectra of the chaotic systems can differ among themselves by level of their complexity.
... Classical evolution on graphs So far we discussed graph dynamics from a quantum mechanical point of view. At the present stage, we would like to study graphs from a different point of view, which provides the classical counterpart of the quantum theory [160,26,194]. Usually the connection between the quantum and the classical description is provided by quantising a classical system. Here we take the process in the reverse direction, for reasons which will be explained below. ...
... This is not a very strong statement on the classical dynamics of a graph. It is known that the classical dynamics for every dynamically connected graph is ergodic [26]. A Markov process may have the stronger dynamical property of being mixing which is defined by lim n→∞ M n ρ(0) = ρ inv (4.14) ...
During the last years quantum graphs have become a paradigm of quantum chaos
with applications from spectral statistics to chaotic scattering and wave
function statistics. In the first part of this review we give a detailed
introduction to the spectral theory of quantum graphs and discuss exact trace
formulae for the spectrum and the quantum-to-classical correspondence. The
second part of this review is devoted to the spectral statistics of quantum
graphs as an application to quantum chaos. Especially, we summarise recent
developments on the spectral statistics of generic large quantum graphs based
on two approaches: the periodic-orbit approach and the supersymmetry approach.
The latter provides a condition and a proof for universal spectral statistics
as predicted by random-matrix theory.
... To conclude this introductory chapter on quantum graphs, we present a brief insight into the classical dynamics which can be associated with graphs (for more detailed reviews see [43,24]). ...
... Spectra of quantum graphs display in general universal statistics characteristic for ensembles of random unitary matrices. This observation by Kottos and Smilansky [1, 2] has led to a variety of studies in this direction [3, 4, 5, 6, 7, 8, 9, 10, 11]. It has became clear that the quantisation scheme of Kottos and Smilansky can be considerably generalised to be applicable also for directed graphs (digraphs) [12, 13, 14]. ...
Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family L
n
(G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.
