Leonardo Ermann - Academia.edu (original) (raw)

Papers by Leonardo Ermann

Physical Review Letters, 2017

We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochroma... more We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle and co-workers in 1995. We allow that the external driving allows us to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads to localization of energy on low energy modes. This is in a drastic contrast to the random phase approximation leading to energy flow to high modes. A critical threshold is determined above which the turbulent flow to high energies becomes possible. We argue that this phenomenon can be studied with ultracold atoms in magneto-optical traps.

Ecological systems have a high level of complexity combined with stability and rich biodiversity.... more Ecological systems have a high level of complexity combined with stability and rich biodiversity. Recently, the analysis of their properties and evolution has been pushed forward on a basis of concept of mutualistic networks that provides a detailed understanding of their features being linked to a high nestedness of these networks. It was shown that the nestedness architecture of mutualistic networks of plants and their pollinators minimizes competition and increases biodiversity. Here, using the United Nations COM-TRADE database for years 1962 -2009, we show that a similar ecological analysis gives a valuable description of the world trade. In fact the countries and trade products are analogous to plants and pollinators, and the whole trade network is characterized by a low nestedness temperature which is typical for the ecological networks. This approach provides new mutualistic features of the world trade highlighting new significance of countries and trade products for the world trade.

The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic po... more The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic potential under the effect of different drivings in contact with a thermal environment is studied. For weak forcing and finiteh, we identify a strong dependence of the current on the structure of the chaotic region. Moreover, we have determined the robustness of the current against thermal fluctuations in the very weak coupling regime. In the case of strong forcing, the current is determined by the shape of a chaotic attractor. In both cases the temperature quickly stabilizes the ratchet, but in the latter it also destroys the asymmetry responsible for the current generation. Finally, applications to isomerization reactions are discussed.

We study families of open chaotic maps that classically share the same asymptotic properties-forw... more We study families of open chaotic maps that classically share the same asymptotic properties-forward and backward trapped sets, repeller dimensions, and escape rate-but differ in their short time behavior. When these maps are quantized we find that the fine details of the distribution of resonances and the corresponding eigenfunctions are sensitive to the initial shape and size of the openings. We study phase space localization of the resonances with respect to the repeller and find strong delocalization effects when the area of the openings is smaller thanh.

Abstract. We study the properties of spectrum and eigenstates of the Google matrix of a directed ... more Abstract. We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν≈ 0.65 that corresponds to the fractal dimension of the network d≈ 1.3.

The appearance of a directed flow induced by a zeromean monochromatic force is a generic nonequil... more The appearance of a directed flow induced by a zeromean monochromatic force is a generic nonequilibrium phenomenon known as the ratchet effect (see, eg,[1–3]). The ratchet transport at the nanoscale has attracted significant interest in recent years (see [4] and references therein). The electron ratchet currents induced by a monochromatic ac driving have been experimentally observed in asymmetric mesoscopic structures at high [5], very low [6], and gigahertz [7] frequencies.

Abstract: The basic ingredients in a semiclassical theory are the classical invariant objects ser... more Abstract: The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for the quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that it is the classical repeller--the set of non escaping orbits in the future and past evolution--the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case.

Abstract: We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wik... more Abstract: We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.

The quantum treatment of scattering situations where the trapped set has a complex fractal struct... more The quantum treatment of scattering situations where the trapped set has a complex fractal structure has recently been the source of much attention, particularly with respect to the confirmation of the fractal Weyl law [1–3] hypothesis concerning the global distribution of long-lived resonances and their localization properties on the classical repeller [4–7]. The use of quantum maps to model these systems has been of paramount importance, allowing extensive numerical, and sometimes analytical, results.

An accurate description of the spectral properties of quantum propagators, which correspond to ch... more An accurate description of the spectral properties of quantum propagators, which correspond to chaotic systems in the classical limit, remains an open problem. The motivations are not only of fundamental nature, but also due to potential applications in complex systems. Indeed, in this thesis we study the transport phenomena in a chain of coupled chaotic maps using its spectral properties. This system models a cavity chain where nonlinearity and quantum mechanics have a fundamental role, and can represent physical systems of interest in a simplified way like nanotubes, Bose-Einstein condensates, molecular motors and quantum circuits. The spectral properties of the internal evolution of each cell are directly related to the asymptotic current of the chain, and therefore are central to the analysis of transport. The thesis is dedicated to both the study of spectral properties of chaotic maps, and the analysis of the transport in chain of coupled maps.

Arxiv preprint arXiv: …, Jan 1, 2011

We study the statistical properties of various directed networks using ranking of their nodes bas... more We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of ingoing links, while CheiRank orders nodes proportionally to a number of outgoing links. In this way the ranking of nodes becomes twodimensional that paves the way for development of two-dimensional search engines of new type. Statistical properties of information flow on PageRank-CheiRank plane are analyzed for networks of British, French and Italian Universities, Wikipedia, Linux Kernel, gene regulation and other networks. A special emphasis is done for British Universities networks using the large database publicly available at UK. Methods of spam links control are also analyzed.

The European Physical …, Jan 1, 2011

Using extensive Monte Carlo simulations we study numerically and analytically a photogalvanic eff... more Using extensive Monte Carlo simulations we study numerically and analytically a photogalvanic effect, or ratchet, of directed electron transport induced by a microwave radiation on a semidisk Galton board of antidots in graphene. A comparison between usual two-dimensional electron gas (2DEG) and electrons in graphene shows that ratchet currents are comparable at very low temperatures. However, a large mean free path in graphene should allow to have a strong ratchet transport at room temperatures. Also in graphene the ratchet transport emerges even for unpolarized radiation. These properties open promising possibilities for room temperature graphene based sensitive photogalvanic detectors of microwave and terahertz radiation.

Arxiv preprint arXiv:1107.0437, Jan 1, 2011

We study the properties of the Arnold cap map on a torus with a several periodic sections using t... more We study the properties of the Arnold cap map on a torus with a several periodic sections using the Ulam method. This approach generates a Markov chain with the Ulam matrix approximant. We study numerically the spectrum and eigenstates of this matrix showing their relation with the Fokker-Plank relaxation and the Kolmogorov-Sinai entropy. We show that, in the frame of the Ulam method, the time reversal property of the map is preserved only on a short Ulam time which grows only logarithmically with the matrix size. Parallels with the evolution in a regime of quantum chaos are also discussed.

Arxiv preprint arXiv:1201.3584, Jan 1, 2012

Ecological systems have a high level of complexity combined with stability and rich biodiversity.... more Ecological systems have a high level of complexity combined with stability and rich biodiversity. Recently, the analysis of their properties and evolution has been pushed forward on a basis of concept of mutualistic networks that provides a detailed understanding of their features being linked to a high nestedness of these networks. It was shown that the nestedness architecture of mutualistic networks of plants and their pollinators minimizes competition and increases biodiversity. Here, using the United Nations COM-TRADE database for years 1962 -2009, we show that a similar ecological analysis gives a valuable description of the world trade. In fact the countries and trade products are analogous to plants and pollinators, and the whole trade network is characterized by a low nestedness temperature which is typical for the ecological networks. This approach provides new mutualistic features of the world trade highlighting new significance of countries and trade products for the world trade.

scholarpedia.org

The quantized baker's map is a simple mathematical model that is used to explore the rel... more The quantized baker's map is a simple mathematical model that is used to explore the relationship between classical and quantum mechanics in the semiclassical limit, when the dynamics is chaotic. The relative simplicity of the classical description-provided by ...

Physical Review-Section E-Statistical Nonlinear …, Jan 1, 2010

Arxiv preprint arXiv:1103.5027, Jan 1, 2011

Using the United Nations Commodity Trade Statistics Database [1] we construct the Google matrix o... more Using the United Nations Commodity Trade Statistics Database [1] we construct the Google matrix of the world trade network and analyze its properties for various trade commodities for all countries and all available years from 1962 to 2009. The trade flows on this network are classified with the help of PageRank and CheiRank algorithms developed for the World Wide Web and other large scale directed networks. For the world trade this ranking treats all countries on equal democratic grounds independent of country richness. Still this method puts at the top a group of industrially developed countries for trade in all commodities. Our study establishes the existence of two solid state like domains of rich and poor countries which remain stable in time, while the majority of countries are shown to be in a gas like phase with strong rank fluctuations. A simple random matrix model provides a good description of statistical distribution of countries in two-dimensional rank plane. The comparison with usual ranking by export and import highlights new features and possibilities of our approach.

Physical Review E, Jan 1, 2011

The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic po... more The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic potential under the effect of different drivings in contact with a thermal environment is studied. For weak forcing and finite , we identify a strong dependence of the current on the structure of the chaotic region. Moreover, we have determined the robustness of the current against thermal fluctuations in the very weak coupling regime. In the case of strong forcing, the current is determined by the shape of a chaotic attractor. In both cases the temperature quickly stabilizes the ratchet, but in the latter it also destroys the asymmetry responsible for the current generation. Finally, applications to isomerization reactions are discussed.

Physical Review E, Jan 1, 2009

Recently, a new mechanism leading to purely quantum directed transport in the asymmetric multibak... more Recently, a new mechanism leading to purely quantum directed transport in the asymmetric multibaker map has been presented. Here, we show a comprehensive characterization of the finite asymptotic current behavior with respect to the h value, the shape of the initial conditions, and the features of the spectrum. We have considered different degrees of asymmetry in these studies and we have also analyzed the classical and quantum phase space distributions for short times in order to understand the mechanisms behind the generation of the directed current.

Physical Review E, Jan 1, 2008

By studying a modified (unbiased) quantum multibaker map, we were able to obtain a finite asympto... more By studying a modified (unbiased) quantum multibaker map, we were able to obtain a finite asymptotic quantum current without a classical analogue. This result suggests a general method for the design of purely quantum ratchets, and sheds light on the investigation of the mechanisms leading to net transport generation by breaking symmetries of quantum systems. Moreover, we propose the multibaker map as a resource to study directed transport phenomena in chaotic systems without bias. In fact, this is a paradigmatic model in classical and quantum chaos, but also in statistical mechanics.

Physical Review Letters, 2017

We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochroma... more We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle and co-workers in 1995. We allow that the external driving allows us to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads to localization of energy on low energy modes. This is in a drastic contrast to the random phase approximation leading to energy flow to high modes. A critical threshold is determined above which the turbulent flow to high energies becomes possible. We argue that this phenomenon can be studied with ultracold atoms in magneto-optical traps.

Ecological systems have a high level of complexity combined with stability and rich biodiversity.... more Ecological systems have a high level of complexity combined with stability and rich biodiversity. Recently, the analysis of their properties and evolution has been pushed forward on a basis of concept of mutualistic networks that provides a detailed understanding of their features being linked to a high nestedness of these networks. It was shown that the nestedness architecture of mutualistic networks of plants and their pollinators minimizes competition and increases biodiversity. Here, using the United Nations COM-TRADE database for years 1962 -2009, we show that a similar ecological analysis gives a valuable description of the world trade. In fact the countries and trade products are analogous to plants and pollinators, and the whole trade network is characterized by a low nestedness temperature which is typical for the ecological networks. This approach provides new mutualistic features of the world trade highlighting new significance of countries and trade products for the world trade.

The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic po... more The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic potential under the effect of different drivings in contact with a thermal environment is studied. For weak forcing and finiteh, we identify a strong dependence of the current on the structure of the chaotic region. Moreover, we have determined the robustness of the current against thermal fluctuations in the very weak coupling regime. In the case of strong forcing, the current is determined by the shape of a chaotic attractor. In both cases the temperature quickly stabilizes the ratchet, but in the latter it also destroys the asymmetry responsible for the current generation. Finally, applications to isomerization reactions are discussed.

We study families of open chaotic maps that classically share the same asymptotic properties-forw... more We study families of open chaotic maps that classically share the same asymptotic properties-forward and backward trapped sets, repeller dimensions, and escape rate-but differ in their short time behavior. When these maps are quantized we find that the fine details of the distribution of resonances and the corresponding eigenfunctions are sensitive to the initial shape and size of the openings. We study phase space localization of the resonances with respect to the repeller and find strong delocalization effects when the area of the openings is smaller thanh.

Abstract. We study the properties of spectrum and eigenstates of the Google matrix of a directed ... more Abstract. We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν≈ 0.65 that corresponds to the fractal dimension of the network d≈ 1.3.

The appearance of a directed flow induced by a zeromean monochromatic force is a generic nonequil... more The appearance of a directed flow induced by a zeromean monochromatic force is a generic nonequilibrium phenomenon known as the ratchet effect (see, eg,[1–3]). The ratchet transport at the nanoscale has attracted significant interest in recent years (see [4] and references therein). The electron ratchet currents induced by a monochromatic ac driving have been experimentally observed in asymmetric mesoscopic structures at high [5], very low [6], and gigahertz [7] frequencies.

Abstract: The basic ingredients in a semiclassical theory are the classical invariant objects ser... more Abstract: The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for the quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that it is the classical repeller--the set of non escaping orbits in the future and past evolution--the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case.

Abstract: We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wik... more Abstract: We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.

The quantum treatment of scattering situations where the trapped set has a complex fractal struct... more The quantum treatment of scattering situations where the trapped set has a complex fractal structure has recently been the source of much attention, particularly with respect to the confirmation of the fractal Weyl law [1–3] hypothesis concerning the global distribution of long-lived resonances and their localization properties on the classical repeller [4–7]. The use of quantum maps to model these systems has been of paramount importance, allowing extensive numerical, and sometimes analytical, results.

An accurate description of the spectral properties of quantum propagators, which correspond to ch... more An accurate description of the spectral properties of quantum propagators, which correspond to chaotic systems in the classical limit, remains an open problem. The motivations are not only of fundamental nature, but also due to potential applications in complex systems. Indeed, in this thesis we study the transport phenomena in a chain of coupled chaotic maps using its spectral properties. This system models a cavity chain where nonlinearity and quantum mechanics have a fundamental role, and can represent physical systems of interest in a simplified way like nanotubes, Bose-Einstein condensates, molecular motors and quantum circuits. The spectral properties of the internal evolution of each cell are directly related to the asymptotic current of the chain, and therefore are central to the analysis of transport. The thesis is dedicated to both the study of spectral properties of chaotic maps, and the analysis of the transport in chain of coupled maps.

Arxiv preprint arXiv: …, Jan 1, 2011

We study the statistical properties of various directed networks using ranking of their nodes bas... more We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of ingoing links, while CheiRank orders nodes proportionally to a number of outgoing links. In this way the ranking of nodes becomes twodimensional that paves the way for development of two-dimensional search engines of new type. Statistical properties of information flow on PageRank-CheiRank plane are analyzed for networks of British, French and Italian Universities, Wikipedia, Linux Kernel, gene regulation and other networks. A special emphasis is done for British Universities networks using the large database publicly available at UK. Methods of spam links control are also analyzed.

The European Physical …, Jan 1, 2011

Using extensive Monte Carlo simulations we study numerically and analytically a photogalvanic eff... more Using extensive Monte Carlo simulations we study numerically and analytically a photogalvanic effect, or ratchet, of directed electron transport induced by a microwave radiation on a semidisk Galton board of antidots in graphene. A comparison between usual two-dimensional electron gas (2DEG) and electrons in graphene shows that ratchet currents are comparable at very low temperatures. However, a large mean free path in graphene should allow to have a strong ratchet transport at room temperatures. Also in graphene the ratchet transport emerges even for unpolarized radiation. These properties open promising possibilities for room temperature graphene based sensitive photogalvanic detectors of microwave and terahertz radiation.

Arxiv preprint arXiv:1107.0437, Jan 1, 2011

We study the properties of the Arnold cap map on a torus with a several periodic sections using t... more We study the properties of the Arnold cap map on a torus with a several periodic sections using the Ulam method. This approach generates a Markov chain with the Ulam matrix approximant. We study numerically the spectrum and eigenstates of this matrix showing their relation with the Fokker-Plank relaxation and the Kolmogorov-Sinai entropy. We show that, in the frame of the Ulam method, the time reversal property of the map is preserved only on a short Ulam time which grows only logarithmically with the matrix size. Parallels with the evolution in a regime of quantum chaos are also discussed.

Arxiv preprint arXiv:1201.3584, Jan 1, 2012

Ecological systems have a high level of complexity combined with stability and rich biodiversity.... more Ecological systems have a high level of complexity combined with stability and rich biodiversity. Recently, the analysis of their properties and evolution has been pushed forward on a basis of concept of mutualistic networks that provides a detailed understanding of their features being linked to a high nestedness of these networks. It was shown that the nestedness architecture of mutualistic networks of plants and their pollinators minimizes competition and increases biodiversity. Here, using the United Nations COM-TRADE database for years 1962 -2009, we show that a similar ecological analysis gives a valuable description of the world trade. In fact the countries and trade products are analogous to plants and pollinators, and the whole trade network is characterized by a low nestedness temperature which is typical for the ecological networks. This approach provides new mutualistic features of the world trade highlighting new significance of countries and trade products for the world trade.

scholarpedia.org

The quantized baker's map is a simple mathematical model that is used to explore the rel... more The quantized baker's map is a simple mathematical model that is used to explore the relationship between classical and quantum mechanics in the semiclassical limit, when the dynamics is chaotic. The relative simplicity of the classical description-provided by ...

Physical Review-Section E-Statistical Nonlinear …, Jan 1, 2010

Arxiv preprint arXiv:1103.5027, Jan 1, 2011

Using the United Nations Commodity Trade Statistics Database [1] we construct the Google matrix o... more Using the United Nations Commodity Trade Statistics Database [1] we construct the Google matrix of the world trade network and analyze its properties for various trade commodities for all countries and all available years from 1962 to 2009. The trade flows on this network are classified with the help of PageRank and CheiRank algorithms developed for the World Wide Web and other large scale directed networks. For the world trade this ranking treats all countries on equal democratic grounds independent of country richness. Still this method puts at the top a group of industrially developed countries for trade in all commodities. Our study establishes the existence of two solid state like domains of rich and poor countries which remain stable in time, while the majority of countries are shown to be in a gas like phase with strong rank fluctuations. A simple random matrix model provides a good description of statistical distribution of countries in two-dimensional rank plane. The comparison with usual ranking by export and import highlights new features and possibilities of our approach.

Physical Review E, Jan 1, 2011

The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic po... more The transitory and stationary behavior of a quantum chaotic ratchet consisting of a biharmonic potential under the effect of different drivings in contact with a thermal environment is studied. For weak forcing and finite , we identify a strong dependence of the current on the structure of the chaotic region. Moreover, we have determined the robustness of the current against thermal fluctuations in the very weak coupling regime. In the case of strong forcing, the current is determined by the shape of a chaotic attractor. In both cases the temperature quickly stabilizes the ratchet, but in the latter it also destroys the asymmetry responsible for the current generation. Finally, applications to isomerization reactions are discussed.

Physical Review E, Jan 1, 2009

Recently, a new mechanism leading to purely quantum directed transport in the asymmetric multibak... more Recently, a new mechanism leading to purely quantum directed transport in the asymmetric multibaker map has been presented. Here, we show a comprehensive characterization of the finite asymptotic current behavior with respect to the h value, the shape of the initial conditions, and the features of the spectrum. We have considered different degrees of asymmetry in these studies and we have also analyzed the classical and quantum phase space distributions for short times in order to understand the mechanisms behind the generation of the directed current.

Physical Review E, Jan 1, 2008

By studying a modified (unbiased) quantum multibaker map, we were able to obtain a finite asympto... more By studying a modified (unbiased) quantum multibaker map, we were able to obtain a finite asymptotic quantum current without a classical analogue. This result suggests a general method for the design of purely quantum ratchets, and sheds light on the investigation of the mechanisms leading to net transport generation by breaking symmetries of quantum systems. Moreover, we propose the multibaker map as a resource to study directed transport phenomena in chaotic systems without bias. In fact, this is a paradigmatic model in classical and quantum chaos, but also in statistical mechanics.