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Galiceanu M, Tota de Carvalho L, Mülken O, Dolgushev M. Dynamics of Dual Scale-Free Polymer Networks. Polymers (Basel) 2017; 9:E577. [PMID: 30965880 PMCID: PMC6418598 DOI: 10.3390/polym9110577] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2017] [Revised: 10/29/2017] [Accepted: 10/31/2017] [Indexed: 01/31/2023] Open
Abstract
We focus on macromolecules which are modeled as sequentially growing dual scale-free networks. The dual networks are built by replacing star-like units of the primal treelike scale-free networks through rings, which are then transformed in a small-world manner up to the complete graphs. In this respect, the parameter γ describing the degree distribution in the primal treelike scale-free networks regulates the size of the dual units. The transition towards the networks of complete graphs is controlled by the probability p of adding a link between non-neighboring nodes of the same initial ring. The relaxation dynamics of the polymer networks is studied in the framework of generalized Gaussian structures by using the full eigenvalue spectrum of the Laplacian matrix. The dynamical quantities on which we focus here are the averaged monomer displacement and the mechanical relaxation moduli. For several intermediate values of the parameters' set ( γ , p ) , we encounter for these dynamical properties regions of constant in-between slope.
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Affiliation(s)
- Mircea Galiceanu
- Departamento de Fisica, Universidade Federal do Amazonas, Manaus 69077-000, Brazil.
| | | | - Oliver Mülken
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany.
| | - Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany.
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2
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Abstract
We study the transport efficiency of excitations on complex quantum networks with loops. For this we consider sequentially growing networks with different topologies of the sequential subgraphs. This can lead either to a universal complete breakdown of transport for complete-graph-like sequential subgraphs or to optimal transport for ringlike sequential subgraphs. The transition to optimal transport can be triggered by systematically reducing the number of loops of complete-graph-like sequential subgraphs in a small-world procedure. These effects are explained on the basis of the spectral properties of the network's Hamiltonian. Our theoretical considerations are supported by numerical Monte Carlo simulations for complex quantum networks with a scale-free size distribution of sequential subgraphs and a small-world-type transition to optimal transport.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
| | - Maxim Dolgushev
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
| | - Mircea Galiceanu
- Departamento de Física, Universidade Federal do Amazonas, 3000 Japiim, 69077-000 Manaus-AM, Brazil.,Institut für Theoretische Physik, Technische Universtät Dresden, 01062 Dresden, Germany
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Abstract
We consider single-particle quantum transport on parametrized complex networks. Based on general arguments regarding the spectrum of the corresponding Hamiltonian, we derive bounds for a measure of the global transport efficiency defined by the time-averaged return probability. For treelike networks, we show analytically that a transition from efficient to inefficient transport occurs depending on the (average) functionality of the nodes of the network. In the infinite system size limit, this transition can be characterized by an exponent which is universal for all treelike networks. Our findings are corroborated by analytic results for specific deterministic networks, dendrimers and Vicsek fractals, and by Monte Carlo simulations of iteratively built scale-free trees.
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Affiliation(s)
- Nikolaj Kulvelis
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
| | - Maxim Dolgushev
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
| | - Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
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Darázs Z, Anishchenko A, Kiss T, Blumen A, Mülken O. Transport properties of continuous-time quantum walks on Sierpinski fractals. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 90:032113. [PMID: 25314401 DOI: 10.1103/physreve.90.032113] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2014] [Indexed: 06/04/2023]
Abstract
We model quantum transport, described by continuous-time quantum walks (CTQWs), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.
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Affiliation(s)
- Zoltán Darázs
- Wigner RCP, SZFKI, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary and Eötvös University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
| | - Anastasiia Anishchenko
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Tamás Kiss
- Wigner RCP, SZFKI, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary
| | - Alexander Blumen
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
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Anishchenko A, Blumen A, Mülken O. Geometrical aspects of quantum walks on random two-dimensional structures. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:062126. [PMID: 24483405 DOI: 10.1103/physreve.88.062126] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2013] [Indexed: 06/03/2023]
Abstract
We study the transport properties of continuous-time quantum walks (CTQWs) over finite two-dimensional structures with a given number of randomly placed bonds and with different aspect ratios (ARs). Here, we focus on the transport from, say, the left side to the right side of the structure where absorbing sites are placed. We do so by analyzing the long-time average of the survival probability of CTQWs. We compare the results to the classical continuous-time random walk case (CTRW). For small ARs (landscape configurations) we observe only small differences between the quantum and the classical transport properties, i.e., roughly the same number of bonds is needed to facilitate the transport. However, with increasing ARs (portrait configurations) a much larger number of bonds is needed in the CTQW case than in the CTRW case. While for CTRWs the number of bonds needed decreases when going from small ARs to large ARs, for CTQWs this number is large for small ARs, has a minimum for the square configuration, and increases again for increasing ARs. We explain our findings by analyzing the average eigenstates of the corresponding structures: The participation ratios allow us to distinguish between localized and nonlocalized (average) eigenstates. In particular, for large ARs we find for CTQWs that the eigenstates are localized for bond numbers exceeding the bond numbers needed to facilitate transport in the CTRW case. Thus, a rather large number of bonds is needed in order for quantum transport to be efficient for large ARs.
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Affiliation(s)
- Anastasiia Anishchenko
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Alexander Blumen
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
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Schijven P, Mülken O. Avoiding dark states in open quantum systems by tailored initializations. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:062102. [PMID: 23005149 DOI: 10.1103/physreve.85.062102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2012] [Revised: 05/30/2012] [Indexed: 06/01/2023]
Abstract
We study the transport of excitations on a network of three coupled two-level systems that are subjected to an environment that induces incoherent hopping between the nodes. The two end nodes are coupled to a source while the central node is coupled to a drain. A common feature of these networks is the existence of a dark state that blocks the transport to the drain. Here we propose a means to avoid this state by a suitable initialization of the system, induced by a source that is common to both coupled nodes.
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Affiliation(s)
- P Schijven
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.
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Mülken O, Bauer M. Directed excitation transfer in vibrating chains by external fields. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 83:061123. [PMID: 21797318 DOI: 10.1103/physreve.83.061123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2010] [Revised: 03/03/2011] [Indexed: 05/31/2023]
Abstract
We study the coherent dynamics of excitations on vibrating chains. By applying an external field and matching the field strength with the oscillation frequency of the chain it is possible to obtain an (average) transport of an initial Gaussian wave packet. We distinguish between a uniform oscillation of all nodes of the chain and the chain being in its lowest eigenmode. Both cases can lead to directed transport.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Freiburg, Germany
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Mülken O, Schmid T. Environment-assisted quantum transport and trapping in dimers. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 82:042104. [PMID: 21230333 DOI: 10.1103/physreve.82.042104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2010] [Revised: 09/28/2010] [Indexed: 05/30/2023]
Abstract
We study the dynamics and trapping of excitations for a dimer with an energy off-set Δ coupled to an external environment. Using a Lindblad quantum master equation approach, we calculate the survival probability Π(t) of the excitation and define different lifetimes τ(s) of the excitation, corresponding to the duration of the decay of Π(t) in between two predefined values. We show that it is not possible to always enhance the overall decay to the trap. However, it is possible, even for not too small environmental couplings and for values of Δ of the order O(1), to decrease certain lifetimes τ(s), leading to faster decay of Π(t) in these time intervals: there is an optimal environmental coupling, leading to a maximal decay for fixed Δ.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
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9
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Mülken O, Mühlbacher L, Schmid T, Blumen A. Dissipative dynamics with trapping in dimers. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:041114. [PMID: 20481684 DOI: 10.1103/physreve.81.041114] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2009] [Indexed: 05/29/2023]
Abstract
The trapping of excitations in systems coupled to an environment allows one to study the quantum to classical crossover by different means. We show how to combine the phenomenological description by a non-Hermitian Liouville-von Neumann equation (LvNE) approach with the numerically exact path integral Monte Carlo (PIMC) method, and exemplify our results for a system of two coupled two-level systems. By varying the strength of the coupling to the environment we are able to estimate the parameter range in which the LvNE approach yields satisfactory results. Moreover, by matching the PIMC results with the LvNE calculations, we have a powerful tool to extrapolate the numerically exact PIMC method to long times.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
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10
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Mülken O, Pernice V, Blumen A. Slow excitation trapping in quantum transport with long-range interactions. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 78:021115. [PMID: 18850794 DOI: 10.1103/physreve.78.021115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2008] [Revised: 07/09/2008] [Indexed: 05/26/2023]
Abstract
Long-range interactions (LRIs) slow down the excitation trapping in quantum transport on a one-dimensional chain with traps at both ends when compared to the case with only nearest-neighbor interactions. This is in contrast to the corresponding classical case, in which LRIs lead to faster excitation trapping. The reason for the slowing down is to be found in subtle changes--due to LRIs--in the spectrum of the Hamiltonian. Pertubation theory allows an analytical analysis of this fact.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.
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11
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Mülken O, Pernice V, Blumen A. Universal behavior of quantum walks with long-range steps. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:021117. [PMID: 18351997 DOI: 10.1103/physreve.77.021117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2007] [Indexed: 05/26/2023]
Abstract
Continuous-time quantum walks with long-range steps R(-gamma) (R being the distance between sites) on a discrete line behave in similar ways for all gamma > or =2 . This is in contrast to classical random walks, which for gamma>3 belong to a different universality class than for gamma < or =3 . We show that the average probabilities to be at the initial site after time t as well as the mean square displacements are of the same functional form for quantum walks with gamma =2, 4, and with nearest neighbor steps. We interpolate this result to arbitrary gamma > or =2 .
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, Freiburg, Germany.
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12
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Mülken O, Pernice V, Blumen A. Quantum transport on small-world networks: a continuous-time quantum walk approach. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 76:051125. [PMID: 18233641 DOI: 10.1103/physreve.76.051125] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2007] [Revised: 07/31/2007] [Indexed: 05/25/2023]
Abstract
We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.
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Mülken O, Blumen A, Amthor T, Giese C, Reetz-Lamour M, Weidemüller M. Survival probabilities in coherent exciton transfer with trapping. Phys Rev Lett 2007; 99:090601. [PMID: 17930996 DOI: 10.1103/physrevlett.99.090601] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2007] [Indexed: 05/25/2023]
Abstract
In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power law, in contrast with the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg i.Br., Germany
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Mülken O, Bierbaum V, Blumen A. Localization of coherent exciton transport in phase space. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:031121. [PMID: 17500682 DOI: 10.1103/physreve.75.031121] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2006] [Revised: 01/31/2007] [Indexed: 05/15/2023]
Abstract
We study numerically the dynamics of excitons on discrete rings in the presence of static disorder. Based on continuous-time quantum walks we compute the time evolution of the Wigner function (WF) both for pure diagonal (site) disorder, as well as for diagonal and off-diagonal (site and transfer) disorder. In both cases, large disorder leads to localization and destroys the characteristic phase space patterns of the WF found in the absence of disorder.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
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15
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Mülken O, Blumen A. Efficiency of quantum and classical transport on graphs. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 73:066117. [PMID: 16906924 DOI: 10.1103/physreve.73.066117] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2006] [Indexed: 05/11/2023]
Abstract
We propose a measure to quantify the efficiency of classical and quantum mechanical transport processes on graphs. The measure only depends on the density of states (DOS), which contains all the necessary information about the graph. For some given (continuous) DOS, the measure shows a power law behavior, where the exponent for the quantum transport is twice the exponent of its classical counterpart. For small-world networks, however, the measure shows rather a stretched exponential law but still the quantum transport outperforms the classical one. Some finite tree graphs have a few highly degenerate eigenvalues, such that, on the other hand, on them the classical transport may be more efficient than the quantum one.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg i.Br., Germany.
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Abstract
We model coherent exciton transport in dendrimers by continuous-time quantum walks. For dendrimers up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For larger dendrimers, the recurrence ceases to be perfect, a fact which resembles results for discrete quantum carpets. Moreover, depending on the initial excitation site, we find that the coherent transport to certain nodes of the dendrimer has a very low probability. When the initial excitation starts from the central node, the problem can be mapped onto a line which simplifies the computational effort. Furthermore, the long time average of the quantum mechanical transition probabilities between pairs of nodes shows characteristic patterns and allows us to classify the nodes into clusters with identical limiting probabilities. For the (space) average of the quantum mechanical probability to be still or to be again at the initial site, we obtain, based on the Cauchy-Schwarz inequality, a simple lower bound which depends only on the eigenvalue spectrum of the Hamiltonian.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.
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Mülken O, Blumen A. Spacetime structures of continuous-time quantum walks. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:036128. [PMID: 15903514 DOI: 10.1103/physreve.71.036128] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2004] [Indexed: 05/02/2023]
Abstract
The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg i.Br., Germany
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van Beijeren H, Mülken O. Thermodynamic formalism for the Lorentz gas with open boundaries in d dimensions. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:036213. [PMID: 15903552 DOI: 10.1103/physreve.71.036213] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2004] [Indexed: 05/02/2023]
Abstract
A Lorentz gas may be defined as a system of fixed dispersing scatterers, with a single light particle moving among these and making specular collisions on encounters with the scatterers. For a dilute Lorentz gas with open boundaries in d dimensions we relate the thermodynamic formalism to a random flight problem. Using this representation we analytically calculate the central quantity within this formalism, the topological pressure, as a function of system size and a temperaturelike parameter beta . The topological pressure is given as the sum of the topological pressure for the closed system and a diffusion term with a beta-dependent diffusion coefficient. From the topological pressure we obtain the Kolmogorov-Sinai entropy on the repeller, the topological entropy, and the partial information dimension.
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Affiliation(s)
- Henk van Beijeren
- Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.
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Abstract
Continuous time quantum walks (CTQWs) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRWs). For one special graph, where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by CTQWs than by CTRWs, we demonstrate that in another direction of propagation the opposite is true. In this case a CTQW initially localized at one site displays a slow transport. We furthermore show that when the CTQW's initial condition is a totally symmetric superposition of states of equivalent sites, the transport gets to be much more rapid.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg i.Br., Germany.
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Mülken O, van Beijeren H. Thermodynamic formalism for field-driven Lorentz gases. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 69:046203. [PMID: 15169086 DOI: 10.1103/physreve.69.046203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2003] [Indexed: 05/24/2023]
Abstract
We analytically determine the dynamical properties of two-dimensional field-driven Lorentz gases within the thermodynamic formalism. For dilute gases subjected to an isokinetic thermostat, we calculate the topological pressure as a function of a temperaturelike parameter beta up to second order in the strength of the applied field. The Kolmogorov-Sinai entropy and the topological entropy can be extracted from a dynamical entropy defined as a Legendre transform of the topological pressure. Our calculations of the Kolmogorov-Sinai entropy exactly agree with previous calculations based on a Lorentz-Boltzmann equation approach. We give analytic results for the topological entropy and calculate the dimension spectrum from the dynamical entropy function.
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Affiliation(s)
- Oliver Mülken
- Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.
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Stamerjohanns H, Mülken O, Borrmann P. Deceptive signals of phase transitions in small magnetic clusters. Phys Rev Lett 2002; 88:053401. [PMID: 11863721 DOI: 10.1103/physrevlett.88.053401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2001] [Indexed: 05/23/2023]
Abstract
We present an analysis of the thermodynamic properties of small transition-metal clusters and show how the commonly used indicators of phase transitions such as peaks in the specific heat or magnetic susceptibility can lead to deceptive interpretations of the underlying physics. The analysis of the distribution of zeros of the canonical partition function in the whole complex temperature plane reveals the nature of the transition. We show that signals in the magnetic susceptibility at positive temperatures have their origin at zeros lying at negative temperatures.
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Affiliation(s)
- Heinrich Stamerjohanns
- Department of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg, Germany
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Mülken O, Stamerjohanns H, Borrmann P. Origins of phase transitions in small systems. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:047105. [PMID: 11690179 DOI: 10.1103/physreve.64.047105] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2001] [Revised: 06/25/2001] [Indexed: 05/23/2023]
Abstract
The identification and classification of phases in small systems, e.g., nuclei, social and financial networks, clusters, and biological systems, where the traditional definitions of phase transitions are not applicable, is important to obtain a deeper understanding of the phenomena observed in such systems. Within a simple statistical model, we investigate the validity and applicability of different classification schemes for phase transtions in small systems. We show that the whole complex temperature plane contains necessary information in order to give a distinct classification.
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Affiliation(s)
- O Mülken
- Department of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg, Germany
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