Random transition-rate matrices for the master equation

Timm, Carsten

doi:10.1103/physreve.80.021140

Cited by 35 publications

(41 citation statements)

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“…This means that the decay time of the corresponding deviations from the stationary state is much shorter than their oscillation period. A similar behavior is found for randomly distributed transition rates, where it is essentially a consequence of different scaling of the real and imaginary parts with the dimension of the molecular Fock space [44]. At zero bias, all eigenvalues are real.…”

Section: Transmitting Regimesupporting

confidence: 69%

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Spectroscopy of the transition-rate matrix for molecular junctions: dynamics in the Franck–Condon regime

Timm

2012

New J. Phys.

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The quantum master equation applied to electronic transport through nanoscopic devices provides information not only on the stationary state but also on the dynamics. The dynamics is characterized by the eigenvalues of the transition-rate matrix, or generator, of the master equation. We propose to use the spectrum of these eigenvalues as a tool for the study of nanoscopic transport. We illustrate this idea by analyzing a molecular quantum dot with an electronic orbital coupled to a vibrational mode, which shows the Franck-Condon blockade if the coupling is strong. Our approach provides complementary information compared to the study of observables in the stationary state.Recent progress in nanotechnology has allowed us to fabricate transistors with a single molecule forming the active region [1][2][3][4]. The transport properties of such devices have been studied extensively both experimentally and theoretically. Theoretical approaches have been reviewed by Andergassen et al [5] and Zimbovskaya and Pederson [6].The transport properties of single-molecule devices are often dominated by strong interactions. Electron-electron interaction, which leads to Coulomb blockade [7,8] and the coupling of electrons to vibrational modes [7-10], have to be taken into account. We assume the nuclear motion to be slow on the timescale of electronic transitions, which is the case for most, but not all, single-molecule devices studied so far [11]. The electron-vibron coupling is then due to the changes of the equilibrium nuclear configuration with the electronic state, i.e. the Franck-Condon effect: even if a molecule is initially in a vibrational eigenstate, after an electronic transition it will be in a superposition of vibrational eigenstates belonging to the new electronic configuration. The probability of ending up in any one of them is determined by the Franck-Condon matrix elements between vibrational eigenstates for the old and new electronic configurations [12][13][14][15][16]. These matrix elements can be very small if the equilibrium value of the relevant normal coordinate changes strongly with the electronic state. Consequently, the rates for electrons tunneling into or out of the molecule and thus the current can be strongly suppressed-this is the Franck-Condon blockade [14][15][16][17][18][19][20][21]. The dynamics is also unusual in this regime: electrons tunnel in avalanches separated by quiet time intervals [14,15]. The avalanchetype transport is self-similar on intermediate timescales and also leads to a strong enhancement of the zero-frequency Fano factor [14][15][16].Strong interactions and states far from equilibrium render the description of molecular devices difficult in general [5,6]. We focus here on the case of a weak hybridization between the molecule and the leads. In this case, master-equation (ME) approaches [13,14,[16][17][18][19][22][23][24] or the equivalent real-time diagrammatic approach [25][26][27] can be employed. In principle, the ME takes into account all the interactions in the molecule but r...

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Section: Transmitting Regimesupporting

confidence: 69%

“…This proves that at least one stationary state exists. This solution is unique if the system is ergodic in the sense that every state can be reached from every other state by a finite number of transitions [41][42][43][44]. This condition is satisfied by our model for non-zero temperature.…”

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confidence: 99%

Spectroscopy of the transition-rate matrix for molecular junctions: dynamics in the Franck–Condon regime

Timm

2012

New J. Phys.

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“…We define a matrix A, as follows: A ij = e −rij /ξ , for i = j and A ii = − j =i A ij , the latter definition expressing a conservation law in the physical problem [7][8][9]. We shall be interested in determining P (λ), the probability density of eigenvalues of the matrix A, for low densities (small values of ǫ).…”

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confidence: 99%

Localization, Anomalous Diffusion, and Slow Relaxations: A Random Distance Matrix Approach

2010

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We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such as the diffusion of particles, slow relaxations in glasses, and scalar phonon localization. Using a combination of a renormalization group procedure and a direct moment calculation, we find the eigenvalue distribution density (i.e., the spectrum), for low densities, and the localization properties of the eigenmodes, for arbitrary dimension. Finally, we discuss the physical implications of the results.PACS numbers: 02.10. Yn, 71.23.Cq, Application of the theory of random matrices whose elements are independent Gaussian variables has proven to be rich mathematically and relevant for many physical systems [1]. In this Letter we study a different class of random matrices where the i, j'th element is a function of the Euclidian distance r ij between pairs of points whose positions are chosen randomly and uniformly in a d-dimensional space. It is natural that in cases where the matrix element is related to an overlap between localized quantum-mechanical wavefunctions, the dependence on the distance will be exponential, i.e., A ij = e −rij /ξ , with ξ being the localization length [2].The exponential matrix is an appropriate model for various physical systems, in this Letter we will concentrate on its application to glasses relaxing to equilibrium, a particle diffusing in random environment and localization of phonons. Most of the results are derived at the low density limit, when ǫ = ξ/r nn ≪ 1, with r nn being the average nearest neighbor distance. To understand the properties of these systems one need to find out the distribution density P (λ) of the eigenvalues λ as well as the structure of the eigenmodes. An intuitive picture of the problem arises in the application to phonon localization with springs constants K ij that depend exponentially on the Euclidean distances between the masses; we therefore use the phonon terminology: eigenmode.The low density limit allows us to find P (λ) analytically employing a direct calculation of its moments, see Eq. (2) and the Supplementary Material (SM). We find that P (λ) ∼ 1/λ in all dimensions over a broad range of λ's. While in one dimension the normalization of P (λ) is assured by an integrable power-law divergence at eigenvalues close to zero, for higher dimensions there is a peak related to a finite cutoff, cf Fig. 1. We use a logarithmic scale to plot P [log(−λ/2)] in order emphasize the deviations from the 1/λ distribution.To comprehend the structure of the eigenmodes we use a renormalization group (RG) approach for random systems that was developed in the context of spin chains [3][4][5][6]. At each RG step, we choose the stiffest spring. Since the spring is large by construction, after finding the eigenvalue associated with the stiffest spring we can glue together the two masses at its ends creating a larger mass. At the next RG ste...

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“…12 . Using a perturbative approach, we calculate the ensemble-averaged CGF as a series in the inverse of system size N .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in Ref. 12 Timm considered random transition-rate matrices (we shall use the terminology Liouvillian here), which are non-Hermitian matrices arising from the description of a dynamical process in terms of a rate or master equation. Timm gave a detailed account of the spectral properties of such matrices.…”

Section: Introductionmentioning

confidence: 99%

Full-counting statistics of random transition-rate matrices

Mordovina

Emary

2013

Phys. Rev. E

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We study the full counting statistics of current of large open systems through the application of random matrix theory to transition-rate matrices. We develop a method for calculating the ensemble-averaged current-cumulant generating functions based on an expansion in terms of the inverse system size. We investigate how different symmetry properties and different counting schemes affect the results.

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Random transition-rate matrices for the master equation

Cited by 35 publications

References 31 publications

Spectroscopy of the transition-rate matrix for molecular junctions: dynamics in the Franck–Condon regime

Spectroscopy of the transition-rate matrix for molecular junctions: dynamics in the Franck–Condon regime

Localization, Anomalous Diffusion, and Slow Relaxations: A Random Distance Matrix Approach

Full-counting statistics of random transition-rate matrices

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