Mott Law as Lower Bound for a Random Walk in a Random Environment

Faggionato, Alessandra; Schulz-Baldes, Hermann; Spehner, Dominique

doi:10.1007/s00220-005-1492-5

Cited by 29 publications

(115 citation statements)

References 27 publications

(21 reference statements)

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“…The above bound is in agreement with Mott law (1.6). We note that the requirement α > 0 in [8][Theorem 1] is due to a typing error. Moreover it is simple to check that, although in [8] the transition rates c x,y (ξ) are defined as in (1.4), all the results in [8] remain true for transition rates c x,y (ξ) defined as in (1.7) and assuming only that α > −1.…”

Section: Model and Resultsmentioning

confidence: 99%

“…The analogous of Mott law (1.3) for Mott variable-range random walk is given by 6) where D(β) denotes the diffusion matrix of the random walk. For dimension d ≥ 2, a lower bound of D(β) in agreement with (1.6) has been recently proven in [8]. The present work addresses to the problem of rigorously deriving an upper bound of D(β) in agreement with (1.6).…”

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confidence: 78%

“…Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8]. …”

mentioning

confidence: 82%

“…For a more detailed discussion about mathematical aspects and physical motivations of Mott random walk we refer to [8], [17] and references therein.…”

Section: 3mentioning

confidence: 99%

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Mott Law as Upper Bound for a Random Walk in a Random Environment

Faggionato

Mathieu

2008

Commun. Math. Phys.

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Abstract. We consider a random walk on the support of an ergodic simple point process on R d , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann-type factor. This is an effective model for the phonon-induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8].

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Section: Model and Resultsmentioning

confidence: 99%

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confidence: 78%

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confidence: 82%

“…For a more detailed discussion about mathematical aspects and physical motivations of Mott random walk we refer to [8], [17] and references therein.…”

Section: 3mentioning

confidence: 99%

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Mott Law as Upper Bound for a Random Walk in a Random Environment

Faggionato

Mathieu

2008

Commun. Math. Phys.

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“…[1][2][3][4][5][6][7][8][9][10][11] As shown already by Mott,12 this mobility can be understood to result from thermally assisted variable-range hopping ͑VRH͒ between localized states of varying energy. Mott thereby considers a single-hop rate of the form exp͑−␣R − ␤͒ ͑␣ is the inverse localization length and ␤ =1/k B T͒ first introduced by Miller and Abrahams, 13 which accounts both for tunneling through the tail of the wave function and for a phonon-assisted thermal activation.…”

Section: Introductionmentioning

confidence: 98%

Understanding Mott’s law from scaling of variable-range-hopping currents and intrinsic current fluctuations

Pasveer

Michels

2006

Phys. Rev. B

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Citation for published version (APA):Pasveer, W. F., & Michels, M. A. J. (2006). Understanding Mott's law from scaling of variable-range-hopping currents and intrinsic current fluctuations. Physical Review B, 74(19), 195129-1/8. [195129] Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policyIf you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim. We have used the master equation to simulate variable-range hopping ͑VRH͒ of charges in a strongly disordered d-dimensional energy landscape ͑d =1,2,3͒. The current distribution over hopping distances and hopping energies gives a clear insight into the difference between hops that occur most frequently, dominate quantitatively in the integral over the mobility distribution, or are critical ones that still need to be considered in that integral to recover the full low-temperature mobility. The recently reported scaling with temperature of the VRH-current distribution over hopping distances and hopping energies is quantitatively analyzed in 1D and 2D, and accurately confirmed. Based on this, we present an analytical scaling theory of VRH, which distinguishes between a scaling part of the distribution and an exponential tail, separated by critical currents that set the scale and that follow self-consistently at each temperature. This naturally renders Mott's law for the low-temperature mobility, in a way and with a physical picture different from that of the established criticalpercolation-network approach to VRH. We argue that current fluctuations as observed in simulations are intrins...

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Einstein relation for reversible diffusions in a random environment

Gantert

Mathieu

Piatnitski

2011

Comm Pure Appl Math

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We consider reversible diffusions in random environment and prove the Einstein relation for this model. It says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift. The Einstein relation is conjectured to hold for a variety of models but it is proved insofar only in particular cases. Our proof makes use of homogenization arguments, the Girsanov transform, and a refinement of the regeneration times introduced in [25].

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Mott Law as Lower Bound for a Random Walk in a Random Environment

Cited by 29 publications

References 27 publications

Mott Law as Upper Bound for a Random Walk in a Random Environment

Mott Law as Upper Bound for a Random Walk in a Random Environment

Understanding Mott’s law from scaling of variable-range-hopping currents and intrinsic current fluctuations

Einstein relation for reversible diffusions in a random environment

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