EinsteinVRH · Einstein Relation for the Variable Range Hopping model
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2016-02-01 → 2018-01-31
- EU contribution
- €173,076
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Einstein Relation for the Variable Range Hopping model
SUMMARY Semiconductors form the basis for modern electronic and a study of their electrical conductivity is paramount both for applications and for our comprehension of their structure. A model for the flux of electrons in this kind of materials, called the Variable-Range Hopping, has been proposed in the 70’s from the physics community and it is still used nowadays. In mathematical terms, it accounts to a long-range reversible stochastic processes on a random point process, the point process representing the impurities of the material where the electrons localize. For experimental purposes, physicists often have to assume the validity of the Einstein relation for this model, i.e. the equivalence between the diffusivity of the electrons and their mobility. Rigorously proving the Einstein relation for a stochastic process requires an extremely accurate control of the its asymptotics, and for this reasons in the mathematical literature authors had always to assume rather restrictive theoretical hypothesis. Our main goal is to give a completely rigorous proof of the first Einstein relation for a realistic model: the Variable-Range Hopping. Besides the importance for its applications in physics, the project aims at attacking the tough and relatively unexplored framework of random walks in a random environment with infinite-range. The classical theory does not cover this kind of processes, and new tools have to be invented to face them. These tools give in turn the possibility of approaching other models from a new perspective. CONCLUSION OF THE ACTION The project “EinsteinVRH” has been successful beyond expectations. Besides accomplishing the main goal (the proof of the Einstein relation for the Variable-Range Hopping model), our work lead to new techniques and tools that we applied to other models. In the two years of the action, five between publications and submitted articles have been produced and several new projects have started. Our results have been disseminated through conferences, seminars and discussions with other academics, raising the interest of the experts and attracting other scientists on the topic.
Data: CORDIS, © European Union
Project objective
The Variable Range Hopping is considered in the Physics literature as an effective model for the analysis of conductivity in semiconductors. Understanding how the macroscopic parameters depend on the small-scale randomness of the environment and proving the Einstein Relation for this model is the ambitious aim of this project.Main objectives:1) Extend recent results (law of large numbers, existence of a stationary state) for long-range reversible random walks on point processes including the possibility of traps.2) Analyze how an external field influences the limiting velocity of the Variable Range Hop- ping, in comparison to similar models from Mathematical Physics.3) Establish the first rigorous Einstein Relation for a physically relevant model, the Variable Range Hopping.The mathematical techniques we have at our disposal nowadays (such as the weak Einstein Relation and the control of long range models) are a solid basis for the investigation of the problem: This would be the first time an Einstein Relation is rigorously proven for a relevant physical model. Furthermore, the richness of the subject guarantees also many intermediate results of great relevance in the field of Probability Theory.Besides the big scientific relevance of the expected results, the project will have a strong impact also on the career of the experienced researcher, completing his international profile of independent scientist, and will also strengthen the interplay between the Probability Theory communities of France, Germany and Italy. Finally, a positive outcome of the action will bring a significant insight on the physical study of semiconductors.
Original text from CORDIS.
Participants
- UNIVERSITE PARIS DAUPHINE · ParisCoordinatorFrance
Links
Data: CORDIS, © European Union