Final Report Summary - MICRONM (Microscopic derivation of non Markovian dynamics) FINAL REPORT FP7-PEOPLE-2012-IEF MICRONM 328600The project concerned the mathematically rigorous derivation of macroscopic Non- Markovian evolutions equations. Such equations are of interest in open quantum systems, chemical physics, and biological systems. Non-Markovian evolution refers to evolutions which incorporate memory effects. The most famous physical memory effect is the hysteresis effect in magnetic material. Non-Markovian behavior is however ubiquitous in systems composed of many constituents and appears when one focuses on a subsystem. A mathematically rigorous treatment of a subsystem description involving memory is extremely hard. The mathematical study of differential equations containing memory (so called delay equations) presents an open mathematical challenge as it is not clear how one should in general classify solutions of such equations. Independent of that, the Non-Markovian behavior of a subsystem as part of a large system composed of many constituents captures the full complexity of the large system and no analytical treatment is possible. To achieve some description which lends itself to analysis and applications one must invoke approximations which are mainly based on probabilistic arguments. The aim is to find an approximative description of the evolution of the subsystem in which the complexity of the full system is captured in few so called transport coefficients. We refer to that as effective descriptions. There are only very few artificial models which can be treated with mathematical rigor. A very famous example, however in terms of a Markovian approximation where the memory can be shown to be approximately negligible is Einstein’s derivation of Brownian motion in 1905. The experimental verification by Perrin in 1907 earned him the Nobel Price. Einstein got his Nobel price for the photo effect, but his discovery of Brownian motion was much more influential than the latter as it reintroduced the understanding and conviction that atoms existed and that they make up matter, nowadays universally accepted and well known. But in recent decades physicists, chemists and biologists understood that for many phenomena Non-Markovian behavior is responsible or at least cannot be ignored, as for example in ultra fast chemical reactions or FMO complexes in biological systems.The Curie project MicroNM was about a mathematical rigorous study of Non-Markovian effective descriptions on a very simple mathematical model, where one particle is interacting with very many particles via harmonic forces. One may picture a ball on which many springs with particles at their ends push an pull. That system can be mathematically treated, even solved in principle, and the task was to express the mathematical description in such a way that a controllable effective Non-Markovian description can emerge. The quality of the Non-Markovian description depends on appropriate times scales and thus the task was also to find the time scale on which the description is effective and effectively correct. The mathematics involved is the transformation of the starting model by an inverse eigenvalue problem into a chain of springs (harmonic oscillators), a so called chain mapping. The results obtained are not spectacular in the sense that now all open problems in chemistry and biology or physics depending on Non-Markovian behavior can be solved. The output of the project is rather a better understanding of how one could possibly proceed to achieve such desired solutions. Mathematical rigor is important in such fundamental approaches to ensure correct understanding. It was important that the system we considered is a finite discrete system as this is closer to real systems. For so called continuous systems similar questions have been successfully addressed but the method in finite systems has its own methodology.The program was carried out in collaborations with other groups, most prominently the group of Angelo Bassi in Trieste and of Lajos Diosi in Budapest. The trainee visited these groups on a regular basis to update the state of the art and discuss further steps. We also controlled the research process by the trainee in regular seminars as well as having the trainee present the results on international conferences, some of which were organized within the European COST Action ”Fundamental Problem in Quantum Physics”. Among the deliverables are publications in peer reviewed journals.No changes to the legal status of any of the beneficiaries were made, no gender nor ethical issues occurred during the project.
