Report
Teaser, summary, work performed and final results
Periodic Reporting for period 2 - LoTGlasSy (Low Temperature Glassy Systems)
Teaser
The behaviour of a heap of near frictionless hard objects (e.g. spherical marble beads) has been the object of scientific investigations for a very long time. A very promising line of attack has been found where very sophisticated statistical mechanics tools has been used:...
Summary
The behaviour of a heap of near frictionless hard objects (e.g. spherical marble beads) has been the object of scientific investigations for a very long time. A very promising line of attack has been found where very sophisticated statistical mechanics tools has been used: indeed, if we neglect friction, we recover the problem of a hard spheres gas in the limit of very large pressure, the so called jamming point.
Analytic tools like mean field theory for disordered systems have been constructed: at the end of the day the approximated theoretical results have been found to be in reasonable agreement with experiments and extensive numerical simulations.
The main issue is to transform these approximated theoretical results (that should be correct in the limit of very high dimensions) into a precise theory in three dimensions where the different approximations involved should be under strict theoretical control.
A precise understanding of the jamming point would strongly benefit the whole the area of study of glasses, that are central in modern science and applications. Trampolining from these results there are many properties of glasses that can be understood, especially studying what happens not far from the jamming point.
The number of applications may be very large. Just an example: the problem of memory capacity of a neural network is deeply related to a phenomenon that is the equivalent of jamming; many of the ideas developed in this context may be extremely useful in constructing a first principle theory of deep learning with neural networks.
The main objectives are:
• We want develop a complete analytic theory of the infinite pressure limit (jamming) of hard spheres in finite dimensions. We will first compute analytically the dimensional dependence of all the critical exponents that characterize jamming and we will compare with numerical simulations.
• We want to derive universal properties of glassy materials in the low temperature regions near the jamming point both in the classical and in the quantum regime. We aim to understand the structure of small (localized or extended) oscillations around them and the tunneling barriers that arise from classical thermal activation and from quantum tunneling.
• We want analyze off-equilibrium features, like plasticity and the time dependence of the specific heat. The subject has been widely discussed and phenomenological laws have been derived. The goal is to derive from first principles the observed phenomenological laws laws using a controlled analytic approach.
Work performed
During this period an important part of the work performed consists in forging the instruments needed to make further progress in the second part of the project. In the meanwhile we have also obtained very important scientific results.
• We have found a simple way to describe hard spheres at very high pressure: this can be done using the appropriate mean field equations that are a great simplification with respect to much complex functional equations that have been used in the past.
• We have started the study of quantum hard spheres near jamming and we have obtained the precise results on their behavior in a particular regime.
• We have started to put under better theoretical and numerical control the properties of the free energy cost of the interfaces among different glassy states.
• We have carefully studied the dynamics in the case of spin glasses and we have developed a methodology that enable us to do a careful and successful comparison with the experimental data.
Final results
We have improved the state of art introducing new methodologies in many respects
• We have shown how to compute the spectrum of small oscillations in more realistic mean field models (Bethe approximation), removing one of the non realistic feature of the usual mean field models (i.e. the diverging mean connectivity). We found impressive differences that need to be better understood in order to build the correct mean field theory.
• We have introduced a new method for computing corrections to mean field models. This is an important step forward in a field where no progress has been done in the last forty years.
• We have used new quite sophisticated mathematics to describe the properties of glassy systems.
As far the future is concerned, until the end of the project we adopt the following strategy:
- We plan to study in more detail mean field models on the Bethe lattice in order to develop the correct mean field theory for jamming and glassy phenomena
- We plan to study out of equilibrium dynamics in some of these mean field models to better understand the connection with the complex energy landscape
- Critical behavior with localized and extended low energy excitation will be also studied to better understand the nature of the critical point in these models
- We plan to do full quantum computations in mean field models.
- We plan to arrive to control the corrections to mean field theories in finite dimensions.
Website & more info
More info: http://chimera.roma1.infn.it/LOTGLASSY/.