Analysis of the effective long time-behaviour of molecular systems
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
Playing golf means: moving the golf ball from the given initial position on the tee through a landscape of hills and valleys to the hole as the final location. Somewhat simplified, the aim of this project is to understand how to play golf not in the three-dimensional space we are accustomed to, but in a complicated landscape with hundreds of dimensions.
While this sounds like a mathematical folly, this is what in fact happens in many complex systems, such as molecules or DNA. For example, molecules can exist in several distinctly different states, sometimes called conformations. These states can be pictured as wells of a complex energy landscape. Let us say that the molecule consists of N atoms, each described by 3 space coordinates and 3 momentum coordinates. Then the wells are in a 6N dimensional space. Starting with one conformation of the molecule (one of the wells, as the chemical equivalent of the tee), the atoms will then spend most of the time jostling around in that well before a rare spontaneous fluctuation occurs that lifts the atoms of the reactant over the barrier into the next valley, the well of the other conformation (corresponding to the hole in the golf analogy). This is an example of a so-called rare event. And while these events are indeed rare, they normally carry crucial information on the system in question. So one would like to understand and predict these transitions and rare events. However, a direct molecular simulation would need to resolve the atomistic timescale, while rare events take place on timescales which can be larger by many orders of magnitude; this renders direct simulations unfeasible even on the largest supercomputers.
Instead, the aim of this project is to derive rigorously reduced models that capture the effective long-time behaviour of
high-dimensional complex systems. A particular focus will be on rare events and transition states. A variety of model problems will be investigated, chosen to capture key challenges present in a number of more complicated problems in various application areas.
While this sounds like a mathematical folly, this is what in fact happens in many complex systems, such as molecules or DNA. For example, molecules can exist in several distinctly different states, sometimes called conformations. These states can be pictured as wells of a complex energy landscape. Let us say that the molecule consists of N atoms, each described by 3 space coordinates and 3 momentum coordinates. Then the wells are in a 6N dimensional space. Starting with one conformation of the molecule (one of the wells, as the chemical equivalent of the tee), the atoms will then spend most of the time jostling around in that well before a rare spontaneous fluctuation occurs that lifts the atoms of the reactant over the barrier into the next valley, the well of the other conformation (corresponding to the hole in the golf analogy). This is an example of a so-called rare event. And while these events are indeed rare, they normally carry crucial information on the system in question. So one would like to understand and predict these transitions and rare events. However, a direct molecular simulation would need to resolve the atomistic timescale, while rare events take place on timescales which can be larger by many orders of magnitude; this renders direct simulations unfeasible even on the largest supercomputers.
Instead, the aim of this project is to derive rigorously reduced models that capture the effective long-time behaviour of
high-dimensional complex systems. A particular focus will be on rare events and transition states. A variety of model problems will be investigated, chosen to capture key challenges present in a number of more complicated problems in various application areas.
Planned Impact
The immediate beneficiaries of the research will be the community of mathematicians working on dynamical systems with several time scales. This includes researchers in applied stochastic processes and applied analysis, and in particular people studying the effective numerical simulation of molecular systems. This represents a substantial international community, with significant groups and keyplayers in the UK.
One aim of the research is to foster the growing community working on the mathematical analysis of problems arising in computational chemistry (thermostatting, free energy calculations, rare events). The research brings together applied analysis and applied stochastic processes in a novel way.
The core challenge of the proposed research is to derive effective descriptions of high-dimensional complex systems. If the objectives of this project are achieved, it will provide an in-depth understanding of model problems which are amenable to a rigorous analysis. The understanding gained there can then be extrapolated to realistic models appearing in a range of application areas. The benefit to the wide community working on problems in this kind, both in mathematics and the sciences, is thus an improved understanding of how to derive effective descriptions of complex systems. This understanding can the inform the design of algorithms, giving scientists a handle how to study systems which are too large and have to disparate time-scales to be simulated directly. We anticipate a number of interdisciplinary follow-on projects, with the overarching tool of developing and applying new tools for understanding, predicting and sometimes controlling the evolution of complex systems.
One aim of the research is to foster the growing community working on the mathematical analysis of problems arising in computational chemistry (thermostatting, free energy calculations, rare events). The research brings together applied analysis and applied stochastic processes in a novel way.
The core challenge of the proposed research is to derive effective descriptions of high-dimensional complex systems. If the objectives of this project are achieved, it will provide an in-depth understanding of model problems which are amenable to a rigorous analysis. The understanding gained there can then be extrapolated to realistic models appearing in a range of application areas. The benefit to the wide community working on problems in this kind, both in mathematics and the sciences, is thus an improved understanding of how to derive effective descriptions of complex systems. This understanding can the inform the design of algorithms, giving scientists a handle how to study systems which are too large and have to disparate time-scales to be simulated directly. We anticipate a number of interdisciplinary follow-on projects, with the overarching tool of developing and applying new tools for understanding, predicting and sometimes controlling the evolution of complex systems.
Organisations
People |
ORCID iD |
Johannes Zimmer (Principal Investigator) |
Publications
Buffoni B
(2015)
Travelling waves for a Frenkel-Kontorova chain
Buffoni B
(2019)
Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential
in Journal de Mathématiques Pures et Appliquées
Buffoni B
(2017)
Travelling waves for a Frenkel-Kontorova chain
in Journal of Differential Equations
Dirr N
(2016)
Entropic and gradient flow formulations for nonlinear diffusion
in Journal of Mathematical Physics
Dirr N
(2017)
Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles.
in Journal of statistical physics
Duhart H
(2017)
The Semi-infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products
in Potential Analysis
Title | Performance "This Moment Now", exploring irreversibility of time |
Description | The artist Sylvia Rimat, http://www.sylviarimat.com, found publications on entropy on the web and contacted me. She then explored the notion of irreversibility of time in a performance "This Moment Now" |
Type Of Art | Performance (Music, Dance, Drama, etc) |
Year Produced | 2015 |
Impact | The performance was in my view very well received and communicated questions of the arrow of time to a wide audience. |
Description | This award has to a number of results including new interdisciplinary collaborations with physics (paper with Jack) and engineering (paper with Reina). Key results include the analysis of links between gradient flows, large deviations and stochastic particle models. Specifically, we have shown that equations of fluctuating hydrodynamics type are distinguished by their compatibility with the geometry of the deterministic evolution. These theoretical findings can be used for modelling, as they lead to thermodynamically consistent models. Another outcome is a novel approach to find transition paths between different paths in a multi-well landscape. Such paths are of interest in chemistry, physics and other disciplines; they can describe the behaviour of one stable configuration into another one. Transition paths are hard to compute, and we developed a numerical method to compute such paths and gave a convergence proof. Another insight is that there is a variational structure for (finite state) Markov chains, which can be useful to study acceleration to equilibrium and other problems. |
Exploitation Route | The curve-shortening approach we developed to compute transition paths has been translated into an open-souce code, which has been downloaded thousands of times. |
Sectors | Chemicals Energy |
Description | We developed an open source code for transition state theory, see http://suttond.github.io/GeometricMD/. |
Sector | Chemicals,Energy |
Impact Types | Economic |