Abstract:
I will present an overview over challenges faced by simulations of many body problems in statistical physics, especially in the vicinity of phase transitions. Large scale simulations are only possible with Monte Carlo methods pioneered by the Metropolis algorithm developed fifty-five years ago. Only Monte Carlo simulations allow the sampling of the exponentially large configuration space in a time that is polynomial (and often linear) in the problem size. While Monte Carlo methods thus seem like the ideal methods they face three big problems close to phase transitions: critical slowing down, exponential tunneling times and the negative sign problem for some quantum simulations.
- Critical slowing down is a divergence of the autocorrelation times as a consequence of diverging length scales at a phase transition: local update schemes are no longer efficient and optimized cluster updates are needed.
- Slow tunneling between local minima of the energy is a challenge at first order phase transitions and in tunneling over barriers such as in protein folding. There, optimized ensemble methods can give exponential speedup in tunneling times through these barriers.
- Finally the biggest challenge for quantum mechanical simulations is the negative sign problem: the weights of configurations can become negative, resulting in a numerical cancellation problem, which scales exponentially with the system size. This problem turns out to be NP-hard and can thus only be solved in special cases.
Coffee and tea will be served 20 minutes prior to the seminar at the Departmental Lobby, 5/F, Department of Physics, Chong Yuet Ming Physics Building.