Efficient path sampling for trajectory ensembles with applications to non-equilibrium systems - Turner, R.M. (2016)
This thesis utilises large deviation methods to study nonequilibrium phenomena in both quantum and classical systems. The dynamical analogues of the ensembles of statistical mechanics are used to explore dynamical phase spaces of systems, quantifying atypical fluctuations that can play a critical role in long term behaviour. A dynamical ensemble based on fixed numbers of dynamical events, allowing trajectory observation time to fluctuate, is introduced. This ensemble, denoted the x-ensemble, is found to be well suited to numerically simulate atypical fluctuations using transition path sampling (TPS). x-ensemble TPS schemes are analysed with reference to existing methods in both quantum and classical stochastic systems, and are found to offer more flexibility and efficiency in a variety of situations. The potential to develop this scheme into a self-optimizing algorithm is discussed with examples. The x-ensemble is then used in three non-equilibrium scenarios. Firstly in plaquette models of glass formers, in an effort to provide insight into the nature of the glass transition. It is shown that a two-dimensional triangular plaquette model (TPM) exhibits both a trajectory phase-transition between dynamical active and inactive phases, and when two replicas are coupled, a thermal phase transition between states of low and high overlap between the replicas. These two transitions are similar to those seen to occur in more realistic glass formers. When the TPM is generalised to a three-dimensional square pyramid plaquette model (SPyM) these dynamical and thermodynamic features of interest remain. It is argued that these models therefore provide an ideal test-bed for competing theories of the glass transition. Secondly the x-ensemble is used to define and analyse the dynamical analogue of the Jarzynski equality, allowing for the computation of dynamical free energy differences with, in principle, arbitrarily fast protocols linking two dynamical states. This relation is tested and found to hold in open quantum systems. Finally the partition sum zeros method of Lee and Yang is used to extract the location of dynamical phase transitions from the high-order, short-time cumulants of the x-ensemble. Results in both classical and open quantum systems are compared with previously studied dynamical ensembles, providing insight into the nature in which dynamical behaviours are encoded by these ensembles.