Efficient characterisation of large deviations using population dynamics

We consider population dynamics as implemented by the cloning algorithm for analysis of large deviations of time-averaged quantities. We use the simple symmetric exclusion process with periodic boundary conditions as a prototypical example and investigate the convergence of the results with respect to the algorithmic parameters, focussing on the dynamical phase transition between homogeneous and inhomogeneous states, where convergence is relatively difficult to achieve. We discuss how the performance of the algorithm can be optimised, and how it can be efficiently exploited on parallel computing platforms

Error in the invariant measure of numerical discretization schemes for canonical sampling of molecular dynamics

Molecular dynamics (MD) computations aim to simulate materials at the atomic level by approximating molecular interactions classically, relying on the Born-Oppenheimer approximation and semi-empirical potential energy functions as an alternative to solving the difficult time-dependent Schrodinger equation. An approximate solution is obtained by discretization in time, with an appropriate algorithm used to advance the state of the system between successive timesteps. Modern MD simulations simulate complex systems with as many as a trillion individual atoms in three spatial dimensions. Many applications use MD to compute ensemble averages of molecular systems at constant temperature. Langevin dynamics approximates the effects of weakly coupling an external energy reservoir to a system of interest, by adding the stochastic Ornstein- Uhlenbeck process to the system momenta, where the resulting trajectories are ergodic with respect to the canonical (Boltzmann-Gibbs) distribution. By solving the resulting stochastic differential equations (SDEs), we can compute trajectories that sample the accessible states of a system at a constant temperature by evolving the dynamics in time. The complexity of the classical potential energy function requires the use of efficient discretization schemes to evolve the dynamics.In this thesis we provide a systematic evaluation of splitting-based methods for the integration of Langevin dynamics. We focus on the weak properties of methods for confiurational sampling in MD, given as the accuracy of averages computed via numerical discretization. Our emphasis is on the application of discretization algorithms to high performance computing (HPC) simulations of a wide variety of phenomena, where configurational sampling is the goal. Our first contribution is to give a framework for the analysis of stochastic splitting methods in the spirit of backward error analysis, which provides, in certain cases, explicit formulae required to correct the errors in observed averages. A second contribution of this thesis is the investigation of the performance of schemes in the overdamped limit of Langevin dynamics (Brownian or Smoluchowski dynamics), showing the inconsistency of some numerical schemes in this limit. A new method is given that is second-order accurate (in law) but requires only one force evaluation per timestep. Finally we compare the performance of our derived schemes against those in common use in MD codes, by comparing the observed errors introduced by each algorithm when sampling a solvated alanine dipeptide molecule, based on our implementation of the schemes in state-of-the-art molecular simulation software. One scheme is found to give exceptional results for the computed averages of functions purely of position

A Molecular Dynamics Study of the Structure-Dynamics Relationships of Supercooled Liquids and Glasses

Central to the field of condensed matter physics is a decades old outstanding problem in the study of glasses – namely explaining the extreme slowing of dynamics in a liquid as it is supercooled towards the so-called glass transition. Efforts to universally describe the stretched relaxation processes and heterogeneous dynamics that characteristically develop in supercooled liquids remain divided in both their approaches and successes. Towards this end, a consensus on the role that atomic and molecular structures play in the liquid is even more tenuous. However, mounting material science research efforts have culminated to reveal that the vast diversity of metallic glass species and their properties are rooted in an equally-broad set of structural archetypes. Herein lies the motivation of this dissertation: the detailed information available regarding the structure-property relationships of metallic glasses provides a new context in which one can study the evolution of a supercooled liquid by utilizing a structural motif that is known to dominate the glass. Cu_64 Zr_36 is a binary alloy whose good glass-forming ability and simple composition makes it a canonical material to both empirical and numerical studies. Here, we perform classical molecular dynamics simulations and conduct a comprehensive analysis of the dynamical regimes of liquid Cu_64 Zr_36, while focusing on the roles played by atomic icosahedral ordering – a structural motif which ultimately percolates the glass’ structure. Large data analysis techniques are leveraged to obtain uniquely detailed structural and dynamical information in this context. In doing so, we develop the first account of the origin of icosahedral order in this alloy, revealing deep connections between this incipient structural ordering, frustration-limited domain theory, and recent important empirical findings that are relevant to the nature of metallic liquids at large. Furthermore, important dynamical landmarks such as the breakdown of the Stokes-Einstein relationship, the decoupling of particle diffusivities, and the development of general “glassy” relaxation features are found to coincide with successive manifestation of icosahedral ordering that arise as the liquid is supercooled. Remarkably, we detect critical-like features in the growth of the icosahedron network, with signatures that suggest that a liquid-liquid phase transition may occur in the deeply supercooled regime to precede glass formation. Such a transition is predicted to occur in many supercooled liquids, although explicit evidence of this phenomenon in realistic systems is scarce. Ultimately this work concludes that icosahedral order characterizes all dynamical regimes of Cu_64 Zr_36, demonstrating the importance and utility of studying supercooled liquids in the context of locally-preferred structure. More broadly, it serves to confirm and inform recent theoretical and empirical findings that are central to understanding the physics underlying the glass transition

Efficient characterisation of large deviations using population dynamics

We consider population dynamics as implemented by the cloning algorithm for analysis of large deviations of time-averaged quantities. Using the simple symmetric exclusion process as a prototypical example, we investigate the convergence of the results with respect to the algorithmic parameters, focussing on the dynamical phase transition between homogeneous and inhomogeneous states, where convergence is relatively difficult to achieve. We discuss how the performance of the algorithm can be optimised, and how it can be efficiently exploited on parallel computing platforms.Comment: 23 pages, final versio

Sorption in disordered porous media

The lattice-gas model of sorption in disordered porous media is studied for a variety of settings, using existing, updated and newly developed numerical techniques. Firstly, we construct an efficient algorithm to calculate the exact partition function for small lattice-gas systems. The exact partition function is used for detailed analysis of the core features exhibited by such systems. We proceed to develop an interactive Monte Carlo (MC) simulation engine, that simulates sorption in a porous media sample and provides real-time visual data of the state space projection and the 3d view of the sample among other parameters of interest, as the external fields are manipulated. The use of such tool provides a more intuitive understanding of the system behaviour. The MC simulations are employed to study sorption in several porous solids: silica aerogel, Vycor glass and soil. We investigate how the phenomena depend on the microstructure of the original samples, how the behaviour varies with the external conditions, and how it is reflected in the paths that the system takes across its state space. Secondly, we develop two methods for estimation of the relative degeneracy (the number of microstates that have the same value of some macroscopic variables) in the systems that are too large to be handled exactly. The methods, based on a restricted infinite temperature sampling, obtain equidegenerate surfaces and the degeneracy gradient across the state space. Combined with the knowledge of an internal energy of a microstate, it enables us to construct the free energy map and thus the equilibrium probability distribution for the studied projection of the state space. Thirdly, the jump-walking Monte-Carlo algorithm is revisited and updated to study the equilibrium properties of systems exhibiting quasi-ergodicity. It is designed for a single processing thread as opposed to currently predominant algorithms for large parallel processing systems. The updated algorithm is tested on the Ising model and applied to the lattice-gas model for sorption in aerogel and Vycor glass at low temperatures, when dynamics of the system is significantly slowed down. It is demonstrated that the updated jump-walking simulations are able to produce equilibrium isotherms which are typically hidden by the hysteresis effect characteristic of the standard single-flip simulations. As a result, we answer the long standing question about the existence of the first-order phase transitions in Vycor. Finally, we investigate sorption in several distinct topology network representations of soil and aerogel samples and demonstrate that the recently developed analytical techniques for random networks can be used to achieve a qualitative understanding of the phenomena in real materials.EPSR

Deterministic Chaos in Digital Cryptography

This thesis studies the application of deterministic chaos to digital cryptography. Cryptographic systems such as pseudo-random generators (PRNG), block ciphers and hash functions are regarded as a dynamic system (X, j), where X is a state space (Le. message space) and f : X -+ X is an iterated function. In both chaos theory and cryptography, the object of study is a dynamic system that performs an iterative nonlinear transformation of information in an apparently unpredictable but deterministic manner. In terms of chaos theory, the sensitivity to the initial conditions together with the mixing property ensures cryptographic confusion (statistical independence) and diffusion (uniform propagation of plaintext and key randomness into cihertext). This synergetic relationship between the properties of chaotic and cryptographic systems is considered at both the theoretical and practical levels: The theoretical background upon which this relationship is based, includes discussions on chaos, ergodicity, complexity, randomness, unpredictability and entropy. Two approaches to the finite-state implementation of chaotic systems (Le. pseudo-chaos) are considered: (i) floating-point approximation of continuous-state chaos; (ii) binary pseudo-chaos. An overview is given of chaotic systems underpinning cryptographic algorithms along with their strengths and weaknesses. Though all conventional cryposystems are considered binary pseudo-chaos, neither chaos, nor pseudo-chaos are sufficient to guarantee cryptographic strength and security. A dynamic system is said to have an analytical solution Xn = (xo) if any trajectory point Xn can be computed directly from the initial conditions Xo, without performing n iterations. A chaotic system with an analytical solution may have a unpredictable multi-valued map Xn+l = f(xn). Their floating-point approximation is studied in the context of pseudo-random generators. A cryptographic software system E-Larm ™ implementing a multistream pseudo-chaotic generator is described. Several pseudo-chaotic systems including the logistic map, sine map, tangent- and logarithm feedback maps, sawteeth and tent maps are evaluated by means of floating point computations. Two types of partitioning are used to extract pseudo-random from the floating-point state variable: (i) combining the last significant bits of the floating-point number (for nonlinear maps); and (ii) threshold partitioning (for piecewise linear maps). Multi-round iterations are produced to decrease the bit dependence and increase non-linearity. Relationships between pseudo-chaotic systems are introduced to avoid short cycles (each system influences periodically the states of other systems used in the encryption session). An evaluation of cryptographic properties of E-Larm is given using graphical plots such as state distributions, phase-space portraits, spectral density Fourier transform, approximated entropy (APEN), cycle length histogram, as well as a variety of statistical tests from the National Institute of Standards and Technology (NIST) suite. Though E-Larm passes all tests recommended by NIST, an approach based on the floating-point approximation of chaos is inefficient in terms of the quality/performance ratio (compared with existing PRNG algorithms). Also no solution is known to control short cycles. In conclusion, the role of chaos theory in cryptography is identified; disadvantages of floating-point pseudo-chaos are emphasized although binary pseudo-chaos is considered useful for cryptographic applications.Durand Technology Limite

The RSB order parameter in finite-dimensional spin glasses: numerical computation at zero temperature

This thesis is focused on the computation of the overlap distribution which characterizes spin glasses with finite connectivity upon an RSB transition at zero temperature. Two models are considered: the J± Bethe lattice spin glass and the Edwards-Anderson spin glass in three dimensions with random regular bond dilution (random dilution with the constraint of fixed connectivity z = 3). The approach is based on the study of the effects of a bulk perturbation on the energy landscape. In ultrametric spin glasses, the distribution of the excited states is known to be related to the order parameter through a universal formula. This formula is used for deriving the order parameter from the experimental distributions. In addition, the finite-size corrections to the ground state energy are computed for the two models