Classical Many-body Systems Research Articles

Temperature Dependence of the Butterfly Effect in a Classical Many-Body System.

We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, μ, and the butterfly speed, v_{b}, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, D, and spin-autocorrelation time, τ. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form D∼v_{b}^{2}/μ and τ^{-1}∼T. The vanishing of μ∼T^{0.48} is parametrically slower than that of the corresponding quantum bound, μ∼T, raising interesting questions regarding the semiclassical limit of such spin systems.

Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay

We study the robustness and asymptotic stability of the unique equilibrium for the Winfree model on a general network under the effect of small time-delay. Time-delay is a ubiquitous mechanism in classical and quantum many-body systems, and it is due to the finite speed of communication. In this paper, we further investigate the robustness of the equilibrium under the effects of small time-delay and general network topology. For this, we present several sufficient frameworks formulated in terms of the network topology, coupling strength, time-delay, and initial configuration. In the proposed framework, we present a positively invariant arc which contains the unique equilibrium independent of strength of time-delay. By using energy estimates, we show that the unique equilibrium in the invariant arc is asymptotically stable. We also provide several numerical simulations and compare them with analytical results.

Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain.

We find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.

The interplay between frustration and entanglement in many-body systems

Frustration of classical many-body systems can be used to distinguish ferromagnetic interactions from anti-ferromagnetic ones via the Toulouse conditions. A quantum version of the Toulouse conditions provides a similar classification based on the local ground states. We compute the global ground states for a family of models with Heisenberg-like interactions and analyse their behaviour with respect to frustration, entanglement and degeneracy. For that we develop analytical and numerical analysing tools capable of quantifying the interplay between those three quantities. We find that the quantum Toulouse conditions provide a proper classification, however, refinements can be found. Our results show how the different local ground states affect the interplay and pave the way for further generalisation and possible applications to other quantum many-body systems.

Quench dynamics of a dissipative Rydberg gas in the classical and quantum regimes

Understanding the non-equilibrium behavior of quantum systems is a major goal of contemporary physics. Much research is currently focused on the dynamics of many-body systems in low-dimensional lattices following a quench, i.e., a sudden change of parameters. Already such a simple setting poses substantial theoretical challenges for the investigation of the real-time post-quench quantum dynamics. In classical many-body systems the Kolmogorov-Mehl-Johnson-Avrami model describes the phase transformation kinetics of a system that is quenched across a first order phase transition. Here we show that a similar approach can be applied for shedding light on the quench dynamics of an interacting gas of Rydberg atoms, which has become an important experimental platform for the investigation of quantum non-equilibrium effects. We are able to gain an analytic understanding of the time-evolution following a sudden quench from an initial state devoid of Rydberg atoms and identify strikingly different behaviors of the excitation growth in the classical and quantum regimes. Our approach allows us to describe quenches near a non-equilibrium phase transition and provides an approximate analytic solution deep in the quantum domain.

Power functional theory for Newtonian many-body dynamics

We construct a variational theory for the inertial dynamics of classical many-body systems out of equilibrium. The governing (power rate) functional depends on three time- and space-dependent one-body distributions, namely, the density, the particle current, and the time derivative of the particle current. The functional is minimized by the true time derivative of the current. Together with the continuity equation, the corresponding Euler-Lagrange equation uniquely determines the time evolution of the system. An adiabatic approximation introduces both the free energy functional and the Brownian free power functional, as is relevant for liquids governed by molecular dynamics at constant temperature. The forces beyond the Brownian power functional are generated from a superpower (above the overdamped Brownian) functional.

Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics

In these four lectures I describe basic ideas and methods applicable to both classical and quantum systems displaying slow relaxation and non-equilibrium dynamics. The first half of these notes considers classical systems, and the second half, quantum systems. In Lecture 1, I briefly review the glass transition problem as a paradigm of slow relaxation and dynamical arrest in classical many-body systems. I discuss theoretical perspectives on how to think about glasses, and in particular how to model them in terms of kinetically constrained dynamics. In Lecture 2, I describe how via large deviation methods it is possible to define a statistical mechanics of trajectories which reveals the dynamical phase structure of systems with complex relaxation such as glasses. Lecture 3 is about closed (i.e. isolated) many-body quantum systems. I review thermalisation and many-body localisation, and consider the possibility of slow thermalisation and quantum non-ergodicity in the absence of disorder, thus connecting with some of the ideas of the first lecture. Lecture 4 is about open quantum systems, that is, quantum systems interacting with an environment. I review the description of open quantum dynamics within the Markovian approximation in terms of quantum master equations and stochastic quantum trajectories, and explain how to extend the dynamical large deviation method to study the statistical properties of ensembles of quantum jump trajectories. My overall aim is to draw analogies between classical and quantum non-equilibrium and find connections in the way we think about problems in these areas.

On the Performance and Energy Consumption of Molecular Dynamics Applications for Heterogeneous CPU-GPU Platforms Based on Gromacs

Abstract High Performance Computing (HPC) accelerates life science discoveries by enabling scientists to analyze large data sets, to develop detailed models of entire biological systems and to simulate complex biological processes. As computational experiments, molecular dynamics simulations are widely used in life sciences to evaluate the equilibrium nature of classical many-body systems The modelling and molecular dynamics study of surfactant, polymer solutions and the stability of proteins and nucleic acids under different conditions, as well as deoxyribonucleic acid proteins are studied. The study aims to understand the scaling behavior of Gromacs (Groningen machine for chemical simulations) on various platforms, and the maximum performance in the prospect of energy consumption that can be accomplished by tuning the hardware and software parameters. Different system sizes (48K, 64K, and 272K) from scientific investigations have been studied show that the GPU (Graphics Processing Unit) scales rather beneficial than other resources, i.e., with GPU support. We track 2-3 times speedup compared to the latest multi-core CPUs. However, the so-called “threading effect” leads to the better results.

Quantum Order-by-Disorder in Strongly Correlated Metals

Entropic forces in classical many-body systems, e.g., colloidal suspensions, can lead to the formation of new phases. Quantum fluctuations can have similar effects: spin fluctuations drive the superfluidity of helium-3, and a similar mechanism operating in metals can give rise to superconductivity. It is conventional to discuss the latter in terms of the forces induced by the quantum fluctuations. However, focusing directly upon the free energy provides a useful alternative perspective in the classical case and can also be applied to study quantum fluctuations. Villain first developed this approach for insulating magnets and coined the term order-by-disorder to describe the observed effect. We discuss the application of this idea to metallic systems, recent progress made in doing so, and the broader prospects for the future.

Momentum-resolved and correlation spectroscopy using quantum probes

We address some key conditions under which many-body lattice models, intended mainly as simulated condensed matter systems, can be investigated via immersed, fully controllable quantum objects, namely quantum probes. First, we present a protocol that, for a certain class of many-body systems, allows for full momentum resolved spectroscopy using one single probe. Furthermore, we demonstrate how one can extract the two-point correlations using two entangled probes. We apply our theoretical proposal to two well-known exactly solvable lattice models, a 1D Kitaev chain and 2D superfluid Bose-Hubbard model, and show its accuracy as well as its robustness against external noise.

Classical integrable many-body systems disconnected with semi-simple Lie algebras

The review of the results in the theory of integrable many-body systems disconnected with semisimple Lie algebras is done. The one-dimensional systems of light Calogero–Sutherland–Moser particles interacting with one particle of infinite mass located at the origin are described in detail. In some cases the exact solutions of the equations of motion are obtained. The general theory of integration of the equations of motion needs the methods of algebraic geometry. The Lax pairs with spectral parameter are constructed for this purpose. The theory still contains many unsolved problems.

Algorithms for tensor network renormalization

We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. Firstly, we recall established techniques for how the partition function of a 2D classical many-body system or the Euclidean path integral of a 1D quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the 2D classical statistical and 1D quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.

Determinism vs. statistics in classical many-body theory: Dynamical origin of irreversibility

The closed equation of motion for microscopic distribution function of classical many-body system with account of interactions retardation between particles is derived. It is shown that interactions retardation leads to irreversible behavior of many-body systems without any probabilistic assumptions.

Self-learning Monte Carlo method

Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of general and efficient update algorithm for large size systems close to phase transition or with strong frustrations, for which local updates perform badly. In this work, we propose a new general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10-20 times speedup.

Nonergodic Phases in Strongly Disordered Random Regular Graphs.

We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson, and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered random regular graphs (RRG) of N sites with the connectivity K=2. By extrapolation of the results of both approaches to N→∞ we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamics exponent D(W) with the accuracy sufficient to claim that they are nontrivial in the broad interval of disorder strength W_{E}<W<W_{c}. The thorough analysis of the exact diagonalization results for RRG with N>10^{5} reveals a singularity in D_{1,2}(W) dependencies which provides clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}≈10.0. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.

Self-consistent phonons: An accurate and practical method to account for anharmonic effects in equilibrium properties of general classical or quantum many-body systems

The self-consistent phonons (SCP) method is a practical approach for computing structural and dynamical properties of a general quantum or classical many-body system while incorporating anharmonic effects. However, a convincing demonstration of the accuracy of SCP and its advantages over the standard harmonic approximation is still lacking. Here we apply SCP to classical Lennard-Jones (LJ) clusters and compare with numerically exact results. The close agreement between the two reveals that SCP accurately describes structural properties of the classical LJ clusters from zero-temperature (where the method is exact) up to the temperatures at which the chosen cluster conformation becomes unstable. Given the similarities between thermal and quantum fluctuations, both physically and within the SCP ansatz, the accuracy of classical SCP over a range of temperatures suggests that quantum SCP is also accurate over a range of quantum de Boer parameter Λ=ℏ/(σmε), which describes the degree of quantum character of the system.

Event-driven Monte Carlo: Exact dynamics at all time scales for discrete-variable models

We present an algorithm for the simulation of the exact real-time dynamics of classical many-body systems with discrete energy levels. In the same spirit of kinetic Monte Carlo methods, a stochastic solution of the master equation is found, with no need to define any other phase-space construction. However, unlike existing methods, the present algorithm does not assume any particular statistical distribution to perform moves or to advance the time, and thus is a unique tool for the numerical exploration of fast and ultra-fast dynamical regimes. By decomposing the problem in a set of two-level subsystems, we find a natural variable step size, that is well defined from the normalization condition of the transition probabilities between the levels. We successfully test the algorithm with known exact solutions for non-equilibrium dynamics and equilibrium thermodynamical properties of Ising-spin models in one and two dimensions, and compare to standard implementations of kinetic Monte Carlo methods. The present algorithm is directly applicable to the study of the real-time dynamics of a large class of classical Markovian chains, and particularly to short-time situations where the exact evolution is relevant.

Classical many-body theory with retarded interactions: Dynamical irreversibility and determinism without probabilities

The exact equations of motion for microscopic density of classical many-body system with account of inter-particle retarded interactions is derived. It is shown that interactions retardation leads to irreversible behavior of many-body systems.

Bill2d — A software package for classical two-dimensional Hamiltonian systems

We present Bill2d, a modern and efficient C++ package for classical simulations of two-dimensional Hamiltonian systems. Bill2d can be used for various billiard and diffusion problems with one or more charged particles with interactions, different external potentials, an external magnetic field, periodic and open boundaries, etc. The software package can also calculate many key quantities in complex systems such as Poincaré sections, survival probabilities, and diffusion coefficients. While aiming at a large class of applicable systems, the code also strives for ease-of-use, efficiency, and modularity for the implementation of additional features. The package comes along with a user guide, a developer’s manual, and a documentation of the application program interface (API). Program summaryProgram title: Bill2dCatalogue identifier: AEYL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 37098No. of bytes in distributed program, including test data, etc.: 1155037Distribution format: tar.gzProgramming language: C++(14).Computer: Tested on x86 and x86 64 architectures.Operating systems: Tested on Linux, and OS X versions 10.9–10.11.Has the code been vectorized or parallelized?: Shared memory parallelization when simulating ensembles of systems. Vectorization of operations with R2 vectors.RAM: Simulation dependent: kilobytes to gigabytesClassification: 4.3, 7.8, 7.9, 7.10, 16.9.External routines: Boost, CMake, GSL, HDF5; and optionally Google-Mock, GoogleTest, and DoxygenNature of problem: Numerical propagation of classical two-dimensional single and many-body systems, possibly in a magnetic field, and calculation of relevant quantities such as Poincaré sections, survival probabilities, diffusion co-efficients, etc.Solution method: Symplectic numerical integration of Hamilton’s equations of motion in Cartesian coordinates, or solution of Newton’s equations of motion if in a magnetic field. The program implements several well-established algorithms.Restrictions: Pointlike particles with equal masses and charges, although the latter restrictions are easy to lift.Unusual features: Program is efficient, extremely modular and easy to extend, and allows arbitrary particle–particle interactions.Additional comments: The source code is also available at https://bitbucket.org/solanpaa/bill2d. See README for locations of user guide, developer manual, and API docs.Running time: From milliseconds to days, depends on type of simulation.

Exact equation for classical many‐body systems: Passage from dynamics to equilibrium

The exact basic equation of motion for the microscopic density of system of interacting particles is derived. In this derivation, no probabilistic hypotheses or assumptions are used. This integro‐differential equation is the equation of motion of nonlinear classical scalar field with respect to the microscopic density. The possible mechanisms of transition of many‐particle system in equilibrium state are discussed. On the basic equation, the exact integral equation for the equilibrium spatial distribution of the particles is obtained. It is shown that the Boltzmann distribution and the Vlasov equation are special cases of this integral equation. The wave equation for almost homogeneous systems with interparticle interactions is obtained. Effect of interparticle interactions on the dispersion law of sound is established. © 2015 Wiley Periodicals, Inc.