Dimensional Phase Space Research Articles

Derivation of Bose–Einstein statistics from the uncertainty principle

Abstract The microstate of any degree of freedom of any classical dynamical system can be represented by a point in its two dimensional phase space. Since infinitely precise measurements are impossible, a measurement can, at best, constrain the location of this point to a region of phase space whose area is finite. This paper explores the implications of assuming that this finite area is bounded from below. I prove that if the same lower bound applied to every degree of freedom of a sufficiently-cold classical dynamical system, the distribution of the system’s energy among its degrees of freedom would be a Bose–Einstein distribution.

Chaotic LLM billiards

We study null geodesics of the ten-dimensional LLM geometries. In particular, we show that there are a subset of these null geodesics that are confined to the LLM plane. The effective dynamics of these in-plane geodesics is that of a Hamiltonian system with two degrees of freedom (a phase space of dimension 4). We show that these are chaotic. In the two-coloring of the LLM plane, if they start in the empty region, they cannot penetrate the filled region and viceversa. The dynamical problem is therefore very similar to that of a billiards problem with fixed obstacles. We study to what extent LLM geometries with many droplets may be treated as an incipient black hole and draw analogies with the fuzzball proposal. We argue that for in-plane null geodesics deep in the interior of a region with a lot of droplets, in order to exit towards the AdS boundary they will need to undergo a process that resembles diffusion. This mechanism can account for signals getting lost in the putative black hole for a very long time.

A reduced order model discretisation of the space-angle phase-space dimensions of the Boltzmann transport equation with application to nuclear reactor problems

This article presents a new reduced order model (ROM) for fast solutions to neutron transport problems. The novelty lies in the construction of optimal basis functions spanning the space-angle phase-space dimensions of the Boltzmann transport equation (BTE). It uses Proper Orthogonal Decomposition and the method of snapshots to form the reduced basis, but here a 2-stage construction is proposed that compresses the angle, then space, dimensions sequentially. The approach alleviates the potentially limiting memory burden for BTE-based ROMs by not processing the full discretised solutions of BTE during the construction stage. The model is both accurate and efficient and is demonstrated here for eigenvalue and fixed source reactor physics problems with assumed uncertainties in material cross-section data. Reductions in problem size and solving times exceeds 5 orders of magnitude in comparison to high fidelity models, and which could potentially improve further for larger scale problems.

Noncooperative Spectrum Sensing Strategy Based on Recurrence Quantification Analysis in the Context of the Cognitive Radio

This paper addresses the problem of noncooperative spectrum sensing in very low signal-to-noise ratio (SNR) conditions. In our approach, detecting an unoccupied bandwidth consists of detecting the presence or absence of a communication signal on this bandwidth. Digital communication signals may contain hidden periodicities, so we use Recurrence Quantification Analysis (RQA) to reveal the hidden periodicities. RQA is very sensitive and offers reliable estimation of the phase space dimension m or the time delay τ. In view of the limitations of the algorithms proposed in the literature, we have proposed a new algorithm to simultaneously estimate the optimal values of m and τ. The new proposed optimal values allow the state reconstruction of the observed signal and then the estimation of the distance matrix. This distance matrix has particular properties that we have exploited to propose a Recurrence-Analysis-based Detector (RAD). The RAD can detect a communication signal in a very low SNR condition. Using Receiver Operating Characteristic curves, our experimental results corroborate the robustness of our proposed algorithm compared with classic widely used algorithms.

Reduced dimensional Monte Carlo method: Preliminary integrations.

A technique for reducing the number of integrals in a Monte Carlo calculation is introduced. For integrations relying on classical or mean-field trajectories with local weighting functions, it is possible to integrate analytically at least half of the integration variables prior to setting up the particular Monte Carlo calculation of interest, in some cases more. Proper accounting of invariant phase space structures shows that the system's dynamics is reducible into composite stable and unstable degrees of freedom. Stable degrees of freedom behave locally in the reduced dimensional phase space exactly as an analogous integrable system would. Classification of the unstable degrees of freedom is dependent upon the degree of chaos present in the dynamics. The techniques for deriving the requisite canonical coordinate transformations are developed and shown to block diagonalize the stability matrix into irreducible parts. In doing so, it is demonstrated how to reduce the amount of sampling directions necessary in a Monte Carlo simulation. The technique is illustrated by calculating return probabilities and expectation values for different dynamical regimes of a two-degrees-of-freedom coupled quartic oscillator within a classical Wigner method framework.

Intermittency analysis of charged particles generated in Xe-Xe~collisions at $\sqrt{s_{\rm{NN}}}$ = 5.44 TeV using the AMPT model

The multiplicity fluctuations are sensitive to QCD phase transition and to the presence of critical point in QCD phase diagram. At critical point a system undergoing phase transition is characterized by large fluctuations in the observables which is an important tool to understand the dynamics of particle production in heavy-ion interactions and phase changes. Multiplicity fluctuations of produced particles is an important observable to characterize the evolving system. Using scaling exponent obtained from the normalized factorial moments of the number of charged hadrons in the two dimensional (\eta,\phiη,ϕ) phase space, one can learn about the dynamics of system created in these collisions. Events generated using Xe-Xe collisions at $ = 5.44 $ TeV with string-melting (SM) version of the AMPT model are analyzed and the scaling exponent (\nu)(ν) for various p_TpT intervals is determined. It is observed that the calculated value of \nuν is larger than the universal value 1.304, as is obtained from Ginzburg-Landau theory for second order phase transition. Here we will also present the results of the dependence of the scaling exponent on the transverse momentum bin width.

Analyzing and identifying predictable time range for stress prediction based on chaos theory and deep learning

ProposeStress is a common problem globally. Prediction of stress in advance could help people take effective measures to manage stress before bad consequences occur. Considering the chaotic features of human psychological states, in this study, we integrate deep learning and chaos theory to address the stress prediction problem.MethodsBased on chaos theory, we embed one’s seemingly disordered stress sequence into a high dimensional phase space so as to reveal the underlying dynamics and patterns of the stress system, and meanwhile are able to identify the stress predictable time range. We then conduct deep learning with a two-layer (dimension and temporal) attention mechanism to simulate the nonlinear state of the embedded stress sequence for stress prediction.ResultsWe validate the effectiveness of the proposed method on the public available Tesserae dataset. The experimental results show that the proposed method outperforms the pure deep learning method and Chaos method in both 2-label and 3-label stress prediction.ConclusionIntegrating deep learning and chaos theory for stress prediction is effective, and can improve the prediction accuracy over 2% and 8% more than those of the deep learning and the Chaos method respectively. Implications and further possible improvements are also discussed at the end of the paper.

Photoplethysmogram beat detection using Symmetric Projection Attractor Reconstruction.

Many methods have been proposed to detect beats in photoplethysmogram (PPG) signals. We present a novel method which uses the Symmetric Projection Attractor Reconstruction (SPAR) method to generate an attractor in a two dimensional phase space from the PPG signal. We can then define a line through the origin of this phase space to be a Poincaré section, as is commonly used in dynamical systems. Beats are detected when the attractor trajectory crosses the Poincaré section. By considering baseline drift, we define an optimal Poincaré section to use. The performance of this method was assessed using the WESAD dataset, achieving median F 1 scores of 74.3% in the Baseline phase, 63.0% during Stress, 93.6% during Amusement, and 97.7% during Meditation. Performance was better than an earlier version of the method, and comparable to one of the best algorithms identified in a recent benchmarking study of 15 beat detection algorithms. In addition, our method performed better than two others in the accuracy of the inter-beat intervals for two resting subjects.

Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

Limits to predictability of the asymptotic state of the Atlantic Meridional Overturning Circulation in a conceptual climate model

Anticipating critical transitions in the Earth system is of great societal relevance, yet there may be intrinsic limitations to their predictability. For instance, the asymptotic state of a dynamical system possessing multiple chaotic attractors depends sensitively on the initial condition in the proximity of a fractal basin boundary. Here, we approach the problem of final-state sensitivity of the Atlantic Meridional Overturning Circulation (AMOC) using a conceptual climate model, composed of a slow bistable ocean coupled to a fast chaotic atmosphere. First, we explore the occurrence of long chaotic transients in the monostable regime, which can mask a loss of stability near bifurcations. In the bistable regime, we explicitly construct the chaotic saddle using the edge tracking technique. We quantify the final-state sensitivity through the maximum Lyapunov exponent and the lifetime of the saddle and find that the system exhibits a fractal basin boundary with almost full phase space dimension, implying vanishing predictability of the second kind near the basin boundary. Our results demonstrate the usefulness of studying non-attracting chaotic sets in the context of predicting climatic tipping points, and provide guidance for the interpretation of critical transitions in higher-dimensional climate models.

Probing reaction channels via reinforcement learning

Chemical reactions are dynamical processes involving the correlated reorganization of atomic configurations, driving the conversion of an initial reactant into a result product. By virtue of the metastability of both the reactants and products, chemical reactions are rare events, proceeding fleetingly. Reaction pathways can be modelled probabilistically by using the notion of reactive density in the phase space of the molecular system. Such density is related to a function known as the committor function, which describes the likelihood of a configuration evolving to one of the nearby metastable regions. In theory, the committor function can be obtained by solving the backward Kolmogorov equation (BKE), which is a partial differential equation (PDE) defined in the full dimensional phase space. However, using traditional methods to solve this problem is not practical for high dimensional systems. In this work, we propose a reinforcement learning based method to identify important configurations that connect reactant and product states along chemical reaction paths. By shooting multiple trajectories from these configurations, we can generate an ensemble of states that concentrate on the transition path ensemble. This configuration ensemble can be effectively employed in a neural network-based PDE solver to obtain an approximation solution of a restricted BKE, even when the dimension of the problem is very high. The resulting solution provides an approximation for the committor function that encodes mechanistic information for the reaction, paving a new way for understanding of complex chemical reactions and evaluation of reaction rates.

Stability analysis of an acted asymmetric rigid body by a gyrostatic moment and a constant body-fixed torque

This work focuses on the dynamical rotatory motion of an asymmetric rigid body (RB) subjected to a constant body-fixed torque and a vector of a gyrostatic moment (GM). The motion is considered in the absence of the first two components of the GM. The novelty of the current work is the conversion of the stability analysis of this body from a two dimensional phase plane to three ones phase space. Specifically, two scenarios of stationary torques are considered, the first one is directed on the minor or major axis, while the latter is presumed to act on the middle axis. Therefore, three dimensional phase space routes are generated. In both scenarios some novel analytical and simulation outcomes are provided regarding equilibrium manifolds, periodic solutions or non-periodic ones, separator surfaces, as well as the extreme periodic solutions. The significance of this work is due to its great applications, especially those that use the gyro’s theory.

Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement PA between the two classifications. We also study the influence of the final number of orbit iterations T, the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T, n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving PA≳90%. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators.

Time quasi-periodic vortex patches of Euler equation in the plane

We prove the existence of time quasi-periodic vortex patch solutions of the 2 $d$ -Euler equations in $\mathbb{R}^{2} $ , close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -as well as the degeneracy of a normal frequency- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.

The thermodynamic efficiency of the Lorenz system

We study the thermodynamic efficiency of the Malkus–Lorenz waterwheel. For this purpose, we derive an exact analytical formula that describes the efficiency of this dissipative structure as a function of the phase space variables and the constant parameters of the dynamical system. We show that, generally, as the machine is progressively driven far from thermodynamic equilibrium by increasing its uptake of matter from the environment, it also tends to increase its efficiency. However, sudden drops in the efficiency are found at critical bifurcation points leading to chaotic dynamics. We relate these discontinuous crises in the efficiency to a reduction of the attractor’s average value projected along the phase space dimensions that contribute to the rate of entropy generation in the system. In this manner, we provide a thermodynamic criterion that, presumably, governs the evolution of far-from-equilibrium dissipative systems towards their self-assembly and synchronization into increasingly complex networks and structures.

Classical Nambu brackets in higher dimensions

We consider n-linear Nambu brackets in dimension N higher than n. Starting from a Hamiltonian system with a Poisson bracket and K Casimir invariants defined in the phase space of dimension N = K + 2M, where M is the number of effective degrees of freedom, we investigate a necessary and sufficient condition for this system to possess n-linear Nambu brackets. For the case of n = 3, by looking for the possible solutions to the fundamental identity, the condition is found to be N = K + 2, i.e., the system should have effectively one degree of freedom. Locally, it is shown that there is only one fundamental solution, up to a local change of variables, and this solution is the canonical Nambu bracket, generated by Levi–Cività tensors. These results generalize to the case of n(≥4)-linear Nambu brackets.

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L2-stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

The harmonium model and its unified system view of psychopathology: a validation study by means of a convolutional neural network

The harmonium model (HM) is a recent conceptualization of the unifying view of psychopathology, namely the idea of a general mechanism underpinning all mental disorders (the p factor). According to HM, psychopathology consists of a low dimensional Phase Space of Meaning (PSM), where each dimension of meaning maps a component of the environmental variability. Accordingly, the lower thenumber of independent dimensions in the PSM, and hence its intrinsic complexity, the more limited the way of interpreting the environment. The current simulation study, based on a Convolutional Neural Network (CNN) framework, aims at validating the HM low-dimensionality hypothesis. CNN-based classifiers were employed to simulate normotypical and pathological cognitive processes. Results revealed that normotypical and pathological CNNs were different in terms of both classification performance and layer activation patterns. Using Principal Component Analysis to characterize the PSM associated with the two algorithms, we found that the performance of the normotypical CNN relies on a larger and more evenly distributed number of components, compared with the pathological one. This finding might be indicative of the fact that psychopathology can be modelled as a low-dimensional, poorly modulable PSM, which means the environment is detected through few components of meaning, preventing complex information patterns from being taken into account.

Chaotic Dynamics of Spatially Homogeneous Spacetimes

In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.

A data-driven approach for instability analysis of thin composite structures

This paper aims to propose a data-driven computing algorithm integrated with model reduction technique to conduct instability analysis of thin composite structures. The data-driven computing method was originally introduced by Kirchdoerfer and Ortiz (2016), whose basic idea lies in directly employing the stress and strain sets to drive the mechanical simulation, thus eliminating the material modeling error and uncertainty. By introducing the Euler–Bernoulli beam theory into data-driven computing, the one-dimensional reduced beam model is adopted by the herein proposed approach, namely structural-genome-driven (SGD) computing. In this manner, not only the integration points number but also the database phase space dimensions will be decreased, thereby enhancing the computational efficiency for structural analysis. Besides, the weight coefficient settings in data-driven penalty function are determined by the locally tangent linear material behavior of the data sets and are updated for each integration point during data-driven iterations. Several demonstrative numerical tests are performed to validate the robustness and effectiveness of the proposed method in predicting buckling path and bifurcation point.