Accessible Phase Space Research Articles

Isles of regularity in a sea of chaos amid the gravitational three-body problem

Context. The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. Aims. This study probes the presence of regular (i.e. non-chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Methods. Using three-body simulations performed with the accurate, regularized code TSUNAMI, we established criteria for identifying regular trajectories and analysed their impact on statistical outcomes. Results. Our analysis reveals that regular trajectories occupy a significant fraction of the phase space, ranging from 28% to 84% depending on the initial setup, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Conclusions. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.

Probing the dynamics and bottleneck of the key atmospheric SO2 oxidation reaction by the hydroxyl radical

SO2 (Sulfur dioxide) is the major precursor to the production of sulfuric acid (H2SO4), contributing to acid rain and atmospheric aerosols. Sulfuric acid formed from SO2 generates light-reflecting sulfate aerosol particles in the atmosphere. This property has prompted recent geoengineering proposals to inject sulfuric acid or its precursors into the Earth's atmosphere to increase the planetary albedo to counteract global warming. SO2 oxidation in the atmosphere by the hydroxyl radical HO to form HOSO2 is a key rate-limiting step in the mechanism for forming acid rain. However, the dynamics of the HO + SO2 → HOSO2 reaction and its slow rate in the atmosphere are poorly understood to date. Herein, we use photoelectron spectroscopy of cryogenically cooled HOSO2- anion to access the neutral HOSO2 radical near the transition state of the HO + SO2 reaction. Spectroscopic and dynamic calculations are conducted on the first ab initio-based full-dimensional potential energy surface to interpret the photoelectron spectra of HOSO2- and to probe the dynamics of the HO + SO2 reaction. In addition to the finding of a unique pre-reaction complex (HO⋯SO2) directly connected to the transition state, dynamic calculations reveal that the accessible phase space for the HO + SO2 → HOSO2 reaction is extremely narrow, forming a key reaction bottleneck and slowing the reaction rate in the atmosphere, despite the low reaction barrier. This study underlines the importance of understanding the full multidimensional potential energy surface to elucidate the dynamics of complex bimolecular reactions involving polyatomic reactants.

A fully automatized framework for accurate phase space characterization of flexible molecules in aqueous environments.

Complete structural characterization of flexible molecules in solution, such as sugars and peptides in aqueous environments, is demanding. Only a collection of structures with defined weights or even continuous regions of accessible phase space can describe their structural complexity. Spectroscopic techniques such as NMR and Raman appear the obvious choice for studying such systems. They are sensitive to structure, and measurements can be performed in aqueous solutions under biologically relevant conditions. Unfortunately, the obtained spectra and responses result from a weighted average of all accessible structures and are hard to deconvolute. When supported with computer simulations, one can deconvolute them and obtain an accurate description of the accessible phase space. Existing simulation protocols can perform such tasks. However, all available workflows are lengthy and require extensive human input, ultimately biasing the results. In this work, we have synthesized the experience of our developed protocols to calculate Raman and NMR spectra for saccharides into a program designed to exclude any bias by reducing human intervention. It only requires the input of the molecular structure and the corresponding experimental spectra to deliver accurate structural predictions, mainly when using spectra from different techniques simultaneously. Currently, it can handle several types of organic compounds in an aqueous solution, including biologically relevant ones (e.g., sugars and peptides). Importantly, it estimates the quality of the prediction independently for each molecular degree of freedom. The information gathered can be used to interpret the experimental spectra and develop more accurate force fields of flexible moieties. Decoding the structural complexity of organic molecules using our tool can potentially shed light on their behavior and function in aqueous environments.

Creating triple-NOON states with ultracold atoms via chaos-assisted tunneling

Triple-NOON states are superpositions of the form $e^{i \varphi_1} |{N,0,0}\rangle + e^{i \varphi_2} |{0,N,0}\rangle + e^{i \varphi_3} |{0,0,N}\rangle$ involving $N$ bosonic quanta distributed over three modes. We theoretically show how such highly entangled states can be generated with interacting ultracold bosonic atoms in a symmetric three-site lattice. The basic protocol consists in preparing all atoms on one site of the lattice and then letting the system evolve during a specific time such that collective tunneling of the atoms to the other two sites takes place. The key point put forward here is that this evolution time can be reduced by several orders of magnitude via the application of a periodic driving of the lattice, thereby rendering this protocol feasible in practice. This driving is suitably tuned such that classical chaos is generated in the entire accessible phase space except for the Planck cells that host the states participating at the above triple-NOON superposition. Chaos-assisted tunneling can then give rise to a dramatic speed-up of this collective tunneling process, without significantly affecting the purity of this superposition. A triple-NOON state containing $N = 5$ particles can thereby be realized with $^{87}$Rb atoms on time scales of the order of a few seconds.

Direct free energy evaluation of classical and quantum many-body systems via field-theoretic simulation

Free energy evaluation in molecular simulations of both classical and quantum systems is computationally intensive and requires sophisticated algorithms. This is because free energy depends on the volume of accessible phase space, a quantity that is inextricably linked to the integration measure in a coordinate representation of a many-body problem. In contrast, the same problem expressed as a field theory (auxiliary field or coherent states) isolates the particle number as a simple parameter in the Hamiltonian or action functional and enables the identification of a chemical potential field operator. We show that this feature leads a “direct” method of free energy evaluation, in which a particle model is converted to a field theory and appropriate field operators are averaged using a field-theoretic simulation conducted with complex Langevin sampling. These averages provide an immediate estimate of the Helmholtz free energy in the canonical ensemble and the entropy in the microcanonical ensemble. The method is illustrated for a classical polymer solution, a block copolymer melt exhibiting liquid crystalline and solid mesophases, and a quantum fluid of interacting bosons.

Tethered-particle model: The calculation of free energies for hard-sphere systems

Two methods for computing the entropy of hard-sphere systems using a spherical tether model are explored, which allow the efficient use of event-driven molecular-dynamics simulations. An intuitive derivation is given, which relates the rate of particle collisions, either between two particles or between a particle and its respective tether, to an associated hypersurface area, which bounds the system's accessible configurational phase space. Integrating the particle-particle collision rates with respect to the sphere diameter (or, equivalently, density) or the particle-tether collision rates with respect to the tether length then directly determines the volume of accessible phase space and, therefore, the system entropy. The approach is general and can be used for any system composed of particles interacting with discrete potentials in fluid, solid, or glassy states. The entropies calculated for the liquid and crystalline hard-sphere states using these methods are found to agree closely with the current best estimates in the literature, demonstrating the accuracy of the approach.

Quantum Black Holes as Solvents

Almost all of the entropy in the universe is in the form of Bekenstein–Hawking (BH) entropy of super-massive black holes. This entropy, if it satisfies Boltzmann’s equation S=log mathcal{N}, hence represents almost all the accessible phase space of the Universe, somehow associated to objects which themselves fill out a very small fraction of ordinary three-dimensional space. Although time scales are very long, it is believed that black holes will eventually evaporate by emitting Hawking radiation, which is thermal when counted mode by mode. A pure quantum state collapsing to a black hole will hence eventually re-emerge as a state with strictly positive entropy, which constitutes the famous black hole information paradox. Expanding on a remark by Hawking we posit that BH entropy is a thermodynamic entropy, which must be distinguished from information-theoretic entropy. The paradox can then be explained by information return in Hawking radiation. The novel perspective advanced here is that if BH entropy counts the number of accessible physical states in a quantum black hole, then the paradox can be seen as an instance of the fundamental problem of statistical mechanics. We suggest a specific analogy to the increase of the entropy in a solvation process. We further show that the huge phase volume (mathcal{N}), which must be made available to the universe in a gravitational collapse, cannot originate from the entanglement between ordinary matter and/or radiation inside and outside the black hole. We argue that, instead, the quantum degrees of freedom of the gravitational field must get activated near the singularity, resulting in a final state of the ‘entangled entanglement’ form involving both matter and gravity.

A Scaling Relation Between Accessible Phase Space Volume and Escape Rates for Random Walks in High-Dimensional Hierarchical Energy Landscapes - Implications for Protein Dynamics

A Scaling Relation Between Accessible Phase Space Volume and Escape Rates for Random Walks in High-Dimensional Hierarchical Energy Landscapes - Implications for Protein Dynamics

Back-mapping based sampling: Coarse grained free energy landscapes as a guideline for atomistic exploration.

One ongoing topic of research in MD simulations is how to enable sampling to chemically and biologically relevant time scales. We address this question by introducing a back-mapping based sampling (BMBS) that combines multiple aspects of different sampling techniques. BMBS uses coarse grained (CG) free energy surfaces (FESs) and dimensionality reduction to initiate new atomistic simulations. These new simulations are started from atomistic conformations that were back-mapped from CG points all over the FES in order to sample the entire accessible phase space as fast as possible. In the context of BMBS, we address relevant back-mapping related questions like where to start the back-mapping from and how to judge the atomistic ensemble that results from the BMBS. The latter is done with the use of the earth mover's distance, which allows us to quantitatively compare distributions of CG and atomistic ensembles. By using this metric, we can also show that the BMBS is able to correct inaccuracies of the CG model. In this paper, BMBS is applied to a just recently introduced neural network (NN) based approach for a radical coarse graining to predict free energy surfaces for oligopeptides. The BMBS scheme back-maps these FESs to the atomistic scale, justifying and complementing the proposed NN based CG approach. The efficiency benefit of the algorithm scales with the length of the oligomer. Already for the heptamers, the algorithm is about one order of magnitude faster in sampling compared to a standard MD simulation.

Prospects for disentangling long- and short-distance effects in the decays B \u2192 K\u2217\u03bc+\u03bc\u2212

Theory uncertainties on non-local hadronic effects limit the New Physics discovery potential of the rare decays B → K∗μ+μ−. We investigate prospects to disentangle New Physics effects in the short-distance coefficients from these effects. Our approach makes use of an event-by-event amplitude analysis, and relies on a particular parametrisation of the non-local contributions. We find that non-standard effects in the short-distance coefficients can be successfully disentangled from non-local hadronic effects. The impact of the truncation on the parametrisation of non-local contributions to the Wilson coefficients are for the first time systematically examined and prospects for their precise determination are discussed. Theoretical inputs on the non-local matrix elements beyond the physically- accessible phase space are crucial to stabilise the determination of Wilson coefficients, while we find that physical observables are unaffected by these uncertainties. Compared to other methods, our approach provides for a more precise extraction of the angular observables from data.

Thermal conductivity for III-V and II-VI semiconductor wurtzite and zinc-blende polytypes: The role of anharmonicity and phase space

We calculate the lattice thermal conductivity ($\kappa$) for cubic (zinc-blende) and hexagonal (wurtzite) phases for 8 semiconductors using $\textit{ab initio}$ calculations and solving the Phonon Boltzmann Transport Equation, explaining the different behavior of the ratio $\kappa_{\rm hex}/\kappa_{\rm cub}$ between the two phases. We show that this behavior depends on the relative importance of two antagonistic factors: anharmonicity, which we find to be always higher in the cubic phase; and the accessible phase space, which is higher for the less symmetric hexagonal phase. Based on that, we develop a method that predicts the most conducting phase---cubic or hexagonal---where other more heuristic approaches fail. We also present results for nanowires made of the same materials, showing the possibility to tune $\kappa_{\rm hex}/\kappa_{\rm cub}$ over a wide range by modifying their diameter, thus making them attractive materials for complex phononic and thermoelectric applications/systems.

Robustness of ballistic transport in antidot superlattices

The magneto-resistance of antidot lattices shows pronounced peaks, which became a hallmark of ballistic electron transport. While most studies agree that they reflect the interplay of regular and chaotic motion in the quasi-classical dynamics, the exact mechanism has been surprisingly controversial. Inspired by recent experiments on graphene antidot lattices showing that the effect survives strong impurity scattering, we give a new explanation of the peaks linked to a fundamental relation between collision times and accessible phase space volumes, accounting for their robustness. Due to the fundamental nature of the mechanism described it will be relevant in many mesoscopic transport phenomena.

Time Domain Simulations of Single Molecule Raman Scattering.

Nonequilibrium chemical phenomena are known to play an important role in single molecule microscopy and spectroscopy. Herein, we explore these effects through ab initio molecular dynamics (AIMD)-based Raman spectral simulations. We target an isolated aromatic thiol (thiobenzonitrile, TBN) as a prototypical molecular system. We first show that the essential features contained in the ensemble-averaged Raman spectrum of TBN can be reproduced by averaging over 18 short AIMD trajectories spanning a total simulation time of ∼60 ps. This involved more than 90 000 polarizability calculations at the B3LYP/def2-TZVP level of theory. We then illustrate that the short trajectories (∼3.3 ps total simulation time), where the accessible phase space is not fully sampled, provide a starting point for understanding key features that are often observed in measurements targeting single molecules. Our results suggest that a complete understanding of single molecule Raman scattering needs to account for molecular conformational flexibility and nonequilibrium chemical phenomena in addition to local optical fields and modified selection rules. The former effects are well-captured using the described AIMD-based single molecule Raman spectral simulations.

Solubility limits in quaternary SnTe-based alloys

A combined theoretical and experimental approach was used to determine the equilibrium as well as non-equilibrium solubility lines in the quaternary Sn1−yMnyTe1−xSex alloy space, revealing a large area of accessible metastable phase space.

Water Dynamics in Cytoplasm-Like Crowded Environment Correlates with the Conformational Transition of the Macromolecular Crowder.

Polyethylene glycol (PEG) is a unique polymer material with enormous applicability in many industrial and scientific fields. Here, its use as macromolecular crowder to mimic the cellular environment in vitro is the focus of the present study. We show that femtosecond mid-IR pump-probe spectroscopy using three different IR probes, HDO, HN3, and azido-derivatized crowder, provides complete and stereoscopic information on water structure and dynamics in the cytoplasm-like macromolecular crowding environment. Our experimental results suggest two distinct subpopulations of water molecules: those that interact with other water molecules and those that are part of a hydration shell of crowder on its surface. Interestingly, water dynamics even in highly crowded environment remains bulk-like in spite of significant perturbation to the tetrahedral H-bonding network of water molecules. That is possible because of the formation of water aggregates (pools) even in water-deficient PEGDME-water solutions. In such a crowded environment, the conformationally accessible phase space of the macromolecular crowder is reduced, similar to biopolymers in highly crowded cytoplasm. Nonetheless, the hydration water on the surface of crowders slows down considerably with increased crowding. Most importantly, we do not observe any coalescing of surface hydration water (of the crowder) with bulk-like water to generate collective hydration dynamics at any crowder concentration, contrary to recent reports. We anticipate that the present triple-IR-probe approach is of exceptional use in studying how conformational states of crowders correlate with structural and dynamical changes of water, which is critical in understanding their key roles in biological and industrial applications.

A new entropy based on a group-theoretical structure

A multi-parametric version of the nonadditive entropy Sq is introduced. This new entropic form, denoted by Sa,b,r, possesses many interesting statistical properties, and it reduces to the entropy Sq for b=0, a=r:=1−q (hence Boltzmann–Gibbs entropy SBG for b=0, a=r→0). The construction of the entropy Sa,b,r is based on a general group-theoretical approach recently proposed by one of us, Tempesta (2016). Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of Sa,b,r with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy Sa,b,r can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles N of the system, or even stabilizes, by increasing N, to a limiting value.This paves the way to the use of this entropy in contexts where the size of the phase space does not increase as fast as the number of its constituting particles (or subsystems) increases.

Long-Time-Step Molecular Dynamics through Hydrogen Mass Repartitioning.

Previous studies have shown that the method of hydrogen mass repartitioning (HMR) is a potentially useful tool for accelerating molecular dynamics (MD) simulations. By repartitioning the mass of heavy atoms into the bonded hydrogen atoms, it is possible to slow the highest-frequency motions of the macromolecule under study, thus allowing the time step of the simulation to be increased by up to a factor of 2. In this communication, we investigate further how this mass repartitioning allows the simulation time step to be increased in a stable fashion without significantly increasing discretization error. To this end, we ran a set of simulations with different time steps and mass distributions on a three-residue peptide to get a comprehensive view of the effect of mass repartitioning and time step increase on a system whose accessible phase space is fully explored in a relatively short amount of time. We next studied a 129-residue protein, hen egg white lysozyme (HEWL), to verify that the observed behavior extends to a larger, more-realistic, system. Results for the protein include structural comparisons from MD trajectories, as well as comparisons of pKa calculations via constant-pH MD. We also calculated a potential of mean force (PMF) of a dihedral rotation for the MTS [(1-oxyl-2,2,5,5-tetramethyl-pyrroline-3-methyl)methanethiosulfonate] spin label via umbrella sampling with a set of regular MD trajectories, as well as a set of mass-repartitioned trajectories with a time step of 4 fs. Since no significant difference in kinetics or thermodynamics is observed by the use of fast HMR trajectories, further evidence is provided that long-time-step HMR MD simulations are a viable tool for accelerating MD simulations for molecules of biochemical interest.

Some Trends in Quantum Thermodynamics

Traditional answers to what the 2nd Law is are well known. Some are based on the microstate of a system wandering rapidly through all accessible phase space, while others are based on the idea of a system occupying an initial multitude of states due to the inevitable imperfections of measurements that then effectively, in a coarse grained manner, grow in time (mixing). What has emerged are two somewhat less traditional approaches from which it is said that the 2nd Law emerges, namely, that of the theory of quantum open systems and that of the theory of typicality. These are the two principal approaches, which form the basis of what today has come to be called quantum thermodynamics. However, their dynamics remains strictly linear and unitary, and, as a number of recent publications have emphasized, “testing the unitary propagation of pure states alone cannot rule out a nonlinear propagation of mixtures”. Thus, a non-traditional approach to capturing such a propagation would be one which complements the postulates of QM by the 2nd Law of thermodynamics, resulting in a possibly meaningful, nonlinear dynamics. An unorthodox approach, which does just that, is intrinsic quantum thermodynamics and its mathematical framework, steepest-entropy-ascent quantum thermodynamics. The latter has evolved into an effective tool for modeling the dynamics of reactive and non-reactive systems at atomistic scales. It is the usefulness of this framework in the context of quantum thermodynamics as well as the theory of typicality which are discussed here in some detail. A brief discussion of some other trends such as those related to work, work extraction, and fluctuation theorems is also presented.

Adaptive Biasing Combined with Hamiltonian Replica Exchange to Improve Umbrella Sampling Free Energy Simulations.

The accurate calculation of potentials of mean force for ligand-receptor binding is one of the most important applications of molecular simulation techniques. Typically, the separation distance between ligand and receptor is chosen as a reaction coordinate along which a PMF can be calculated with the aid of umbrella sampling (US) techniques. In addition, restraints can be applied on the relative position and orientation of the partner molecules to reduce accessible phase space. An approach combining such phase space reduction with flattening of the free energy landscape and configurational exchanges has been developed, which significantly improves the convergence of PMF calculations in comparison with standard umbrella sampling. The free energy surface along the reaction coordinate is smoothened by iteratively adapting biasing potentials corresponding to previously calculated PMFs. Configurations are allowed to exchange between the umbrella simulation windows via the Hamiltonian replica exchange method. The application to a DNA molecule in complex with a minor groove binding ligand indicates significantly improved convergence and complete reversibility of the sampling along the pathway. The calculated binding free energy is in excellent agreement with experimental results. In contrast, the application of standard US resulted in large differences between PMFs calculated for association and dissociation pathways. The approach could be a useful alternative to standard US for computational studies on biomolecular recognition processes.

Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.