One-point Correlation Functions Research Articles

A Universal Method for Modeling and Characterizing Non-Circular Packing Systems Based on n-Point Correlation Functions.

A universal method for modeling and characterizing non-circular particles is developed. The n-point correlation functions (n = 1, 2 and 3) are efficiently computed with a GPU parallel computing procedure. An algorithm for dynamic packing of impenetrable non-circular particles is developed based on the fast estimation of overlap information using a one-point correlation function. The packing algorithm is independent of particle shape and proved to be reliable by examples of polygons and super-ellipses. In addition, penetrable packings are generated in an efficient and precise way. Using a two-point correlation function, these non-circular packs are accurately characterized and compared in terms of features such as penetrable and impenetrable, packing fraction and particle shape. In addition, three-point correlation functions are also illustrated and discussed.

Taylor's frozen-in hypothesis for magnetohydrodynamic turbulence and solar wind

In hydrodynamics, Taylor's frozen-in hypothesis connects the wavenumber spectrum to the frequency spectrum of a time series measured in real space. In this paper, we generalize Taylor's frozen-in hypothesis to magnetohydrodynamic turbulence. We analytically derive one-point two-time correlation functions for Elsässer variables whose Fourier transform yields the corresponding frequency spectra, E±(f). We show that for isotropic turbulence, E±(f)∝|U0 ∓ B0|2/3 in the Kolmogorov-like model and E±(f)∝(B0|U0 ∓ B0|)1/2 in the Iroshnikov–Kraichnan model, where U0 and B0 are the mean velocity and mean magnetic fields, respectively, and f±=k|U0 ∓ B0|/(2π) are the respective frequencies for a wavenumber k. However, for anisotropic magnetohydrodynamic turbulence, E±(f)∝B02/3 when U0≪B0. These results are important for the analysis of solar wind, in particular, those measured by Parker Solar Probe.

Six-Vertex Model as a Grassmann Integral, One-Point Function, and the Arctic Ellipse

We formulate the six-vertex model with domain wall boundary conditions in terms of an integral over Grassmann variables. Relying on this formulation, we propose a method of calculation of correlation functions of the model for the case of the weights satisfying the free-fermion condition. We consider here in details the one-point correlation function describing the probability of a given state on an arbitrary edge of the lattice. We show that in the thermodynamic limit performed in such a way that the lattice is scaled to the square of unit side length, this function exhibits the “arctic ellipse” phenomenon, in agreement with the previous studies on random domino tilings of Aztec diamonds: it approaches its limiting values outside of an ellipse inscribed into this square, and takes continuously intermediate values inside the ellipse. We derive also the scaling properties of the one-point function in the vicinity of an arbitrary point of the arctic ellipse and in the vicinities of the points where the ellipse touches the boundary.

Dynamics of Characteristic and One-Point Correlation Functions of Multi-Mode Bosonic Systems: Exactly Solvable Model

In this communication we study dynamics of the open quantum bosonic system governed by the generalized Lindblad equation with both dynamical and environment induced intermode couplings taken into account. By using the method of characteristics we deduce the analytical expression for the normally ordered characteristic function. Analytical results for one-point correlation functions describing temporal evolution of the covariance matrix are obtained.

Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

S2D2: Small-scale Significant substructure DBSCAN Detection

Context.The spatial and dynamical structure of star-forming regions can offer insights into stellar formation patterns. The amount of data from current and upcoming surveys calls for robust and objective procedures for detecting structures in order to statistically analyse the various regions and compare them.Aims.We aim to provide the community with a tool capable of detecting, above random expectations, the small-scale significant structure in star-forming regions that could serve as an imprint of the stellar formation process. The tool makes use of the one-point correlation function to determine an appropriate length scale forϵand uses nearest-neighbour statistics to determine a minimum number of pointsNminfor the DBSCAN algorithm in the neighbourhood ofϵ.Methods.We implemented the procedure and applied it to synthetic star-forming regions of different nature and characteristics to obtain its applicability range. We also applied the method to observed star-forming regions to demonstrate its performance in realistic circumstances and to analyse its results.Results.The procedure successfully detects significant small-scale substructures in heterogeneous regions, fulfilling the goals it was designed for and providing very reliable structures. The analysis of regions close to complete spatial randomness (Q ∈ [0.7, 0.87]) shows that even when some structure is present and recovered, it is hardly distinguishable from spurious detection in homogeneous regions due to projection effects. Thus, any interpretation should be done with care. For concentrated regions, we detect a main structure surrounded by smaller ones, corresponding to the core plus some Poisson fluctuations around it. We argue that these structures do not correspond to the small compact regions we are looking for. In some realistic cases, a more complete hierarchical, multi-scale analysis would be needed to capture the complexity of the region.Conclusions.We carried out implementations of our procedure and devised a catalogue of the Nested Elementary STructures (NESTs) detected as a result in four star-forming regions (Taurus, IC 348, Upper Scorpius, and Carina). This catalogue is being made publicly available to the community. Implementations of the 3D versionsof the procedure, as well as up to 6D versions, including proper movements, are in progress and will be provided in a future work.

Entanglement spectrum of geometric states

The reduced density matrix of a given subsystem, denoted by ρA, contains the information on subregion duality in a holographic theory. We may extract the information by using the spectrum (eigenvalue) of the matrix, called entanglement spectrum in this paper. We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state ρA,m associated with an eigenvalue λ for some examples, including a single interval and two intervals in vacuum state of 2D CFTs. We find there exists a microcanonical ensemble state with λ0 which can be seen as an approximate state of ρA. The parameter λ0 is obtained in the two examples. For a general geometric state, the approximate microcanonical ensemble state also exists. The parameter λ0 is associated with the entanglement entropy of A and Rényi entropy in the limit n → ∞. As an application of the above conclusion we reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs as conditions of Holevo information. We show the constraints on the eigenstates. Finally, we point out some unsolved problems and their significance on understanding the geometric states.

Critical properties of the Floquet time crystal within the Gaussian approximation

The periodically driven $O(N)$ model is studied near the critical line separating a disordered paramagnetic phase from a period doubled phase, the latter being an example of a Floquet time crystal. The time evolution of one-point and two-point correlation functions are obtained within the Gaussian approximation and perturbatively in the drive amplitude. The correlations are found to show not only period doubling, but also power-law decays at large spatial distances. These features are compared with the undriven $O(N)$ model, within the Gaussian approximation, in the vicinity of the paramagnetic-ferromagnetic critical point. The algebraic decays in space are found to be qualitatively different in the driven and the undriven cases. In particular, the spatiotemporal order of the Floquet time crystal leads to position-momentum and momentum-momentum correlation functions which are more long-ranged in the driven than in the undriven model. The light-cone dynamics associated with the correlation functions is also qualitatively different as the critical line of the Floquet time crystal shows a light cone with two distinct velocities, with the ratio of these two velocities scaling as the square-root of the dimensionless drive amplitude. The Floquet unitary, which describes the time evolution due to a complete cycle of the drive, is constructed for modes with small momenta compared to the drive frequency, but having a generic relationship with the square-root of the drive amplitude. At intermediate momenta, which are large compared to the square-root of the drive amplitude, the Floquet unitary is found to simply rotate the modes. On the other hand, at momenta which are small compared to the square-root of the drive amplitude, the Floquet unitary is found to primarily squeeze the modes, to an extent which increases upon increasing the wavelength of the modes, with a power-law dependence on it.

Predicting permeability via statistical learning on higher-order microstructural information

Quantitative structure–property relationships are crucial for the understanding and prediction of the physical properties of complex materials. For fluid flow in porous materials, characterizing the geometry of the pore microstructure facilitates prediction of permeability, a key property that has been extensively studied in material science, geophysics and chemical engineering. In this work, we study the predictability of different structural descriptors via both linear regressions and neural networks. A large data set of 30,000 virtual, porous microstructures of different types, including both granular and continuous solid phases, is created for this end. We compute permeabilities of these structures using the lattice Boltzmann method, and characterize the pore space geometry using one-point correlation functions (porosity, specific surface), two-point surface-surface, surface-void, and void-void correlation functions, as well as the geodesic tortuosity as an implicit descriptor. Then, we study the prediction of the permeability using different combinations of these descriptors. We obtain significant improvements of performance when compared to a Kozeny-Carman regression with only lowest-order descriptors (porosity and specific surface). We find that combining all three two-point correlation functions and tortuosity provides the best prediction of permeability, with the void-void correlation function being the most informative individual descriptor. Moreover, the combination of porosity, specific surface, and geodesic tortuosity provides very good predictive performance. This shows that higher-order correlation functions are extremely useful for forming a general model for predicting physical properties of complex materials. Additionally, our results suggest that artificial neural networks are superior to the more conventional regression methods for establishing quantitative structure–property relationships. We make the data and code used publicly available to facilitate further development of permeability prediction methods.

The Fyodorov–Bouchaud formula and Liouville conformal field theory

In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk defined by Huang, Rhodes and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (BPZ equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF and tail expansions of GMC.

Closed-Form Expressions of Ergodic Capacity and MMSE Achievable Sum Rate for MIMO Jacobi and Rayleigh Fading Channels

Multimode/multicore fibers are expected to provide an attractive solution to overcome the capacity limit of the current optical communication system. In the presence of strong crosstalk between modes and/or cores, the squared singular values of the input/output transfer matrix follow the law of the Jacobi ensemble of random matrices. Assuming that the channel state information is only available at the receiver, we derive a new expression for the ergodic capacity of the MIMO Jacobi fading channel. The proposed expression involves double integrals which can be easily evaluated for a high-dimensional MIMO scenario. Moreover, the method used in deriving this expression does not appeal to the classical one-point correlation function of the random matrix model. Using a limiting transition between Jacobi and associated Laguerre polynomials, we derive a similar formula for the ergodic capacity of the MIMO Rayleigh fading channel. Moreover, we derive a new exact closed-form expressions for the achievable sum rate of MIMO Jacobi and Rayleigh fading channels employing linear minimum mean squared error (MMSE) receivers. The analytical results are compared to the results obtained by Monte Carlo simulations and the related results available in the literature, which shows perfect agreement.

Multiplicity and clustering in Taurus star forming region

Context. Multiplicity and clustering of young pre-main sequence stars appear as critical clues to understand and constrain the star formation process. Taurus is the archetypical example of the most quiescent star forming regions that may still retain primeval signatures of star formation. Aims. This work identifies local overdense stellar structures as a critical scale between wide pairs and loose groups in Taurus. Methods. Using the density-based spatial clustering of applications with noise (dbscan) algorithm, and setting its free parameters based on the one-point correlation function and the k-nearest neighbor statistics, we have extracted reliably overdense structures from the sky-projected spatial distribution of stars. Results. Nearly half of the entire stellar population in Taurus is found to be concentrated in 20 very dense, tiny and prolate regions called NESTs (for Nested Elementary STructures). They are regularly spaced (≈2 pc) and mainly oriented along the principal gas filaments axes. Each NEST contains between four and 23 stars. Inside NESTs, the surface density of stars may be as high as 2500 pc−2 and the mean value is 340 pc−2. Nearly half (11) of these NESTs contain about 75% of the class 0 and I objects. The balance between Class I, II, and, III fraction within the NESTs suggests that they may be ordered as an evolutionary temporal scheme, some of them getting infertile with time, while other still giving birth to young stars. We have inferred that only 20% of stars in Taurus do not belong to any kind of stellar groups (either multiple system, ultra wide pairs or NESTs). The mass-size relation for stellar NESTs is very close to the Bonnor–Ebert expectation. The range in mass is about the same as that of dense molecular cores. The distribution in size is bimodal peaking at 12.5 and 50 kAU and the distribution of the number of YSOs in NESTs as a function of size exhibits two regimes. Conclusions. We propose that the NESTs in their two size regimes represent the spatial imprints of stellar distribution at birth as they may have emerged within few millions years from their natal cloud either from a single core or from a chain of cores. We have identified them as the preferred sites of star formation in Taurus. These NESTs are the regions of highest stellar density and intermediate spatial scale structures between ultra-wide pairs and loose groups.

From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

We derive exact formulas for the expectation value of local observables in a one-dimensional gas of bosons with point-wise repulsive interactions (Lieb–Liniger model). Starting from a recently conjectured expression for the expectation value of vertex operators in the sinh-Gordon field theory, we derive explicit analytic expressions for the one-point K-body correlation functions in the Lieb–Liniger gas, for arbitrary integer K. These are valid for all excited states in the thermodynamic limit, including thermal states, generalized Gibbs ensembles and non-equilibrium steady states arising in transport settings. Our formulas display several physically interesting applications: most prominently, they allow us to compute the full counting statistics for the particle-number fluctuations in a short interval. Furthermore, combining our findings with the recently introduced generalized hydrodynamics, we are able to study multi-point correlation functions at the Eulerian scale in non-homogeneous settings. Our results complement previous studies in the literature and provide a full solution to the problem of computing one-point functions in the Lieb–Liniger model.

Correlators of mixed symmetry operators in defect CFTs

We use the embedding formalism technique to study correlation functions of a d-dimensional Euclidean CFT in the presence of a q co-dimensional defect. The defect breaks the global conformal group SO(d + 1, 1) into SO(d − q + 1, 1) × SO(q). We calculate all possible invariant structures that can appear in one-point, two-point and three-point correlation functions of bulk and defect operators in mixed symmetry representation. Their generalization to n-point correlation functions are also worked out. Correlation functions in the presence of a defect, in arbitrary representation of SO(q), are also calculated.

Coherence effects in disordered geometries with a field-theory dual

We investigate the holographic dual of a probe scalar in an asymptotically Anti-de-Sitter (AdS) disordered background which is an exact solution of Einstein’s equations in three bulk dimensions. Unlike other approaches to model disorder in holography, we are able to explore quantum wave-like interference effects between an oscillating or random source and the geometry. In the weak-disorder limit, we compute analytically and numerically the one-point correlation function of the dual field theory for different choices of sources and backgrounds. The most interesting feature is the suppression of the one-point function in the presence of an oscillating source and weak random background. We have also computed analytically and numerically the two-point function in the weak disorder limit. We have found that, in general, the perturbative contribution induces an additional power-law decay whose exponent depends on the distribution of disorder. For certain choices of the gravity background, this contribution becomes dominant for large separations which indicates breaking of perturbation theory and the possible existence of a phase transition induced by disorder.

Anomalous diffusion in viscoelastic media with active force dipoles.

With the use of the "two-fluid model," we discuss anomalous diffusion induced by active force dipoles in viscoelastic media. Active force dipoles, such as proteins and bacteria, generate nonthermal fluctuating flows that lead to a substantial increment of the diffusion. Using the partial Green's function of the two-fluid model, we first obtain passive (thermal) two-point correlation functions such as the displacement cross-correlation function between the two-point particles separated by a finite distance. We then calculate active (nonthermal) one-point and two-point correlation functions due to active force dipoles. The time correlation of a force dipole is assumed to decay exponentially with a characteristic time scale. We show that the active component of the displacement cross-correlation function exhibits various crossovers from super-diffusive to subdiffusive behaviors depending on the characteristic time scales and the particle separation. Our theoretical results are intimately related to the microrheology technique to detect fluctuations in nonequilibrium environment.

Bell Violation in Primordial Cosmology

In this paper, we have worked on the possibility of setting up an Bell’s inequality violating experiment in the context of primordial cosmology following the fundamental principles of quantum mechanics. To set up this proposal, we have introduced a model-independent theoretical framework using which we have studied the creation of new massive particles for the scalar fluctuations in the presence of an additional time-dependent mass parameter. Next we explicitly computed the one-point and two-point correlation functions from this setup. Then, we comment on the measurement techniques of isospin breaking interactions of newly introduced massive particles and its further prospects. After that, we give an example of the string theory-originated axion monodromy model in this context. Finally, we provide a bound on the heavy particle mass parameter for any arbitrary spin field.

Quantum quenches in the sinh-Gordon model: steady state and one-point correlation functions

.We consider quantum quenches to the sinh-Gordon integrable quantum field theory from a particular class of initial states. Our analysis includes the case of mass and interaction quenches starting from a non-interacting theory. By means of the recently developed quench action method, we fully characterize the stationary state reached at long times after the quench in terms of the corresponding rapidity distribution. We also provide exact results for the expectation values of arbitrary vertex operators in the post-quench stationary state by proposing a formula based on the analogy with the standard thermodynamic Bethe ansatz. Finally, we comment on the behavior of the post-quench stationary state under the mapping between the sinh-Gordon field theory and the one-dimensional Lieb–Liniger model.

Exterior statistics based boundary conditions for representative volume elements of elastic composites

Statistically equivalent representative volume elements or SERVEs are representations of the microstructure that are used for micromechanical simulations to generate homogenized material constitutive responses and properties (Swaminathan et al., 2006a; Ghosh, 2011). Typically, a SERVE is generated from the parent microstructure as a statistically equivalent region, whose size is determined from the requirements of convergence of macroscopic properties. Standard boundary conditions, such as affine transformation-based displacement boundary conditions (ATDBCs), uniform traction boundary conditions (UTBCs) or periodic boundary conditions (PBCs) are conventionally applied on the SERVE boundary for micromechanical simulations. However, when the microstructure is characterized by arbitrary, nonuniform distributions of heterogeneities, these simple boundary conditions do not represent the effect of regions exterior to the SERVE. Improper boundary conditions can result in significantly larger than optimal SERVE domains, needed for converged properties. In an attempt to overcome the limitations of the conventional boundary conditions on the SERVE, this paper explores the effect of boundary conditions that incorporate the statistics of the exterior region on the SERVE of elastic composites. Using Green's function based interaction kernels, coupled with statistical functions of the microstructural characteristics like one-point and two-point correlation functions, a novel exterior statistics-based boundary condition or ESBC is derived for the SERVE. The advantages of the ESBC are established by comparing with results of simulations using conventional boundary conditions. Results of the SERVE simulations subjected to ESBCs are also compared with those from other popular methods like statistical volume element (SVE) and weighted statistical volume element (WSVE). The proposed ESBCs offer significant advantages over other methods in the SERVE-based analysis of heterogeneous materials.

Nonlocal probes of thermalization in holographic quenches with spectral methods

We describe the application of pseudo-spectral methods to problems of holographic thermal quenches of relevant couplings in strongly coupled gauge theories. We focus on quenches of a fermionic mass term in a strongly coupled N=4 supersymmetric Yang-Mills plasma, and the subsequent equilibration of the system. From the dual gravitational perspective, we study the gravitational collapse of a massive scalar field in asymptotically anti-de Sitter geometry with a prescribed boundary condition for its non-normalizable mode. Access to the full background geometry of the gravitational collapse allows for the study of nonlocal probes of the thermalization process. We discuss the evolution of the apparent and the event horizons, the two-point correlation functions of operators of large conformal dimensions, and the evolution of the entanglement entropy of the system. We compare the thermalization process from the viewpoint of local (the one-point) correlation functions and these nonlocal probes, finding that the thermalization time as measured by the probes is length dependent, and approaches the thermalization time of the one-point function for longer probes. We further discuss how the different energy scales of the problem contribute to its thermalization.