Position Correlation Function Research Articles

Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes

We investigate the observables of the one-dimensional model for anomalous transport in semiconductor devices where diffusion arises from scattering at dislocations at fixed random positions, known as the Lévy-Lorentz gas. To gain insight into the microscopic properties of such a stochastically complex system, deterministic dynamics known as the slicer map and fly-and-die dynamics are used. We analytically derive the generalized position autocorrelation function of these dynamics and study the special case, the 3-point position correlation function. For this, we derive single-parameter-dependent scaling and compare it with the numerically estimated 3-point position autocorrelation of the Lévy-Lorentz gas, for which the analytical expression is still an open question. Here we obtained a remarkable agreement between them, irrespective of any functional relationship with time. Moreover, we demonstrate that the position moments and the position autocorrelations of these systems scale in the same fashion, provided the times are large enough and far enough apart. Other observables, such as velocity moments and correlations, are reported to distinguish the systems. Published by the American Physical Society 2024

Comparative analysis of molecular dynamics and method of moments in two-dimensional concentric circular layers

In this manuscript, we undertake an examination of a classical plasma deployed on two finite co-planar surfaces: a circular region Ωin into an annular region Ωout with a gap in between. It is studied both from the point of view of statistical mechanics and the electrostatics of continua media. We employ a dual perspective: the first one is by using molecular dynamics (MD) simulations to find the system’s positional correlation functions and velocity distributions. That by modeling the system as a classical two-dimensional Coulomb plasma of point-like charged particles q 1 and q 2 on the layers Ωin and Ωout respectively with no background density. The second one corresponds to a finite Surface Electrode (SE) composed of planar metallic layers displayed on the regions Ωin , Ωout at constant voltages Vin , Vout considering axial symmetry. The surface charge density is calculated by the Method of Moments (MoM) under the electrostatic approximation. Point-like and differential charges elements interact via a 1/r —electric potential in both cases. The thermodynamic averages of the number density, and electric potential due to the plasma depend on the coupling and the charge ratio ξ=q1/q2 once the geometry of the layers is fixed. On the other hand, the fields due to the SE depend on the layer’s geometry and their voltage. In the document, is defined a protocol to properly compare the systems. We show that there are values of the coupling parameter, where the thermodynamic averages computed via MD agree with the results of MoM for attractive ξ=−1 and repulsive layers ξ = 1.

An Empirical Model For Intrinsic Alignments: Insights From Cosmological Simulations

We extend current models of the halo occupation distribution (HOD) to include a flexible, empirical framework for the forward modeling of the intrinsic alignment (IA) of galaxies. A primary goal of this work is to produce mock galaxy catalogs for the purpose of validating existing models and methods for the mitigation of IA in weak lensing measurements. This technique can also be used to produce new, simulation-based predictions for IA and galaxy clustering. Our model is probabilistically formulated, and rests upon the assumption that the orientations of galaxies exhibit a correlation with their host dark matter (sub)halo orientation or with their position within the halo. We examine the necessary components and phenomenology of such a model by considering the alignments between (sub)halos in a cosmological dark matter only simulation. We then validate this model for a realistic galaxy population in a set of simulations in the Illustris-TNG suite. We create an HOD mock with Illustris-like correlations using our method, constraining the associated IA model parameters, with the between our model’s correlations and those of Illustris matching as closely as 1.4 and 1.1 for orientation–position and orientation–orientation correlation functions, respectively. By modeling the misalignment between galaxies and their host halo, we show that the 3-dimensional two-point position and orientation correlation functions of simulated (sub)halos and galaxies can be accurately reproduced from quasi-linear scales down to . We also find evidence for environmental influence on IA within a halo. Our publicly-available software provides a key component enabling efficient determination of Bayesian posteriors on IA model parameters using observational measurements of galaxy-orientation correlation functions in the highly nonlinear regime.

Variance sum rule: proofs and solvable models

We derive, in more general conditions, a recently introduced variance sum rule (VSR) (Di Terlizzi et al 2024 Science 383 971) involving variances of displacement and force impulse for overdamped Langevin systems in a nonequilibrium steady state (NESS). This formula allows visualising the effect of nonequilibrium as a deviation of the sum of variances from normal diffusion 2Dt, with D the diffusion constant and t the time. From the VSR, we also derive formulas for the entropy production rate σ that, differently from previous results, involve second-order time derivatives of position correlation functions. This novel feature gives a criterion for discriminating strong nonequilibrium regimes without measuring forces. We then apply and discuss our results to three analytically solved models: a stochastic switching trap, a Brownian vortex, and a Brownian gyrator. Finally, we compare the advantages and limitations of known and novel formulas for σ in an overdamped NESS.

Correlation functions for confined wormlike chains

Polymer models describing the statistics of biomolecules under confinement have applications to a wide range of single-molecule experimental techniques and give insight into biologically relevant processes in vivo. In this paper, we determine the transverse position and bending correlation functions for a wormlike chain confined within slits and cylinders (with one and two confined dimensions, respectively) using a mean-field approach that enforces rigid constraints on average. We show the theoretical predictions accurately capture the statistics of a wormlike chain from Monte Carlo simulations in both confining geometries for both weak and strong confinement. We also show that the longitudinal correlation function is accurately computed for a chain confined to a slit and leverages the accuracy of the model to suggest an experimental technique to infer the (often unobservable) transverse statistics from the (directly observable) longitudinal end-to-end distance.

ANALISIS FUNGSI KORELASI KLASIK DAN KUANTUM UNTUK MODEL CINCIN: SEBUAH REVIEW

A review of the journal entitled 'Classical And Quantum Correlation Functions For A Ring Model' has been carried out which discusses the solutions of classical and quantum correlation functions for the ring model. Classical and quantum correlation functions are derived for systems of non-interacting particles moving in a circle. demonstrated that the decay behavior of the classical expression for the correlation function can be recovered from a strictly periodic quantum mechanical expression by taking the limit ℏ→0, after the appropriate transformation. The aim of this study is to present clearly and in detail how the position correlation function for a system of particles moving in a circle having a certain average energy and written in a form which shows the nature of the transformation. Next, we examine the use of Poisson addition to represent F(z), and how the correlation function becomes identical to the form given by classical statistical mechanics, which exhibits Gaussian decay. The results of this study indicate that the positional correlation function for a system of particles moving in a circle, and having a certain average energy, can be written in a form which shows the nature of the transformation. Then using the Poisson addition formula to represent F(z), the expression is rewritten to enable the limit ℏ→0 to be taken. It was finally found that the correlation function became identical to the form given by classical statistical mechanics, which exhibits a Gaussian decay.

Hazardous Shortcuts in Standard Binding Free Energy Calculations.

Calculating the standard binding free energies of protein-protein and protein-ligand complexes from atomistic molecular dynamics simulations in explicit solvent is a problem of central importance in computational biophysics. A rigorous strategy for carrying out such calculations is the so-called "geometrical route". In this method, two molecular objects are progressively separated from one another in the presence of orientational and conformational restraints serving to control the change in configurational entropy that accompanies the dissociation process, thereby allowing the computations to converge within simulations of affordable length. Although the geometrical route provides a rigorous theoretical framework, a tantalizing computational shortcut consists of simply leaving out such orientational and conformational degrees of freedom during the separation process. Here the accuracy and convergence of the two approaches are critically compared in the case of two protein-ligand complexes (Abl kinase-SH3:p41 and MDM2-p53:NVP-CGM097) and three protein-protein complexes (pig insulin dimer, SARS-CoV-2 spike RBD:ACE2, and CheA kinase-P2:CheY). The results of the simulations that strictly follow the geometrical route match the experimental standard binding free energies within chemical accuracy. In contrast, simulations bereft of geometrical restraints converge more poorly, yielding inconsistent results that are at variance with the experimental measurements. Furthermore, the orientational and positional time correlation functions of the protein in the unrestrained simulations decay over several microseconds, a time scale that is far longer than the typical simulation times of the geometrical route, which explains why those simulations fail to sample the relevant degrees of freedom during the separation process of the complexes.

Dynamics in condensed phase for systems involving phase functions obeying Gaussian statistics

The study of non-equilibrium processes in condensed phase has been fascinating and has attracted considerable attention of experimentalists as well as theoreticians during the last few decades. Since the theoretical treatment using Liouville equation involves multi-dimensional phase space functions, it is very difficult to visualize and also to obtain analytical expressions for the time-dependent distribution functions in Liouville space, which has led to formulation of various reduced space descriptions. The bottleneck in describing the non-equilibrium processes in the reduced space lies in the appearance of coupled velocity and position correlation functions in the equations, which are difficult to evaluate in general. In this work, we have developed a scheme to decouple the complicated multi-point coupled correlation function into two-point velocity and position correlation functions for situations characterized by a time-dependent phase function obeying Gaussian distribution, by making use of the Novikov's theorem. As an illustrative application, we have investigated the problem of optically controlled solvation dynamics and obtained exact expressions for the non-equilibrium solvation time correlation functions showing explicit dependence on the excitation frequency, as observed experimentally. We have shown that the linear response theory result is valid as long as the energy gap obeys the Gaussian distribution, rescuing the linear response theory based result from the domain of validity for small fluctuations in solvation coordinate.

Ultralarge-scale approximations and galaxy clustering: Debiasing constraints on cosmological parameters

ABSTRACT Upcoming galaxy surveys will allow us to probe the growth of the cosmic large-scale structure with improved sensitivity compared to current missions, and will also map larger areas of the sky. This means that in addition to the increased precision in observations, future surveys will also access the ultralarge-scale regime, where commonly neglected effects such as lensing, redshift-space distortions, and relativistic corrections become important for calculating correlation functions of galaxy positions. At the same time, several approximations usually made in these calculations such as the Limber approximation break down at those scales. The need to abandon these approximations and simplifying assumptions at large scales creates severe issues for parameter estimation methods. On the one hand, exact calculations of theoretical angular power spectra become computationally expensive, and the need to perform them thousands of times to reconstruct posterior probability distributions for cosmological parameters makes the approach unfeasible. On the other hand, neglecting relativistic effects and relying on approximations may significantly bias the estimates of cosmological parameters. In this work, we quantify this bias and investigate how an incomplete modelling of various effects on ultralarge scales could lead to false detections of new physics beyond the standard ΛCDM model. Furthermore, we propose a simple debiasing method that allows us to recover true cosmologies without running the full parameter estimation pipeline with exact theoretical calculations. This method can therefore provide a fast way of obtaining accurate values of cosmological parameters and estimates of exact posterior probability distributions from ultralarge-scale observations.

Velocity auto correlation function of a confined Brownian particle

Motivated by the simple models of molecular motor obeying a linear force-velocity relation, we have studied the stochastic dynamics of a Brownian particle in the presence of a linear velocity dependent force, $$f_s(1-\frac{v}{v_0})$$ where $$f_{s}$$ is a constant. The position and velocity auto correlation functions in different situations of the dynamics are calculated exactly. We observed that the velocity auto correlation function shows an exponentially decaying behaviour with time and saturates to a constant value in the time asymptotic limit, for a fixed $$f_s$$ . It attains saturation faster with increase in the $$f_{s}$$ value. When the particle is confined in a harmonic well, the spectral density exhibits a symmetric behaviour and the corresponding velocity auto correlation function shows a damped oscillatory behaviour before decaying to zero in the long time limit. With viscous coefficient, a non-systematic variation of the velocity auto correlation function is observed. Further, in the presence of a sinusoidal driving force, the correlation in velocities increases with increase in the amplitude of driving in the transient regime. For the particle confined in a harmonic well, the correlation corresponding to the shift relative to the average position is basically the thermal contribution to the total position correlation. Moreover, in the athermal regime, the total correlation is entirely due to the velocity dependent force.

Brownian Motion of Magnetic Skyrmions in One- and Two-Dimensional Systems

Magnetic skyrmions, which are topologically protected particle-like spin textures, exhibit Brownian motion at ambient temperatures in a ferromagnetic thin film. The potential applications of a skyrmions’ diffusive motion to the ultra-low energy stochastic and Brownian computation make research on its thermal dynamics necessary. Herein we experimentally demonstrate the mobility anomaly of the diffusion coefficients in one-dimensional diffusion processes. In addition, we observed a chiral property in the velocity–position correlation functions. The results are discussed in the context of Brownian motion in the shallow potential fluctuations. The data indicate the importance of the chiral property of skyrmions in Brownian motion.

DECaNT: Simulation tool for diffusion of excitons in carbon nanotube films

We present the numerical tool DECaNT (Diffusion of Excitons in Carbon NanoTubes) that simulates exciton transport in thin films of carbon nanotubes. Through a mesh of nanotubes generated using the Bullet Physics C++ library, excitons move according to an ensemble Monte Carlo algorithm, with the scattering rates that account for tube chirality, orientation, and distance. We calculate the diffusion tensor from the position–position correlation functions and analyze its anisotropy and dependence on the film composition, morphology, and defect density.

Noninteracting trapped fermions in double-well potentials: Inverted-parabola kernel

We study a system of $N$ noninteracting spinless fermions in a confining, double-well potential in one dimension. When the Fermi energy is close to the value of the potential at its local maximum we show that physical properties, such as the average density and the fermion position correlation functions, display a universal behavior that depends only on the local properties of the potential near its maximum. This behavior describes the merging of two Fermi gases, which are disjoint at sufficiently low Fermi energies. We describe this behavior in terms of a new correlation kernel that we compute analytically and we call it the inverted parabola kernel". As an application, we calculate the mean and variance of the number of particles in an interval of size $2L$ centered around the position of the local maximum, for sufficiently small $L$. Finally, we discuss the possibility of observing our results in experiments, as well as the extensions to nonzero temperature and to higher space dimensions.

Critical force in active microrheology.

Soft solids like colloidal glasses exhibit a yield stress, above which the system starts to flow. The microscopic analogon in microrheology is the untrapping or depinning of a tracer particle subject to an external force exceeding a threshold value in a glassy host. We characterize this delocalization transition based on a bifurcation analysis of the corresponding mode-coupling theory equations. A schematic model that allows analytical progress is presented first, and the full physical model is studied numerically next. This analysis yields a continuous dynamic transition with a critical power-law decay of the probe correlation functions with exponent -1/2. To compare with simulations with a limited duration, a finite-time analysis is performed, which yields reasonable results for not-too-small wave vectors. The theoretically predicted findings are verified by Langevin dynamics simulations. For small wave vectors we find anomalous behavior for the probe position correlation function, which can be traced back to a wave-vector divergence of the critical amplitude. In addition, we propose and test three methods to extract the critical force from experimental data, which provide the same value of the critical force when applied to the finite-time theory or simulations.

Generalized diffusion of quantum Brownian motion.

This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wigner function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic, and overdamped. It is found in the periodic case that the generalized diffusion is a discontinuous function exhibiting negative values during short time periods of time. This counterintuitive result, found theoretically in other systems and waiting for its experimental confirmation, can be perfectly explained in the framework of the quantum Langevin equation. Its oscillatory behavior is primordially due to the response to the external field while its quantum origin contributes only in its magnitude. The results are compared to those of the continuum limit which exhibits similar behavior.

Event-chain Monte Carlo simulations of the liquid to solid transition of two-dimensional decagonal colloidal quasicrystals

In event-chain Monte Carlo simulations, we model colloidal particles in two dimensions that interact according to an isotropic short-ranged pair potential which supports the two typical length scales present in decagonal quasicrystals. We investigate the assembled structures as we vary the density and temperature. Our special interest is related to the transition from quasicrystal to liquid. In contrast to the KTHNY melting theory for quasicrystals which predicts an intermediate pentahedratic phase, we find a one-step first-order melting transition. However, we discover that the slow relaxation of phasonic flips, i.e. rearrangements of the particles due to additional degrees of freedom in quasicrystals, changes the positional correlation functions, to the extent that structures with long-range orientational correlations, but exponentially decaying positional correlations, are observed.

Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation.

We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power-law index of the sojourn-time distributions. In contrast, at long times, a power-law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or nonequilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.

Dynamics of heavy quarkonia in memory-dependent dissipative environment from Bohmian trajectory perspective

In this work, we study the dynamics of particles coupled to a dissipative environment from Bohmian trajectory perspective. The dissipation is modeled using the concept of memory-dependent derivative (MDD), which is characterized by its time-delay constant [Formula: see text] and nonsingular kernel [Formula: see text] of two parameters [Formula: see text], [Formula: see text]. By assuming a Gaussian packet wave function, we derived a MDD-Langevin equation (MDDLE). The general behavioral solution [Formula: see text] of the MDDLE is investigated for the case of Gaussian fluctuation force. Based on the miscellaneous choices of [Formula: see text], [Formula: see text], [Formula: see text], the findings are that [Formula: see text] can exhibit distinct behaviors, such as monotonic and nonmonotonic decay without zero crossings, oscillatory-like without zero and with zero crossing. Therefore, we have either diffusion or oscillatory dominate based on the problem parameters. For a harmonically bound heavy quarkonium, characterized by the angular frequency [Formula: see text], the position correlation function [Formula: see text] is then obtained and analyzed numerically. The analysis shows that this correlation function is also sensitive to the various choices of [Formula: see text] and kernel parameters. Based on these choices, the correlation function exhibits distinct behaviors: oscillation without damping, damping, and enhanced. This wide range of behavior coverage increases the versatility to fit nonlinear or memory-dependent experimental findings. The results are compared with the fractional Langevin equation.

Molecular dynamics simulations of nematic phases formed by cyano-biphenyl dimers

ABSTRACTMolecular dynamics simulations of selected members of the cyano-biphenyl (CB) series of dimers (CBnCB) have been set up using atomistic detail for the description of the alkyl spacer and coarse-grained representation of the CB end-segments. Detailed results are presented for the CB7CB dimers, showing an isotropic fluid phase and two nematic phases. The positional and orientational correlation functions extracted from the simulations are used to elucidate the structure of the low-temperature nematic phase. Polar molecular ordering is clearly identified along a direction undergoing helical twisting at right angles to the helical axis, with a constant pitch of about of 8 nm. The local ordering of the various molecular segments is calculated and found to be in excellent agreement with experimental NMR measurements. Key findings of the simulation are shown to be correctly predicted by the theoretical model of the polar-twisted nematic (NPT) phase [A.G. Vanakaras, D.J. Photinos, Soft Matter. 12 (2016) 2208–2220]. The complete failure of the usual twist bend model (NTB) to account for these findings is demonstrated.

Exact density matrix of an oscillator-bath system: Alternative derivation

Starting from a total Lagrangian describing an oscillator-bath system, an alternative derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The modified quantum propagator is obtained in the generalised case where the main oscillator is under the influence of a classical external force. By introducing auxiliary classical external fields, the generalised quantum propagator or generating functional of position correlation functions is obtained.