Particle-particle Correlation Function Research Articles

Laughlin topology on fractal lattices without area law entanglement

Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin type topology. Here, we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.

Dissipation-induced first-order decoherence phase transition in a noninteracting fermionic system

We consider a quantum wire connected to the leads and subjected to dissipation along its length. The dissipation manifests as tunneling into (out of) the chain from (to) a memoryless environment. The evolution of the system is described by the Lindblad equation. Already infinitesimally small dissipation along the chain induces a quantum phase transition (QPT). This is a decoherence QPT: the reduced density matrix of a subsystem in the nonequilibrium steady state (far from the ends of the chain) can be represented as the tensor product of single-site density matrices. The QPT is identified from the jump of the current and the entropy per site as the dissipation becomes nonzero. We also explore the properties of the boundaries of the chain close to the transition point and observe that the boundaries behave as if they undergo a second-order phase transition as a function of the dissipation strength: the particle-particle correlation functions and the response to the electric field exhibit a power-law divergence. Disorder is known to localize one-dimensional systems, but the coupling to the memoryless environment pushes the system back into the delocalized state even in the presence of disorder. Interestingly, we observe a similar transition in the classical dissipative counterflow model: the current has a jump at the ends of the chain introducing an infinitely small dissipation.

Two-point correlation function of an exclusion process with hole-dependent rates.

We consider an exclusion process on a ring in which a particle hops to an empty neighboring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well-known mapping of this model to the zero-range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n)=1+(b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b>2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b-2. We also calculate the two-point correlation function above the critical density and find that the correlation length diverges with a critical exponent ν=1/(b-2) for b<3 and 1 for b>3. These results are compared with those obtained using an exact series expansion for finite systems.

T-Matrix Approximation to Two-Dimensional Hubbard Model

The low-density regime of the two-dimensional Hubbard model has been investigated by the t-matrix approximation, i.e., by taking into account the particle-particle ladder processes represented in terms of particle-particle correlation function, K(q, x+iδ). The q-dependence of K(q, x+iδ) in the region of small x has been examined in detail and it is found that the imaginary part of self-energy function of electrons depends on excitation energy, ε, as ε2ln|ε| which are due to contributions involving K(q, x+iδ) with q≃2kF, kF being the fermi momentum. The existence of the upper Hubbard band for the case of large U has also been demonstrated.

Interpretation of Light Scattering and Turbidity Measurements in Aggregated Systems: Effect of Intra-Cluster Multiple-Light Scattering

In this work we studied the effect of intracluster multiple-light scattering on the scattering properties of a population of fractal aggregates. To do so, experimental data of diffusion-limited aggregation for three polystyrene latexes with similar surface properties but different primary particle diameters (equal to 118, 420, and 810 nm) were obtained by static light scattering and by means of a spectrophotometer. In parallel, a population balance equation (PBE) model, which takes into account the effect of intracluster multiple-light scattering by solving the T-matrix and the mean-field version of T-matrix, was formulated and validated against time evolution of the root mean radius of gyration, <R(g)>, of the zero angle intensity of scattered light, I(0), and of the turbidity, tau. It was found that the mean-field version of the T-matrix theory is able to correctly predict the time evolution of all measured light scattering quantities for all sizes of primary particles without any adjustable parameter. The structure of the aggregates, characterized by fractal dimension, d(f), was independent of the primary particle size and equal to 1.7, which is in agreement with values found in literature. Since the mean-field version of the T-matrix theory used is rather complicated and requires advanced knowledge of cluster structure (i.e., the particle-particle correlation function), a simplified version of the light scattering model was proposed and tested. It was found that within the range of operating conditions investigated, the simplified version of the light scattering model was able to describe with reasonable accuracy the time evolution of all measured light scattering quantities of the cluster mass distribution (CMD) for all three sizes of primary particles and two values of the laser wavelength.

Number of arm selection in two-dimensional diffusion processes

We introduce an algorithm to generate two-dimensional diffusion-limited star-branched aggregates (DLSA) attaching bi-functional monomers successively to a central colloidal particle with any desired number of reactive sites. The proposed algorithm produces star-shaped aggregates that grow forever and show a power law polydispersity in the chemical length of the arms near the central colloid. More interestingly, it gives rise to a number of arm selection consisting in that only a small number of arms (around five) define the final structure at relatively large distances from the central colloid, independently of the initial number of reactive sites and the size of the central colloid. We characterize the structure of the aggregates by means of the particle-particle correlation function, analyze its scaling properties and obtain the fractal dimension.

Depletion from a hard wall induced by aggregation and gelation

Diffusion-limited cluster aggregation and gelation of hard spheres is simulated using off-lattice Monte Carlo simulations. A comparison is made of the wall-particle correlation function with the particle-particle correlation function over a range of volume fractions, both for the initial system of randomly distributed spheres and for the final gel state. For randomly distributed spheres the correlation functions are compared with theoretical results using the Ornstein-Zernike equation and the Percus-Yevick closure. At high volume fractions (phi > 40%) gelation has little influence on the correlation function, but for phi < 10% it is a universal function of the distance normalized by correlation length (xi) of the bulk. The width of the depletion layer is about 0.5xi. The concentration increases as a power law from the wall up to r approximately = xi, where it reaches a weak maximum before decreasing to the bulk value.

Structure, surface excess and effective interactions in polymer nanocomposite melts and concentrated solutions.

The Polymer Reference Interaction Site Model (PRISM) theory is employed to investigate structure, effective forces, and thermodynamics in dense polymer-particle mixtures in the one and two particle limit. The influence of particle size, degree of polymerization, and polymer reduced density is established. In the athermal limit, the surface excess is negative implying an entropic dewetting interface. Polymer induced depletion interactions are quantified via the particle-particle pair correlation function and potential of mean force. A transition from (nearly) monotonic decaying, attractive depletion interactions to much stronger repulsive-attractive oscillatory depletion forces occurs at roughly the semidilute-concentrated solution boundary. Under melt conditions, the depletion force is extremely large and attractive at contact, but is proceeded by a high repulsive barrier. For particle diameters larger than roughly five monomer diameters, division of the force by the particle radius results in a nearly universal collapse of the depletion force for all interparticle separations. Molecular dynamics simulations have been employed to determine the depletion force for nanoparticles of a diameter five times the monomer size over a wide range of polymer densities spanning the semidilute, concentrated, and melt regimes. PRISM calculations based on the spatially nonlocal hypernetted chain closure for particle-particle direct correlations capture all the rich features found in the simulations, with quantitative errors for the amplitude of the depletion forces at the level of a factor of 2 or less. The consequences of monomer-particle attractions are briefly explored. Modification of the polymer-particle pair correlations is relatively small, but much larger effects are found for the surface excess including an energetic driven transition to a wetting polymer-particle interface. The particle-particle potential of mean force exhibits multiple qualitatively different behaviors (contact aggregation, steric stabilization, local bridging attraction) depending on the strength and spatial range of the polymer-particle attraction.

Model of correlated sequential adsorption of colloidal particles.

We present results of a model of sequential adsorption in which the adsorbing particles are correlated with the particles attached to the substrate. The strength of the correlations is measured by a tunable parameter sigma. The model interpolates between free ballistic adsorption in the limit sigma-->infinity and a strongly correlated phase, appearing for sigma-->0 and characterized by the emergence of highly ordered structures. The phenomenon is manifested through the analysis of several magnitudes, as the jamming limit and the particle-particle correlation function. The effect of correlations in one dimension manifests in the increased tendency to particle chaining in the substrate. In two dimensions the correlations induce a percolation transition, in which a spanning cluster of connected particles appears at a certain critical value sigma(c). Our study could be applicable to more general situations in which the coupling between correlations and disorder is relevant, as for example, in the presence of strong interparticle interactions.

On the Concentration Dependence of the Cluster Fractal Dimension in Colloidal Aggregation

We have undertaken the task to calculate, by means of extensive numerical simulations and by different procedures, the cluster fractal dimension (d) of colloidal aggregates at different initial colloid concentrations. Our first approach consists in obtaining d from the slope of the log-log plots of the radius of gyration versus size of all the clusters formed during the aggregation time. In this way, for diffusion-limited colloidal aggregation, we have found a square root type of increase of the fractal dimension with concentration, from its zero-concentration value: d = d0 f + a φβ, with d0 f = 1.80 ± 0.01, a = 0.91 ± 0.03 and β = 0.51 ± 0.02, and where φ is the volume fraction of the colloidal particles. In our second procedure, we get the d via the particle-particle correlation function gcluster(r) and the structure function Scluster(q) of individual clusters. We first show that the stretched exponential law gcluster(r) = Ard −3e−(r/ξ) gives an excellent fit to the cutoff of the g(r). Here, A, a and ξ are parameters characteristic of each of the clusters. From the corresponding fits we then obtain the cluster fractal dimension. In the case of the structure function Scluster (q), using its Fourier transform relation with gcluster(r) and introducing the stretched exponential law, it is exhibited that at high q values it presents a length scale for which it is linear in a log-log plot versus q, and the value of the d extracted from this plot coincides with the d of the stretched exponential law. The concentration dependence of this new estimate of d, using the correlation functions for individual clusters, agrees perfectly well with that from the radius of gyration versus size. It is however shown that the structure factor S(q) of the whole system (related to the normalized scattering intensity) is not the correct function to use when trying to obtain a cluster fractal dimension in concentrated suspensions. The log-log plot of S(q) vs. q proportions a value higher than the true value. Nevertheless, it is also shown that the true value can be obtained from the initial slope of the particle-particle correlation function g(r), of the whole system. A recipe is given on how to obtain approximately this g(r) from a knowledge of the S(q), up to a certain maximum q value.

Structure function and fractal dimension of diffusion-limited colloidal aggregates

On a three-dimensional lattice and at different concentrations we perform extensive numerical simulations of diffusion-limited colloidal aggregation (DLCA). In a previous work, we showed that the fractal dimension ${d}_{f}$ of the DLCA aggregates in the flocculation limit presents a square root type of dependence with the initial colloidal concentration. The ${d}_{f}$ was obtained from the slope of a standard log-log plot of the number of particles versus size of the formed aggregates. In this work we confirm the concentration dependency using the particle-particle correlation function $g(r)$ and the structure function $S(q)$ of individual aggregates. We demonstrate that the ${g(r)=Ar}^{{d}_{f}\ensuremath{-}3}{e}^{\ensuremath{-}(r/\ensuremath{\xi}{)}^{a}},$ where $A,$ $a,$ and \ensuremath{\xi} are parameters characteristic of the aggregates, and $a&gt;1.$ This stretched exponential law gives an excellent fit to the cutoff of the $g(r).$ The structure function reveals the ${d}_{f}$ from the slope of a log-lot plot of $S(q)$ versus $q$ for high $q$ values. We also analyze $g(r)$ and $S(q),$ at different times during the reaction, for the whole aggregating system composed of many clusters of different sizes. We observe that the ${d}_{f}$ calculated from the $g(r)$ agrees well with that obtained from individual clusters. However, caution should be observed to extract a ${d}_{f}$ from the corresponding $S(q).$ Our results indicate that for finite concentrations a ${d}_{f}$ systematically larger than the true value is obtained from such analysis.

Mixing in reaction dynamics under enhanced diffusion conditions

We first review the Lévy-flight approach for the description of enhanced diffusion and then study the effects of mixing on the transient A + B → 0 reaction at stoichiometric conditions in d = 1−3 dimensions under enhanced diffusion conditions. The particles are assumed to follow Lévy walks, characterized by an index γ, which result in enhanced diffusion for 1 < γ < 2. Particle density decays and particle-particle correlation functions are calculated using deterministic reaction diffusion equations and computer simulations of this multi-particle problem. We concentrate on the segregation phenomenon and demonstrate that, depending on γ, segregation may disappear in d = 3 dimensions under enhanced diffusion conditions. Hence, Lévy walks lead to an efficient mixing of reacting particles.

Breakdown of Ovchinnikov-Zeldovich Segregation in the A+B-->0 Reaction under Lévy Mixing.

Density fluctuations in the transient $A+B\ensuremath{\rightarrow}0$ reaction, at stoichiometric conditions, cause the Ovchinnikov-Zeldovich segregation which slows down the reaction process. Continuous mixing that homogenizes the reactants can suppress the role of the fluctuations. L\'evy walks which lead to an anomalous diffusion provide such a mixing mechanism. For L\'evy walks with the number of steps distributed as $P(n)\ensuremath{\sim}{n}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\gamma}},n&gt;0,1&lt;\ensuremath{\gamma}&lt;2$, we demonstrate that segregation disappears in $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ dimensions for $\ensuremath{\gamma}&lt;3/2$. Particle densities and particle-particle correlation functions are presented. The erosion of the segregation is an example of L\'evy mixing.

Driven granular media in one dimension: Correlations and equation of state.

We study a one-dimensional granular system in which each particle is excited by white noise, with inelastic interactions between the particles. When the coefficient of restitution, $\ensuremath{\eta}$, is one, the particles are uncorrelated. As $\ensuremath{\eta}$ decreases, long-range correlations between the particles develop. A computer simulation of the system shows a steady-state, power-law particle-particle correlation function, which depends strongly on $\ensuremath{\eta}$. We give simple analytic arguments for the correlations. We also present an "equation of state" for the system of particles, which relates the noise amplitude to the particle density and the average particle speed.

Simulation of Ostwald Ripening in Two Dimensions: Spatial and Nearest Neighbor Correlations

Recent experimental results on late stage coarsening (Ostwald ripening) in two dimensions were reinvestigated by means of numerical simulation using the Cahn-Hilliard equation (model B). We determine the spatial particle-particle and charge-charge correlation functions according to the experimental results of Krichevsky and Stavans. We find that our numerical results correspond well with these experiments. We also investigate particle-particle correlations between nearest neighbors (defined by the Voronoi diagram of particle centroids) following the experiments of Seul, Morgan and Sire and compare our numerical results with their Maximum Entropy model of Ostwald ripening.

Azimuthal correlation functions and the energy of vanishing flow in nucleus-nucleus collisions

A method is proposed for estimating the energy of vanishing flow (also called balance energy, E bal) without reconstructing the reaction plane. This method is based on the shapes of experimental particle-particle azimuthal correlation functions. It is used for determining E bal for three systems: Ar + Al, Zn + Ti, Zn + Ni. The results are compared with estimations using flow-parameter analysis and with theoretical expectations.

Id Excited Granular Media: Clustering and Equation of State

AbstractWe study a one-dimensional granular system, excited by white noise, with inelastic interactions between the particles. When the coefficient of restitution,η, is one, the particles are totally uncorrelated. As η decreases the particles cluster. A computer simulation of the system shows equilibrium clustering with a power law particle-particle correlation function. This correlation function is independent of the average kinetic energy of the particles but depends strongly on η We give simple analytical arguments to describe this clustering. We also present an “equation of state” for the particles, which relates the noise amplitude to the particle density and the average particle speed.

Two Proton Interferometry in 46.7 MeV/u 12C Induced Reactions

The particle-particle (P-P) correlation functions are investigated in 46.7 MeV/u 12C induced reactions, the correlation functions are greatly enhanced at q ∼ 20 MeV/c when the sum of the energies of two correlated protons are high and almost independent on the targets. On the contrast, a strongly target dependent effect is observed for the P-P correlation functions when the sum energies are low.

Lateral diffusion and aggregation. A Monte Carlo study

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.

Two-dimensional Hubbard model in low electron density limit

The two dimensional Hubbard model with repulsive interaction is studied in the tz. sbnd;matrixapproximation. At T = 0 the particle-particle correlation function and the tz. sbnd;matrix are obtained analytically so that it is shown that the tz. sbnd;matrix and hence the self-energy have no singularities in the upper half phase of the complex energy variable. The phase angle of the tz. sbnd;matrix δ(2 k F + q, ω), in the limit q→0 and ω→0, is found to depend on the ratio ω/q and is not always zero. This non-vanishing phase angle results in the logarithmic correction to the imaginary part of the self-energy as ImΣ( k F ,ϵ) ∝ ϵ 2 lm |ϵ|. However, the renormalization factor Z is finite at the Fermi surface suggesting that the stability of the Fermi liquid state at least within the conventional perturbation theory. The case of the attractive interaction has also been studied and the crossover from the Cooper pairing to the real-space pairing is indicated.