Splitting Probabilities Research Articles

Granular segregation driven by particle interactions.

We report the results of an experimental study of particle-particle interactions in a horizontally shaken granular layer that undergoes a second order phase transition from a binary gas to a segregation liquid as the packing fraction C is increased. By focusing on the behavior of individual particles, the effect of C is studied on (1)the process of cluster formation, (2)cluster dynamics, and (3)cluster destruction. The outcomes indicate that the segregation is driven by two mechanisms: attraction between particles with the same properties and random motion with a characteristic length that is inversely proportional to C. All clusters investigated are found to be transient and the probability distribution functions of the separation times display a power law tail, indicating that the splitting probability decreases with time.

Do tree split probabilities determine the branch lengths?

The evolution of aligned DNA sequence sites is generally modeled by a Markov process operating along the edges of a phylogenetic tree. It is well known that the probability distribution on the site patterns at the tips of the tree determines the tree topology, and its branch lengths. However, the number of patterns is typically much larger than the number of edges, suggesting considerable redundancy in the branch length estimation. In this paper we ask whether the probabilities of just the 'edge-specific' patterns (the ones that correspond to a change of state on a single edge) suffice to recover the branch lengths of the tree, under a symmetric 2-state Markov process. We first show that this holds provided the branch lengths are sufficiently short, by applying the inverse function theorem. We then consider whether this restriction to short branch lengths is necessary. We show that for trees with up to four leaves it can be lifted. This leaves open the interesting question of whether this holds in general. Our results also extend to certain Markov processes on more than 2-states, such as the Jukes-Cantor model.

Moment equations for a piecewise deterministic PDE

We analyze a piecewise deterministic PDE consisting of the diffusion equation on a finite interval Ω with randomly switching boundary conditions and diffusion coefficient. We proceed by spatially discretizing the diffusion equation using finite differences and constructing the Chapman–Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We show how the CK equation can be used to generate a hierarchy of equations for the r-th moments of the stochastic field, which take the form of r-dimensional parabolic PDEs on that couple to lower order moments at the boundaries. We explicitly solve the first and second order moment equations (r = 2). We then describe how the r-th moment of the stochastic PDE can be interpreted in terms of the splitting probability that r non-interacting Brownian particles all exit at the same boundary; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment. Hence the stochastic diffusion equation describes two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment. Finally, in the limit of fast switching, we use a quasi-steady state approximation to reduce the piecewise deterministic PDE to an SPDE with multiplicative Gaussian noise in the bulk and a stochastically-driven boundary.

Spin-orbit interaction in multiple quantum wells

In this paper, we investigate how the structure of multiple quantum wells affects spin-orbit interactions. To increase the interface-related Rashba spin splitting and the strength of the interface-related Rashba spin-orbit interaction, we designed three kinds of multiple quantum wells. We demonstrate that the structure of the multiple quantum wells strongly affected the interface-related Rashba spin-orbit interaction, increasing the interface-related Rashba spin splitting to up to 26% larger in multiple quantum wells than in a stepped quantum well. We also show that the cubic Dresselhaus spin-orbit interaction similarly influenced the spin relaxation time of multiple quantum wells and that of a stepped quantum well. The increase in the interface-related Rashba spin splitting originates from the relationship between interface-related Rashba spin splitting and electron probability density. Our results suggest that multiple quantum wells can be good candidates for spintronic devices.

Tensor gluons and proton structure

We recently considered a possibility that inside a proton and, more generally, inside hadrons there could be additional partons, tensor gluons, which carry a part of the proton momentum. Tensor gluons have zero electric charge, like gluons, but have a larger spin, and we therefore call them tensor gluons. A nonzero density of tensor gluons can be generated by the emission of tensor gluons by gluons. Tensor gluons can further split into pairs of tensor gluons through different channels. To describe all these processes, we must know the general splitting probabilities for tensor gluons. These probabilities should satisfy very general symmetry relations, which we can resolve by introducing a splitting index. This approach allows finding the general form of the splitting functions, deriving the corresponding DGLAP evolution equations, and calculating the one-loop Callan-Symanzik beta function for tensor gluons of a given spin. Our results provide a nontrivial consistency check of the theory and of the Callan-Symanzik beta function calculations because the theory has a unique coupling constant and its high-energy behavior should be universal for all particles of the spectrum. We argue that the contribution of all spins to the beta function vanishes, leading to a conformal invariant theory.

Asymptotic analysis of first passage time problems inspired by ecology.

A hybrid asymptotic-numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is isotropic but possibly spatially heterogeneous. Our singular perturbation methodology, which is based on the assumption that the ratio [Formula: see text] of the radius of a typical prey patch to that of the overall landscape is asymptotically small, leads to the derivation of an algebraic system that determines the MFPT in terms of parameters characterizing the shapes of the small prey patches together with a certain Green's function, which in general must be computed numerically. The expected error in approximating the MFPT by our semi-analytical procedure is smaller than any power of [Formula: see text], so that our approximation of the MFPT is still rather accurate at only moderately small prey patch radii. Overall, our hybrid approach has the advantage of eliminating the difficulty with resolving small spatial scales in a full numerical treatment of the partial differential equation (PDE). Similar semi-analytical methods are also developed and implemented to accurately calculate related quantities such as the variance of the mean first passage time (VMFPT) and the splitting probability. Results for the MFPT, the VMFPT, and splitting probability obtained from our hybrid methodology are validated with corresponding results computed from full numerical simulations of the underlying PDEs.

Multidimensional reaction rate theory with anisotropic diffusion.

An analytical expression is derived for the rate constant that describes diffusive transitions between two deep wells of a multidimensional potential. The expression, in contrast to the Kramers-Langer formula for the rate constant, is valid even when the diffusion is highly anisotropic. Our approach is based on a variational principle for the reactive flux and uses a trial function for the splitting probability or commitor. The theoretical result is validated by Brownian dynamics simulations.

Parton distribution functions in the context of parton showers

When the initial state evolution of a parton shower is organized according to the standard "backward evolution'' prescription, ratios of parton distribution functions appear in the splitting probabilities. The shower thus organized evolves from a hard scale to a soft cutoff scale. At the end of the shower, one expects that only the parton distributions at the soft scale should affect the results. The other effects of the parton distributions should have cancelled. This means that the kernels for parton evolution should be related to the shower splitting functions. If the initial state partons can have non-zero masses, this requires that the evolution kernels cannot be the usual MSbar kernels. We work out what the parton evolution kernels should be to match the shower evolution contained in the parton shower event generator Deductor, in which the b and c quarks have non-zero masses.

Splitting probabilities and interfacial territory covered by two-dimensional and three-dimensional surface-mediated diffusion

We consider the mean territory covered by a particle that performs surface-mediated diffusion inside a spherical confining domain (in two and three dimensions) before exit through an opening on the surface. This quantity can be expressed in terms of the splitting probability between two targets on the surface. We derive a general formula that relates this splitting probability to the mean first passage time to a single target that has been recently calculated for such a surface-mediated diffusion process. This formula is exact for pointlike targets and is shown to be accurate for extended targets. The mean covered territory is then found and analyzed for an arbitrary extension of the exit region in both two- and three-dimensional spherical domains.

Diffusion along the Splitting/Commitment Probability Reaction Coordinate

The splitting or commitment probabilities of states in the region of configuration space that separates reactants and products play an important role in the theory of chemical reactions. Assuming that the splitting probability changes more slowly than any other coordinate, we project multidimensional diffusive dynamics onto it. The resulting one-dimensional diffusion equation is not exact because the assumed separation of time scales does not hold in general. Nevertheless, this equation has the remarkable property that it always predicts the exact value of the number of transitions between reactants and products per unit time at equilibrium and hence the exact reaction rate. In the special case of two deep basins separated by a harmonic saddle, this equation is equivalent to the one that describes diffusion along a coordinate perpendicular to the transition state, defined as the surface starting from which reactants and products are reached with equal probability.

Density-based clustering algorithm combined with limited regional sampling

Concerning the inefficient time performance and lower clustering accuracy revealed by the traditional density-based algorithms of DBSCAN and DBRS,this paper proposed an improved density-based clustering algorithm called DBLRS,which is combined with limited regional sampling technique.The algorithm used the parameter Eps to search for the neighborhood and expanded points of a core point without increasing time and space complexity,and implemented data sampling in a limited area(Eps,2Eps).The experimental results confirm that DBLRS can reduce the probability of large clusters' splitting and improve the algorithmic efficiency and clustering accuracy by selecting representative points to expand a cluster.

Sub-pixel resolution of a pnCCD for X-ray white beam applications

A new approach to achieve sub-pixel spatial resolution in a pnCCD detector with 75 × 75 μm2 pixel size is proposed for X-ray applications in single photon counting mode. The approach considers the energy dependence of the charge cloud created by a single photon and its split probabilities between neighboring pixels of the detector based on a rectangular model for the charge cloud density. For cases where the charge of this cloud becomes distributed over three or four pixels the center position of photon impact can be reconstructed with a precision better than 2 μm. The predicted charge cloud sizes are tested at selected X-ray fluorescence lines emitting energies between 6.4 keV and 17.4 keV and forming charge clouds with size (rms) varying between 8 μm and 10 μm respectively. The 2 μm enhanced spatial resolution of the pnCCD is verified by means of an x-ray transmission experiment throughout an optical grating.

Exact calculations of first-passage quantities on recursive networks

We present general methods to exactly calculate mean first-passage quantities on self-similar networks defined recursively. In particular, we calculate the mean first-passage time and the splitting probabilities associated to a source and one or several targets; averaged quantities over a given set of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities highlights the dependency of first-passage processes with respect to the source-target distance, which has recently revealed to be a key parameter in characterizing transport in complex media. We explicitly perform calculations for different classes of recursive networks [finitely ramified fractals, scale-free (trans)fractals, nonfractals, mixtures between fractals and nonfractals, nondecimable hierarchical graphs] of arbitrary size. Our approach unifies and significantly extends the available results in the field.

A Freight Mode Choice Analysis Using a Binary Logit Model and GIS: The Case of Cereal Grains Transportation in the United States

Mode choice is important in shipping commodities efficiently. This paper develops a binary logit model and a regression model to study the cereal grains movement by truck and rail in the United States using the publically available Freight Analysis Framework (FAF2.2) database and U.S. highway and networks and TransCAD, a geographic information system with strong transportation modeling capabilities. The binary logit model and the regression model both use the same set of generic variables, including mode split probability, commodity weight, value, network travel time, and fuel cost. The results show that both the binary logit and regression models perform well for cereal grains transportation in the United States, with the binary logit model yielding overall better estimates with respect to the observed truck and rail mode splits. The two models can be used to study other commodities between two modes and may produce better results if more mode specific variables are used.

Markov chain modeling of polymer translocation through pores

We solve the Chapman-Kolmogorov equation and study the exact splitting probabilities of the general stochastic process which describes polymer translocation through membrane pores within the broad class of Markov chains. Transition probabilities, which satisfy a specific balance constraint, provide a refinement of the Chuang-Kantor-Kardar relaxation picture of translocation, allowing us to investigate finite size effects in the evaluation of dynamical scaling exponents. We find that (i) previous Langevin simulation results can be recovered only if corrections to the polymer mobility exponent are taken into account and (ii) the dynamical scaling exponents have a slow approach to their predicted asymptotic values as the polymer's length increases. We also address, along with strong support from additional numerical simulations, a critical discussion which points in a clear way the viability of the Markov chain approach put forward in this work.

Splitting Probabilities as a Test of Reaction Coordinate Choice in Single-Molecule Experiments

To explain the observed dynamics in equilibrium single-molecule measurements of biomolecules, the experimental observable is often chosen as a putative reaction coordinate along which kinetic behavior is presumed to be governed by diffusive dynamics. Here, we invoke the splitting probability as a test of the suitability of such a proposed reaction coordinate. Comparison of the observed splitting probability with that computed from the kinetic model provides a simple test to reject poor reaction coordinates. We demonstrate this test for a force spectroscopy measurement of a DNA hairpin.

Family dissolution and precautionary savings: an empirical analysis

The main research question of this paper is whether or not the risk of family disruption has an impact on the consumption/saving decisions of households. Although little empirical work exists in this area, often presenting indirect evidence, the theory is divided over the effect of family risk over saving and wealth accumulation. By using data from the Italian Survey on Households Income and Wealth, we build a probabilistic model to assess the probability of marital splitting, and then we insert this probability as a distinct or interacted regressor, in a statistically consistent way, into a linear model of consumption. Furthermore, we study the differential behaviour, in terms of consumption/saving choices, of couples experiencing marital splitting over the subsequent 2 years. The main result of our analysis is that family disruption risk generates precautionary savings, reducing current consumption. In fact, according to our estimates, on average, the risk of divorce generates an amount of additional yearly precautionary savings of around 800 euros at constant prices of the year 2000, which represents 11% of overall household savings.

Analytic and numerical analysis of some statistical features of fragmentation processes

In this work, we investigate the following inhomogeneous model of fragmentation with an exponential splitting probability: suppose that an initial particle having a mass x 0 > 0 (large enough) is broken, with a size-dependent probability p ( x 0 ) ≔ 1 − e − x 0 , into b > 1 pieces. With probability 1 − p ( x 0 ) , this fragment is left unchanged for ever. The process of fragmentation is repeated recursively for all pieces. Some statistical features are studied, chiefly the fragment size distribution, number of fragments and partition function of the random fragmentation tree. Also, we give details of the numerical analysis of these features.

Real-time traffic estimation using data expansion

This paper presents a method for estimating missing real-time traffic volumes on a road network using both historical and real-time traffic data. The method was developed to address urban transportation networks where a non-negligible subset of the network links do not have real-time link volumes, and where that data is needed to populate other real-time traffic analytics. Computation is split between an offline calibration and a real-time estimation phase. The offline phase determines link-to-link splitting probabilities for traffic flow propagation that are subsequently used in real-time estimation. The real-time procedure uses current traffic data and is efficient enough to scale to full city-wide deployments. Simulation results on a medium-sized test network demonstrate the accuracy of the method and its robustness to missing data and variability in the data that is available. For traffic demands with a coefficient of variation as high as 40%, and a real-time feed in which as much as 60% of links lack data, we find the percentage root mean square error of link volume estimates ranges from 3.9% to 18.6%. We observe that the use of real-time data can reduce this error by as much as 20%.

First-passage quantities of Brownian motion in a bounded domain with multiple targets: a unified approach

In this paper, we introduce a general computation method to systematically determine the mean first-passage time (MFPT), the global mean first-passage time (GMFPT) and splitting probabilities for a continuous Brownian motion in a confined 2D or 3D domain with multiple absorbing targets in the bulk or on the boundary. This method is applied to spherically symmetric domains in the limit of small-sized targets and asymptotic expansions of the MPFT, GMFPT and splitting probabilities are obtained in four distinct cases: 3D domains with targets in the bulk, 3D domains with targets on the boundary, 2D domains with targets on the bulk and 2D domains with targets on the boundary. This approach gives a unified description of existing exact results which were obtained using specific technics, and also yields new results, in particular for N targets splitting probabilities.