Arbitrary Initial Distribution Research Articles

A Projection Integration Method for X-ray Collimator Design.

One of the recent trends in radiation therapy is to increaseconformal and accurate dose deliverysuch as in stereotactic radiosurgery (SRS). Treating small lesions and brain disorders requires the accurate placement of small radiation fields deep inside the human cranium.To design a collimator meeting these requirements, a new numerical concept was developed, which is presented here. The algorithm proposed here can generate beam profiles of plural collimation apertures and arbitrary initial beam spot distributions in a time-efficient method. It is an ideal tool to optimize collimator design for penumbra, dose rate, and field size. The intensity of the source beam spot is divided into slices, and each slice is projected onto the treatment plane at the isocenter through the collimator apertures.The illuminated field range and intensity are determined by geometry and the intensity of that slice of beam source, respectively.By integrating the projected intensity across all the slicesof the source profile, the profile on the treatment plane is obtained. The algorithm is used to generate beam profiles of a conical pencil beam collimator system and compare them to the Monte Carlo simulation as well as measurements. It canalso be used to demonstrate the impact of collimator shape on the beam penumbra, dose rate, and field size. The projection integration method provides a quick and informative tool for collimator design. The results were validated with the Monte Carlo simulation and measurements. This method was demonstrated to be effective for optimizing beam characteristics.

Mode-cleaning in antisymmetrically modulated non-Hermitian waveguides

Abstract We demonstrate all-optical spatial mode-cleaning in non-Hermitian waveguides. The effect is accounted by a unidirectional coupling among the modes resulting from a simultaneous modulation of the refractive index and the gain/loss along graded index multimodal waveguides. Depending on the spatial delay between the real and imaginary part of the potential modulation, higher or lower order modes are favored, which in latter case eventually leads to an nearly-monomode propagation. In this way, for any arbitrary initial field distribution an antisymmetric non-Hermitian modulation results in an effective mode-cleaning. The effect is demonstrated analytically, based on coupled mode theory in 1D waveguides, and numerically proven by solving the wave propagation equation with the antisymmetric non-Hermitian potential. The proposal is also generalized to the more involved case of 2D waveguides, leading to a significant reduction of the beam quality factor and improvement of beam spatial quality.

A Monte Carlo Method for Calculating Lynden-Bell Equilibrium in Self-Gravitating Systems.

We present a Monte Carlo approach that allows us to easily implement Lynden-Bell (LB) entropy maximization for an arbitrary initial particle distribution. The direct maximization of LB entropy for an arbitrary initial distribution requires an infinite number of Lagrange multipliers to account for the Casimir invariants. This has restricted studies of Lynden-Bell's violent relaxation theory to only a very small class of initial conditions of a very simple waterbag form, for which the entropy maximization can be performed numerically. In the present approach, an arbitrary initial distribution is discretized into density levels which are then evolved using an efficient Monte Carlo algorithm towards the final equilibrium state. A comparison is also made between the LB equilibrium and explicit Molecular Dynamics simulations. We find that for most initial distributions, relaxation is incomplete and the system is not able to reach the state of maximum LB entropy. In particular, we see that the tail of the stationary particle distribution is very different from the one predicted by the theory of violent relaxation, with a hard cutoff instead of an algebraic decay predicted by LB's theory.

Quantum Equilibrium in Stochastic de Broglie–Bohm–Bell Quantum Mechanics

This paper investigates dynamical relaxation to quantum equilibrium in the stochastic de Broglie–Bohm–Bell formulation of quantum mechanics. The time-dependent probability distributions are computed as in a Markov process with slowly varying transition matrices. Numerical simulations, supported by exact results for the large-time behavior of sequences of (slowly varying) transition matrices, confirm previous findings that indicate that de Broglie–Bohm–Bell dynamics allows an arbitrary initial probability distribution to relax to quantum equilibrium; i.e., there is no need to make the ad-hoc assumption that the initial distribution of particle locations has to be identical to the initial probability distribution prescribed by the system’s initial wave function. The results presented in this paper moreover suggest that the intrinsically stochastic nature of Bell’s formulation, which is arguable most naturally formulated on an underlying discrete space-time, is sufficient to ensure dynamical relaxation to quantum equilibrium for a large class of quantum systems without the need to introduce coarse-graining or any other modification in the formulation.

MODEL OF FORMATION AND PROPAGATION OF SLIP BANDS IN METALS

A theoretical study of the processes of localization of plastic deformation in metals has been carried out. Within the framework of the system of evolutionary equations for dislocation density, taking into account the multiplication and annihilation of dislocations, the possibility of a running solution for the slip strip is established. It is shown that the initial system has two equilibrium states. For the total dislocation density and dislocation charge normalized to a stationary homogeneous solution for dislocation density, these are the states (0,0) and (1,0) on the phase plane of the above variables in dimensionless form. From the analysis of singular points, it follows that the point (0,0) is a stable node, and the equilibrium state (1,0) is a saddle. In this case, the desired solution of the initial system of evolutionary equations is a separatrix going from point (0,0) to point (1,0), which corresponds to solutions in the form of a drop wave for the dislocation density forming the slip band and a momentum for the dislocation charge. It is shown that the dislocation charge propagates in the front of the slip band, which moves at a velocity u = kV (V is the drift velocity of dislocations due to external load, the proportionality coefficient k satisfies the condition 0 < k < 1). After analyzing the existence of self-similar solutions, it follows that non-homogeneous wave solutions occur only at A = t1/t2 > 1, where t1 and t2 are, respectively, relaxation times of the total dislocation density and dislocation charge to a homogeneous state. Estimates show that for a given dislocation kinetics (dislocation generation and annihilation processes) A > 1, i.e. satisfy the conditions for the formation of a slip strip of a given type. The stability of the obtained wave stationary self-similar solutions is considered. Assuming that deviations from stationary solutions for the dislocation density and dislocation charge are limited to a given domain, and considering the deviations small, we obtain the Sturm – Liouville problem for own functions and eigenvalues with zero boundary conditions at the boundary of this domain. The appropriate transformation reduces the problem to an equation of the Schrodinger equation type. It is shown that under certain conditions the spectrum of the Schrodinger operator is in the left half-plane, i.e. deviations of dislocation density and dislocation charge decay exponentially over time and the desired stationary solutions are asymptotically stable. The issue related to the determination of the steady-state velocity of wave propagation is considered. Linearization of the initial system of equations for dislocation density and dislocation charge allowed us to obtain a linear sine-Gordon equation, from the solution of which the steady-state wave velocity is determined. It is shown that for the initial system of the system of equations for A > 1, arbitrary initial distributions of the desired variables acquire the form of self-similar solutions over time and move at the lowest possible speed u = 2VA1/2/(1 + A).

Homogeneous generalized master equation for an arbitrary initial state of a many-particle classical system

A special projection operator is introduced, which exactly transforms the Nakajima–Zwanzig inhomogeneous Generalized Master Equation (GME) with an irrelevant initial condition term (a source) into a completely closed homogeneous GME (with no initial correlations term) for the relevant part of an N-particle distribution function FN(t) for an arbitrary initial distribution function FN(0) of a system of classical particles. It resolves the problem of the derivation of a closed linear evolution equation from the Liouville equation with no “molecular chaos” type approximation. The obtained equation is equivalent to the completely closed equation for s-particle (s≤N) distribution function Fs(t). The initial correlations are accounted for in this equation in the modified by the introduced projection operator kernel governing the evolution of Fs(t). This equation is further simplified by presenting its kernel in the linear in the particles’ density n approximation. In this approximation the equations for one-particle F1(t) and two-particle F2(t) distribution functions are derived. Special attention is given to analyzing the equation for F1(t) and showed that this equation in the Markov approximation can be written as the linear Boltzmann or Landau (for a small interparticle interaction) equations but with additional terms caused by initial correlations.

New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller–Robert–Sommeria theory

New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller–Robert–Sommeria’s (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.

Born’s rule in multiqubit Bohmian systems

We study the Bohmian trajectories of a 4-qubit system made of coherent states of the quantum harmonic oscillator. We find, as in the 2-qubit and 3-qubit cases, that all the chaotic Bohmian trajectories are also ergodic. However, the chaotic trajectories are effectively ordered for longer times than in the 2 and 3-qubit cases. Then we study the long time dynamical establishing of Born’s rule with arbitrary initial distributions. We find that the higher the dimensionality of the system, the larger the proportion of the chaotic trajectories within the Born distribution, for any non-zero value; additionally, the proportion becomes practically total for a larger range of entanglements than in the 2-qubit and 3-qubit cases. Thus we conclude that in multiqubit systems Born’s rule will be accessible by practically all initial distributions.

BTE-Barna: An extension of almaBTE for thermal simulation of devices based on 2D materials

We present BTE-Barna (Boltzmann Transport Equation - Beyond the RTA for NAnosystems), a software package that extends the Monte Carlo (MC) module of the almaBTE solver of the Peierls-Boltzmann transport equation for phonons (PBTE) to work with nanosystems based on 2D materials with complex geometries. To properly capture how the phonon occupations evolve in momentum space as a result of scattering, we have supplemented the relaxation-time approximation with an implementation of the propagator for the full linearized version of the PBTE. The code can now find solutions for finite and extended devices under the effect of a thermal gradient, with isothermal reservoirs or with an arbitrary initial temperature distribution in space and time, writing out the temperature and heat flux distributions as well as their spectral decompositions. Besides the full deviational MC solver, a number of useful approximations for highly symmetric devices are also included. Program summaryProgram Title::BTE-BarnaCPC Library link to program files:https://doi.org/10.17632/t58z289tpc.1Developer's repository link:https://github.com/sousaw/BTE-BarnaCode Ocean capsule:https://codeocean.com/capsule/4171806Licensing provisions: Apache-2.0Programming language: C++Nature of problem: Calculation of temperature profiles, thermal flux and/or effective thermal conductivities for nanosystems based on 2D materials.Solution method: For highly symmetric systems confined along some direction(s), the linearized phonon Boltzmann transport equation is solved iteratively by partially suppressing the phonon lifetimes due to boundaries. For more complex geometries, an energy-based deviational Monte Carlo method including off-diagonal terms in the linearized scattering operator arising from the phonon-phonon interaction is used.Additional comments including restrictions and unusual features: Depends on the following external libraries: boost, Eigen, HDF5, MessagePack, MPI, oneTBB, spglib, and almaBTE.

Constrained Discounted Stochastic Games

In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state–action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in a locally convex topological vector space. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on an approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using the so-called weighted norm approach.

Collective guiding of acoustically propelled nano- and microparticles.

One of the most important potential applications of motile nano- and microdevices is targeted drug delivery. To realize this, biocompatible particles that can be guided collectively towards a target inside a patient's body are required. Acoustically propelled nano- and microparticles constitute a promising candidate for such biocompatible, artificial motile particles. The main remaining obstacle to targeted drug delivery by motile nano- and microdevices is to also achieve a reliable and biocompatible method for guiding them collectively to their target. Here, we propose such a method. As we confirm by computer simulations, it allows for the remote guiding of large numbers of acoustically propelled particles to a prescribed target by combining a space- and time-dependent acoustic field and a time-dependent magnetic field. The method works without detailed knowledge about the particle positions and for arbitrary initial particle distributions. With these features, it paves the way for the future application of motile particles as vehicles for targeted drug delivery in nanomedicine.

Strong-field control by reverse engineering

Based on the idea of reverse engineering, we design an optimal laser pulse to control strong-field multiphoton atomic transitions. Starting from the time-dependent Schr\"odinger equation of the full system, we adiabatically eliminate the nonessential states and apply the rotating-wave approximation to arrive at an effective two-state representation that involves dynamic Stark shifts and multiphoton coupling. Solving this equation inversely for the field, we obtain an analytical laser pulse shape that is expected to induce the full system's evolution according to user-defined quantum pathways. In our procedure, the amplitude and phase of the laser pulse are engineered such that the dynamically shifted electronic states are resonantly coupled during the action of the pulse at each moment of time. As a result, the driven system evolves from an arbitrary initial population distribution to any desired final quantum state superposition at a predefined rate. The proposed scheme is demonstrated using the example of the $3s\ensuremath{\rightarrow}4s$ two-photon transition of atomic sodium. By solving the time-dependent Schr\"odinger equation of the single-active electron with two different methods, either propagating time-dependent coefficients of many field-free states or directly propagating the three-dimensional electronic wave packet on a grid, we demonstrate the robustness as well as the limitations of the presented reverse engineering scheme.

Kinetics of nonisothermal phase change with arbitrary temperature-time history and initial transformed phase distributions.

This paper describes the extension of the classic Avrami equation to nonisothermal systems with arbitrary temperature-time history and arbitrary initial distributions of transformed phase. We start by showing that through examination of phase change in Fourier space, we can decouple the nucleation rate, growth rate, and transformed fraction, leading to the derivation of a nonlinear differential equation relating these three properties. We then consider a population balance partial differential equation (PDE) on the phase size distribution and solve it analytically. Then, by relating this PDE solution to the transformed fraction of phase, we are able to derive initial conditions to the differential equation relating nucleation rate, growth rate, and transformed fraction.

Simulated Annealing Applied to HIO Method for Phase Retrieval

Phase retrieval utilizing Fourier amplitudes plays a significant role in image recovery. Iterative phase retrieval algorithms have been developed to retrieve phase information that cannot be recorded by detectors directly. However, iterative algorithms face the problem of being trapped in local minima due to the nonconvexity of phase retrieval, and most existing works addressed this by optimizing in multiple runs parallelly to improve the possibility that one of these could reach the global minimum. Alternatively, we propose in this work to increase the probability of reaching the global minimum with one arbitrary initial distribution by adapting simulated annealing in the standard hybrid input-output (HIO) algorithm. Numerical and experimental results demonstrate that the proposed method reconstructs images with mean square errors 50.12% smaller than those reconstructed by HIO. More importantly, the proposed method can be applied to any HIO-based algorithm with multiple runs to further improve the performance.

Ergodicity and Born's rule in an entangled three-qubit Bohmian system.

We study in detail the interplay between chaos and entanglement in the Bohmian trajectories of three entangled qubits, made of coherent states of the quantum harmonic oscillator. We find that all the three-dimensional (3D) chaotic trajectories are ergodic; namely, they have a common long time distribution of points regardless of the initial conditions, and for any nonzero entanglement, their number is much larger than in the corresponding two-qubit system. Furthermore, the range of entanglements for which practically all the trajectories are chaotic and ergodic is much larger than in the two-qubit case. Thus, as the dimensionality of the system increases, Born's rule becomes accessible to a wider range of arbitrary initial distributions than in the 2D case. Our numerical results lead to the conjecture that, for multiqubit systems, Born's rule is the limit of almost all initial distributions of particles.

Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions

We study Bessel and Dunkl processes (Xt,k)t≥0 on RN with possibly multivariate coupling constants k≥0. These processes describe interacting particle systems of Calogero–Moser–Sutherland type with N particles. For the root systems AN−1 and BN these Bessel processes are related with β-Hermite and β-Laguerre ensembles. Moreover, for the frozen case k=∞, these processes degenerate to deterministic or pure jump processes.We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for N→∞ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type B new non-symmetric semicircle-type limit distributions on R appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko–Pastur limit laws for the empirical measures of the zeros of Hermite and Laguerre polynomials respectively.

On existence and uniqueness of solutions to a pantograph type equation

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution. doi:10.1017/S144618112100002X

Irreversibility in Active Matter: General Framework for Active Ornstein-Uhlenbeck Particles

Active matter systems are driven out of equilibrium by conversion of energy into directed motion locally on the level of the individual constituents. In the spirit of a minimal description, active matter is often modeled by so-called active Ornstein-Uhlenbeck particles an extension of passive Brownian motion where activity is represented by an additional fluctuating non-equilibrium “force” with simple statistical properties (Ornstein-Uhlenbeck process). While in passive Brownian motion, entropy production along trajectories is well-known to relate to irreversibility in terms of the log-ratio of probabilities to observe a certain particle trajectory forward in time in comparison to observing its time-reversed twin trajectory, the connection between these concepts for active matter is less clear. It is therefore of central importance to provide explicit expressions for the irreversibility of active particle trajectories based on measurable quantities alone, such as the particle positions. In this technical note we derive a general expression for the irreversibility of AOUPs in terms of path probability ratios (forward vs. backward path), extending recent results from [PRX 9, 021009 (2019)] by allowing for arbitrary initial particle distributions and states of the active driving.

On typical encodings of multivariate ergodic sources

We show that the typical coordinate-wise encoding of multivariate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular polymatroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymptotically mean stationary processes with ergodic mean, ergodic processes, irreducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the output variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution.

Mean field approach to stochastic control with partial information

In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussanet al., Mean Field Games and Mean Field Type Control Theory.Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandiniet al.[Stochastic Process. Appl.129(2019) 674–711], which is fundamentally different from our present proposed framework.