Weak Galilean invariance as a selection principle for coarse-grained diffusive models

How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac–Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call “weak Galilean invariance.” Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.

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Mesoscale methods

10.1093/oso/9780198803195.003.0012 ◽

2017 ◽

Author(s):

Michael P. Allen ◽

Dominic J. Tildesley

Keyword(s):

Large Scale ◽

Equations Of Motion ◽

Dissipative Particle Dynamics ◽

Coarse Graining ◽

Coarse Grained ◽

Collision Dynamics ◽

Force Matching ◽

Multiparticle Collision Dynamics ◽

Long Time ◽

Boltzmann Method

Coarse-graining is an increasingly commonplace approach to study, as economically as possible, large-scale, and long-time phenomena. This chapter covers the main methods. Brownian and Langevin dynamics are introduced, with practical details of the solution of the modified equations of motion. Several techniques which aim to bridge the gap to the hydrodynamic regime are described: these include dissipative particle dynamics, multiparticle collision dynamics, and the lattice Boltzmann method. Several examples of program code are provided. In the last part of the chapter, the derivation of a coarse-grained potential from an atomistic one is considered using force-matching and structure-matching, and the limitations of these approaches are discussed.

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The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

Reports in Mechanical Engineering ◽

10.31181/rme200101093s ◽

2020 ◽

Vol 1 (1) ◽

pp. 93-102

Author(s):

Carsten Strzalka ◽

◽

Manfred Zehn ◽

Keyword(s):

Finite Element ◽

Degrees Of Freedom ◽

Equations Of Motion ◽

A Priori ◽

High Stress ◽

Von Mises Stress ◽

Detailed Knowledge ◽

Variable Loading ◽

Numerical Effort ◽

Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

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On the three-dimensional spatial correlations of curved dislocation systems

Materials Theory ◽

10.1186/s41313-020-00026-w ◽

2021 ◽

Vol 5 (1) ◽

Author(s):

Joseph Pierre Anderson ◽

Anter El-Azab

Keyword(s):

Correlation Function ◽

Slip System ◽

Correlation Functions ◽

Mean Field ◽

Coarse Graining ◽

Dislocation Dynamics ◽

Coarse Grained ◽

Spatial Correlation Function ◽

Line System ◽

Dislocation Densities

AbstractCoarse-grained descriptions of dislocation motion in crystalline metals inherently represent a loss of information regarding dislocation-dislocation interactions. In the present work, we consider a coarse-graining framework capable of re-capturing these interactions by means of the dislocation-dislocation correlation functions. The framework depends on a convolution length to define slip-system-specific dislocation densities. Following a statistical definition of this coarse-graining process, we define a spatial correlation function which will allow the arrangement of the discrete line system at two points—and thus the strength of their interactions at short range—to be recaptured into a mean field description of dislocation dynamics. Through a statistical homogeneity argument, we present a method of evaluating this correlation function from discrete dislocation dynamics simulations. Finally, results of this evaluation are shown in the form of the correlation of dislocation densities on the same slip-system. These correlation functions are seen to depend weakly on plastic strain, and in turn, the dislocation density, but are seen to depend strongly on the convolution length. Implications of these correlation functions in regard to continuum dislocation dynamics as well as future directions of investigation are also discussed.

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THE SIMULATION OF POLYSTYRENE/NANOPARTICLES COMPOSITE MICROSPHERES USING DISSIPATIVE PARTICLE DYNAMICS

Journal of Theoretical and Computational Chemistry ◽

10.1142/s0219633612501118 ◽

2013 ◽

Vol 12 (02) ◽

pp. 1250111 ◽

Cited By ~ 4

Author(s):

HAILONG XU ◽

QIUYU ZHANG ◽

HEPENG ZHANG ◽

BAOLIANG ZHANG ◽

CHANGJIE YIN

Keyword(s):

Dissipative Particle Dynamics ◽

Sodium Dodecyl ◽

Coarse Graining ◽

Particle Dynamics ◽

Coarse Grained ◽

Polystyrene Nanoparticles ◽

Coarse Grained Model ◽

Composite Microspheres ◽

Polystyrene Matrix ◽

Materials Studio

Dissipative particle dynamics (DPD) was initially used to simulate the polystyrene/nanoparticle composite microspheres (PNCM) in this paper. The coarse graining model of PNCM was established. And the DPD parameterization of the model was represented in detail. The DPD repulsion parameters were calculated from the cohesive energy density which could be calculated by amorphous modules in Materials Studio. The equilibrium configuration of the simulated PNCM shows that the nanoparticles were actually "modified" with oleic acid and the modified nanoparticles were embedded in the bulk of polystyrene. As sodium dodecyl sulfate (SDS) was located in the interface between water and polystyrene, the hydrophilic head of SDS stretched into water while the hydrophobic tailed into polystyrene. All simulated phenomena were consistent with the experimental results in preparation of polystyrene/nanoparticles composite microspheres. The effect of surface modification of nanoparticles on its dispersion in polystyrene matrix was also studied by adjusting the interaction parameters between the OA and NP beads. The final results indicated that the nanoparticles removed from the core of composite microsphere to the surface with increase of a OA-NP . All the simulated results demonstrated that our coarse–grained model was reasonable.

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Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

Journal of Navigation ◽

10.1017/s0373463311000415 ◽

2011 ◽

Vol 64 (S1) ◽

pp. S3-S18 ◽

Cited By ~ 20

Author(s):

Yuanxi Yang ◽

Jinlong Li ◽

Junyi Xu ◽

Jing Tang

Keyword(s):

Least Squares ◽

Stochastic Models ◽

Influence Function ◽

A Priori ◽

Functional Model ◽

Global Navigation Satellite Systems ◽

Parameter Estimates ◽

Model Parameters ◽

Integrated Navigation ◽

Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

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A new spectral coarse-graining algorithm based on K-means clustering in complex networks

Modern Physics Letters B ◽

10.1142/s0217984918504213 ◽

2019 ◽

Vol 33 (01) ◽

pp. 1850421 ◽

Cited By ~ 1

Author(s):

Lang Zeng ◽

Zhen Jia ◽

Yingying Wang

Keyword(s):

Complex Networks ◽

Large Scale ◽

Dynamic Properties ◽

Coarse Graining ◽

Kuramoto Model ◽

Coarse Grained ◽

Original Network ◽

Topological Information ◽

Real World Applications ◽

Large Scale Networks

Coarse-graining of complex networks is one of the important algorithms to study large-scale networks, which is committed to reducing the size of networks while preserving some topological information or dynamic properties of the original networks. Spectral coarse-graining (SCG) is one of the typical coarse-graining algorithms, which can keep the synchronization ability of the original network well. However, the calculation of SCG is large, which limits its real-world applications. And it is difficult to accurately control the scale of the coarse-grained network. In this paper, a new SCG algorithm based on K-means clustering (KCSCG) is proposed, which cannot only reduce the amount of calculation, but also accurately control the size of coarse-grained network. At the same time, KCSCG algorithm has better effect in keeping the network synchronization ability than SCG algorithm. A large number of numerical simulations and Kuramoto-model example on several typical networks verify the feasibility and effectiveness of the proposed algorithm.

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Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

Quantum Information and Computation ◽

10.26421/qic13.5-6-3 ◽

2013 ◽

Vol 13 (5&6) ◽

pp. 393-429

Author(s):

Matthew Hastings

Keyword(s):

Ground State ◽

Coding Theory ◽

Energy State ◽

Ground States ◽

Coarse Graining ◽

Quantum Circuit ◽

Technical Condition ◽

Two Dimensions ◽

Low Energy ◽

Coarse Grained

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

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A scaled MP-PIC method for bubbling fluidization

10.22541/au.163253496.64565288/v1 ◽

2021 ◽

Author(s):

Xing Zhao ◽

Yong Jiang ◽

Fei Li ◽

Wei Wang

Keyword(s):

Fluidized Beds ◽

Volume Fraction ◽

Phase Particle ◽

Scaling Law ◽

Coarse Graining ◽

Coarse Grained ◽

Particle In Cell ◽

Wide Range ◽

Pic Method ◽

Multi Phase

Coarse-grained methods have been widely used in simulations of gas-solid fluidization. However, as a key parameter, the coarse-graining ratio, and its relevant scaling law is still far from reaching a consensus. In this work, a scaling law is developed based on a similarity analysis, and then it is used to scale the multi-phase particle-in-cell (MP-PIC) method, and validated in the simulation of two bubbling fluidized beds. The simulation result shows this scaled MP-PIC can reduce the errors of solids volume fraction and velocity distributions over a wide range of coarse-graining ratios. In future, we expect that a scaling law with consideration of the heterogeneity inside a parcel or numerical particle will further improve the performance of coarse-grained modeling in simulation of fluidized beds.

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Coarse-graining for fast dynamics of order parameters in the phase-field model

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2017.0203 ◽

2018 ◽

Vol 376 (2113) ◽

pp. 20170203 ◽

Cited By ~ 4

Author(s):

D. Jou ◽

P. K. Galenko

Keyword(s):

Master Equation ◽

Thermodynamic Equilibrium ◽

Phase Field ◽

Order Parameter ◽

Field Model ◽

Local Thermodynamic Equilibrium ◽

Phase Field Model ◽

Coarse Graining ◽

Order Parameters ◽

Coarse Grained

In standard descriptions, the master equation can be obtained by coarse-graining with the application of the hypothesis of full local thermalization that is equivalent to the local thermodynamic equilibrium. By contrast, fast transformations proceed in the absence of local equilibrium and the master equation must be obtained with the absence of thermalization. In the present work, a non-Markovian master equation leading, in specific cases of relaxation to local thermodynamic equilibrium, to hyperbolic evolution equations for a binary alloy, is derived for a system with two order parameters. One of them is a conserved order parameter related to the atomistic composition, and the other one is a non-conserved order parameter, which is related to phase field. A microscopic basis for phenomenological phase-field models of fast phase transitions, when the transition is so fast that there is not sufficient time to achieve local thermalization between two successive elementary processes in the system, is provided. In a particular case, when the relaxation to local thermalization proceeds by the exponential law, the obtained coarse-grained equations are related to the hyperbolic phase-field model. The solution of the model equations is obtained to demonstrate non-equilibrium phenomenon of solute trapping which appears in rapid growth of dendritic crystals. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

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Accurate ephemeris reconstruction for comet 67P/Churyumov-Gerasimenko from Rosetta data analysis

10.5194/egusphere-egu21-14765 ◽

2021 ◽

Author(s):

Riccardo Lasagni Manghi ◽

Marco Zannoni ◽

Paolo Tortora ◽

Michael Küppers ◽

Laurence O'Rourke ◽

...

Keyword(s):

Stochastic Models ◽

Orbit Determination ◽

Piecewise Linear ◽

A Priori ◽

Physical Models ◽

Unbiased Estimation ◽

Comet Nucleus ◽

Optical Images ◽

Relative Orbit ◽

Comet 67P

Following its arrival at 67P/Churyumov-Gerasimenko in August 2014, the Rosetta spacecraft successfully navigated in proximity of the comet for two years, using a combination of radiometric measurements and optical images collected by the onboard navigation cameras.The reconstructed spacecraft and comet trajectories were obtained combining several long-arc and short-arc orbit determination solutions generated by ESOC Flight Dynamics during the Rosetta operations. Several discontinuities are present within these trajectories, due to the lack of a dynamical model for the representation of the comet Non-Gravitational Accelerations (NGA).The work presented in this study represents an effort to produce an accurate and continuous ephemeris reconstruction for comet 67P/Churyumov-Gerasimenko for the period between July 2014 and October 2016, through a complete reanalysis of the Range and &#916;DOR measurements collected by Rosetta during its proximity phase with the comet.Using as input the reconstructed relative orbit of Rosetta, the radiometric observables were mapped to the comet nucleus and used to estimate the comet state and some key physical and observational parameters within a Square Root Information batch filter implemented in MONTE, most notably the NGA acting on the comet nucleus due to surface outgassing.Several orbit determination solutions were generated by varying the model used to represent the NGA. More specifically, empirical and stochastic models were compared by evaluating the reduced &#967;2&#160;statistics of the measurement residuals to identify the most suitable trajectory estimations for each of the proposed models. From this narrow list of solutions, a preliminary selection for the final ephemeris reconstruction is proposed, based on its adherence to the original ESOC trajectory and on the consistency of the formal state uncertainties with the estimated solutions.It will be shown that the selected ephemeris solution, using a piecewise linear stochastic NGA model with intervals between 3 and 4 weeks, produces a continuous ephemeris reconstruction for 67P/Churyumov-Gerasimenko with maximum formal uncertainties around perihelion of &#963;pos &#8773; [20 km, 30 km, 200 km] in the Radial-Tangential-Normal reference frame. The advantage of using simple stochastic models, with limited a-priori assumptions on the involved physical processes, is that they allow to produce an unbiased estimation of the NGA variations around perihelion, which represent a valuable input for further investigations involving detailed physical models of the cometary activity.

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