Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method

2016 ◽  
Vol 314 ◽  
pp. 661-681 ◽  
Author(s):  
Claudio Bierig ◽  
Alexey Chernov
Keyword(s):  
Monte Carlo ◽  
Probability Density ◽  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Probability Density Functions ◽  
Density Functions ◽  
Multilevel Monte Carlo

Maximum Entropy Methods in the Mechanics of Quasi-Static Deformation of Granular Materials

2002 ◽  
Author(s):  
N. P. Kruyt ◽  
L. Rothenburg
Keyword(s):  
Probability Density ◽  
Maximum Entropy ◽  
Granular Materials ◽  
Probability Density Functions ◽  
Contact Forces ◽  
Static Deformation ◽  
Density Functions ◽  
Two Dimensional ◽  
Entropy Methods ◽  
Theoretical Predictions

In statistical physics of dilute gases maximum entropy methods are widely used for theoretical predictions of macroscopic quantities in terms of microscopic quantities. In this study an analogous approach to the mechanics of quasi-static deformation of granular materials is proposed. The reasoning is presented that leads to the definition of an entropy that is appropriate to quasi-static deformation of granular materials. This entropy is formulated in terms of contact quantities, since contacts constitute the relevant microscopic level for granular materials that consist of semirigid particles. The proposed maximum entropy approach is then applied to two cases. The first case deals with the probability density functions of contact forces in a two-dimensional assembly with frictional contacts under prescribed hydrostatic stress. The second case deals with the elastic behaviour of two-dimensional assemblies of non-rotating particles with bonded contacts. For both cases the probability density functions of contact forces are determined from the proposed maximum entropy method, under the constraints appropriate to the case. These constraints form the macroscopic information available about the system. With the probability density functions for contact forces thus determined, theoretical predictions of macroscopic quantities can be made. These theoretical predictions are then compared with results obtained from two-dimensional Discrete Element simulations and from experiments.

CalcOPP: a program for the calculation of one-particle potentials (OPPs)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dennis Wiedemann
Keyword(s):  
Maximum Entropy Method ◽  
Three Dimensional ◽  
Entropy Method ◽  
Probability Density Functions ◽  
Free Form ◽  
Density Functions ◽  
Command Line Interface ◽  
Ion Migration ◽  
Density Maps ◽  
Microsoft Windows

Abstract In recent years, one-particle potentials (OPPs) derived from neutron-diffraction data have become a popular means to estimate activation energies of ion migration in solids. Computer programs for their calculation, however, have mostly been private in-house solutions. The software CalcOPP presented herein permits calculating two- or three-dimensional OPPs either from probability density functions put out by the crystallographic suite Jana2006/ Jana2020 (including error maps) or from scattering-density maps reconstructed using the maximum entropy method (MEM) implementation Dysnomia. The title program is open-source, written in modern free-form Fortran and Python 3, and available free of charge under the permissive MIT License. Executables are published for 64-bit Microsoft Windows and Linux platforms and can be controlled via an intuitive graphical user interface or via command-line interface. Depending on the kind of input, CalcOPP’s output is readily visualized with standard crystallographic software or plotting applications. The release of the program not only makes the rather powerful OPP method more transparent, but it also opens it up to a broader, less programming-oriented public.

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