ARTICULOS

Wang-Landau algorithm for entropic sampling of arch-based microstates in the volume ensemble of static granular packings

 

D. Slobinsky,^1,2* Luis A. Pugnaloni^1,2†

*E-mail: dslobinsky@frlp.utn.edu.ar
^†E-mail: liiis.pugnaloni@frlp.utn.edu.ar
Departamento de Ingeniería Mecánica, Facultad Regional La Plata, Universidad Tecnológica Nacional, Av. 60 Esq. 124, 1900 La Plata, Argentina.
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina.

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Received: 19 January 2015, Accepted: 25 February 2015
Edited by: C. S. O'Hern
Reviewed by: M. Pica Ciamarra, Nanyang Technological University, Singapore.
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070001

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We implement the Wang–Landau algorithm to sample with equal probabilities the static configurations of a model granular system. The “non-interacting rigid arch model” used is based on the description of static configurations by means of splitting the assembly of grains into sets of stable arches. This technique allows us to build the entropy as a function of the volume of the packing for large systems. We make a special note of the details that have to be considered when defining the microstates and proposing the moves for the correct sampling in these unusual models. We compare our results with previous exact calculations of the model made at moderate system sizes. The technique opens a new opportunity to calculate the entropy of more complex granular models.

 

I. Introduction

In the study of static packings there exists still a lack of predictive capabilities of the available theories. Assemblies of objects that pack (such as grains) can generally sample such packed configurations only by the external excitation of the system. These packings can be built by repeating a given packing protocol (e.g., homogeneous compression or deposition under an external field against a confining boundary) on an initial random configuration. Also, a Markovian or non-Markovian series can be constructed by exciting the system from the previous packing configuration. To what extent the series of packings obtained (using either type of protocol) can be modeled without information on the dynamics that drives the system to the packed configuration is still uncertain. The main reason for this is that the few statistical approaches that attempt to do this are strongly hinder by the poor current ability to generate such packed structures without using a dynamics to build the packings.

One might expect that the packing fraction and its fluctuations, among other properties, could be obtained from basic statistics without resourcing to a full molecular dynamic type of simulations (also known as “discrete element method”, DEM). Although these types of simulations are powerful enough to predict the behaviour of most systems that pack, it is desirable to find a description that could neglect the detailed dynamics between consecutive packed configurations.

In the present work, we will calculate the entropy of the NIRA model in the microcanonical ensemble [7] using entropic sampling through the Wang– Landau (WL) algorithm [8–11]. This approach allows us to obtain the entire entropy function for all possible volumes of the system in one single simulation for larger systems and potentially for more complex models. All derived properties, such as compactivity and volume fluctuations, can then be calculated through numerical differentiation. We pay particular attention to the different descriptions that can be realized for the NIRA model. Some of these representations do not provide a difrect way of sampling the configuration space uniformly.

This work is organized as follows: in section ., we will review the WL algorithm. In section III., we will review the NIRA model and discuss different ways of representing it, along with the issues related to uniform sampling of the configurations. We then present a representation that allows very fast calculations of the entropy and we compare the results with the exact counting of all configurations for systems of moderate size. Finally, we discuss future directions to refine the arch-based ensemble volume function towards capturing detailed features of more realistic systems.

I Propose a new configuration and calculate its energy E₁. The new configuration is generally derived from the previous configuration by a change in the value of one of its DF.

Where N is the total number of disks of diameter d in the system.

It is important to mention that, typically, the máximum size that an arch can take is physically bounded (e.g., due to the size of the container that holds the granular sample). Henee, we will put a cutoff C to the largest arch allowed in the system.

Figure 1: (a) Entropy as a function of volume for the NIRA model in 2D calculated by counting all possible configurations for 500 disks [6] using different arch size cutoffs. (b) The corresponding compactivity calculated by numerical differentiation.

The configurations of the NIRA model are compatible with different representations. In the following subsections we will discuss some of these representations and their suitability for the implementation of the WL algorithm.

Figure 2 shows the entropy for the NIRA model using representation (7) for dierent maximum arch sizes C compared with the exact result obtained by counting all configurations and all permutations [6]. As we can see, not including all distinguishable permutations and including the new "fractional arches" gives a wrong entropy function.

Figure 2: Entropy as a function of volume for the NIRA model calculated using the WL algorithm (symbols) for representation (7) for 4 C 8. The full lines represent the exact results for 200 grains.

The arch listing representation

Figure 3: Entropy as a function of volume for the NIRA model with cutoff 4 = C = 8 (symbols as in Fig. 2) using the binary arch representation to carry out the entropic sampling through the WL algorithm for 200 grains. The solid lines correspond to the exact counting of microstates from Ref. [6].

IV. Conclusions

We have been able to compute the entropy of a system of non-interacting rigid arches using a WL algorithm in the volume ensemble in different representations.

We have exposed the dificulties in dealing with different representations of the configurations of arches and the mechanisms used to propose trial moves for the WL algorithm. These difficulties appear during the choice of a simple sampling scheme that ensure a symmetric selection probability of configurations. Additionally, the degeneracy due to distinguishable permutations of arches pose a further complication in the use of WL. The most suitable representation that we found for a non-interacting system of rigid arches resulted in a binary vector.

We believe that entropic sampling of arches through the WL algorithm has a great potential for testing the granular statistical mechanics hypothesis (such as equiprobability and ergodicity). Having a sampling algorithm like WL adapted for these types of models is crucial to continue the road map towards the refinement of an archbased framework for static granular packs. In particular, the non-interacting condition is clearly a crude approximation and should be lifted, along with the introduction of a more accurate volume function.

 

1 One is tempted to add to the degeneracy factor (5) to correct the entropy in step III of the algorithm. However, the algorithm compensates this factor in order to obtain a fíat histogram. Therefore this is not a viable solution.

 

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