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# Random walks on jammed networks: Spectral properties

## Abstract

Using random walk analyses we explore diffusive transport on networks obtained from contacts between isotropically compressed, monodisperse, frictionless sphere packings generated over a range of pressures in the vicinity of the jamming transition p → 0 . For conductive particles in an insulating medium, conduction is determined by the particle contact network with nodes representing particle centers and edges contacts between particles. The transition rate is not homogeneous, but is distributed inhomogeneously due to the randomness of packing and concomitant disorder of the contact network, e.g., the distribution of the coordination number. A narrow escape time scale is used to write a Markov process for random walks on the particle contact network. This stochastic process is analyzed in terms of spectral density of the random, sparse, Euclidean and real, symmetric, positive, semidefinite transition rate matrix. Findings reveal network structures derived from jammed particles have properties similar to ordered, euclidean lattices but also some unique properties that distinguish them from other structures that are in some sense more homogeneous. Specifically, the distribution of eigenvalues of the transition rate matrix follow a power law with spectral dimension 3. Yet, quantitative details of the statistics of the eigenvectors show subtle differences with homogeneousmore »

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Central New Mexico Community College, Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Scientific User Facilities Division

- OSTI Identifier:
- 1559545

- Alternate Identifier(s):
- OSTI ID: 1542529

- Report Number(s):
- SAND-2018-9709J

Journal ID: ISSN 2470-0045; PLEEE8; 667602

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 1; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING

### Citation Formats

```
Lechman, Jeremy B., Bond, Stephen D., Bolintineanu, Dan S., Grest, Gary S., Yarrington, Cole D., and Silbert, Leonardo E. Random walks on jammed networks: Spectral properties. United States: N. p., 2019.
Web. doi:10.1103/PhysRevE.100.012905.
```

```
Lechman, Jeremy B., Bond, Stephen D., Bolintineanu, Dan S., Grest, Gary S., Yarrington, Cole D., & Silbert, Leonardo E. Random walks on jammed networks: Spectral properties. United States. doi:10.1103/PhysRevE.100.012905.
```

```
Lechman, Jeremy B., Bond, Stephen D., Bolintineanu, Dan S., Grest, Gary S., Yarrington, Cole D., and Silbert, Leonardo E. Mon .
"Random walks on jammed networks: Spectral properties". United States. doi:10.1103/PhysRevE.100.012905.
```

```
@article{osti_1559545,
```

title = {Random walks on jammed networks: Spectral properties},

author = {Lechman, Jeremy B. and Bond, Stephen D. and Bolintineanu, Dan S. and Grest, Gary S. and Yarrington, Cole D. and Silbert, Leonardo E.},

abstractNote = {Using random walk analyses we explore diffusive transport on networks obtained from contacts between isotropically compressed, monodisperse, frictionless sphere packings generated over a range of pressures in the vicinity of the jamming transition p → 0 . For conductive particles in an insulating medium, conduction is determined by the particle contact network with nodes representing particle centers and edges contacts between particles. The transition rate is not homogeneous, but is distributed inhomogeneously due to the randomness of packing and concomitant disorder of the contact network, e.g., the distribution of the coordination number. A narrow escape time scale is used to write a Markov process for random walks on the particle contact network. This stochastic process is analyzed in terms of spectral density of the random, sparse, Euclidean and real, symmetric, positive, semidefinite transition rate matrix. Findings reveal network structures derived from jammed particles have properties similar to ordered, euclidean lattices but also some unique properties that distinguish them from other structures that are in some sense more homogeneous. Specifically, the distribution of eigenvalues of the transition rate matrix follow a power law with spectral dimension 3. Yet, quantitative details of the statistics of the eigenvectors show subtle differences with homogeneous lattices and allow us to distinguish between topological and geometric sources of disorder in the network.},

doi = {10.1103/PhysRevE.100.012905},

journal = {Physical Review E},

number = 1,

volume = 100,

place = {United States},

year = {2019},

month = {7}

}

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