# On extreme points of the diffusion polytope

## Abstract

Here, we consider a class of diffusion problems defined on simple graphs in which the populations at any two vertices may be averaged if they are connected by an edge. The diffusion polytope is the convex hull of the set of population vectors attainable using finite sequences of these operations. A number of physical problems have linear programming solutions taking the diffusion polytope as the feasible region, e.g. the free energy that can be removed from plasma using waves, so there is a need to describe and enumerate its extreme points. We also review known results for the case of the complete graph Kn, and study a variety of problems for the path graph Pn and the cyclic graph Cn. Finall, we describe the different kinds of extreme points that arise, and identify the diffusion polytope in a number of simple cases. In the case of increasing initial populations on Pn the diffusion polytope is topologically an n-dimensional hypercube.

- Authors:

- Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences
- Bar-IIan Univ., Ramat Gan (Israel). Dept. of Mathematics
- Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences; Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24); USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1350094

- Grant/Contract Number:
- AC02-09CH11466; 274-FG52-08NA28553

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physica. A

- Additional Journal Information:
- Journal Volume: 473; Journal Issue: C; Journal ID: ISSN 0378-4371

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Combinatorics; Optimization; Diffusion; Plasma; Networks; Algebra

### Citation Formats

```
Hay, M. J., Schiff, J., and Fisch, N. J..
```*On extreme points of the diffusion polytope*. United States: N. p., 2017.
Web. doi:10.1016/j.physa.2017.01.038.

```
Hay, M. J., Schiff, J., & Fisch, N. J..
```*On extreme points of the diffusion polytope*. United States. doi:10.1016/j.physa.2017.01.038.

```
Hay, M. J., Schiff, J., and Fisch, N. J.. Wed .
"On extreme points of the diffusion polytope". United States.
doi:10.1016/j.physa.2017.01.038. https://www.osti.gov/servlets/purl/1350094.
```

```
@article{osti_1350094,
```

title = {On extreme points of the diffusion polytope},

author = {Hay, M. J. and Schiff, J. and Fisch, N. J.},

abstractNote = {Here, we consider a class of diffusion problems defined on simple graphs in which the populations at any two vertices may be averaged if they are connected by an edge. The diffusion polytope is the convex hull of the set of population vectors attainable using finite sequences of these operations. A number of physical problems have linear programming solutions taking the diffusion polytope as the feasible region, e.g. the free energy that can be removed from plasma using waves, so there is a need to describe and enumerate its extreme points. We also review known results for the case of the complete graph Kn, and study a variety of problems for the path graph Pn and the cyclic graph Cn. Finall, we describe the different kinds of extreme points that arise, and identify the diffusion polytope in a number of simple cases. In the case of increasing initial populations on Pn the diffusion polytope is topologically an n-dimensional hypercube.},

doi = {10.1016/j.physa.2017.01.038},

journal = {Physica. A},

number = C,

volume = 473,

place = {United States},

year = {Wed Jan 04 00:00:00 EST 2017},

month = {Wed Jan 04 00:00:00 EST 2017}

}