# Entropic and gradient flow formulations for nonlinear diffusion

## Abstract

Nonlinear diffusion ∂{sub t}ρ = Δ(Φ(ρ)) is considered for a class of nonlinearities Φ. It is shown that for suitable choices of Φ, an associated Lyapunov functional can be interpreted as thermodynamic entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role.

- Authors:

- School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG (United Kingdom)
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22596517

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 8; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFUSION; ENTROPY; HYDRODYNAMICS; METRICS; NONLINEAR PROBLEMS

### Citation Formats

```
Dirr, Nicolas, E-mail: DirrNP@cardiff.ac.uk, Stamatakis, Marios, E-mail: M.G.Stamatakis@bath.ac.uk, and Zimmer, Johannes, E-mail: zimmer@maths.bath.ac.uk.
```*Entropic and gradient flow formulations for nonlinear diffusion*. United States: N. p., 2016.
Web. doi:10.1063/1.4960748.

```
Dirr, Nicolas, E-mail: DirrNP@cardiff.ac.uk, Stamatakis, Marios, E-mail: M.G.Stamatakis@bath.ac.uk, & Zimmer, Johannes, E-mail: zimmer@maths.bath.ac.uk.
```*Entropic and gradient flow formulations for nonlinear diffusion*. United States. doi:10.1063/1.4960748.

```
Dirr, Nicolas, E-mail: DirrNP@cardiff.ac.uk, Stamatakis, Marios, E-mail: M.G.Stamatakis@bath.ac.uk, and Zimmer, Johannes, E-mail: zimmer@maths.bath.ac.uk. Mon .
"Entropic and gradient flow formulations for nonlinear diffusion". United States.
doi:10.1063/1.4960748.
```

```
@article{osti_22596517,
```

title = {Entropic and gradient flow formulations for nonlinear diffusion},

author = {Dirr, Nicolas, E-mail: DirrNP@cardiff.ac.uk and Stamatakis, Marios, E-mail: M.G.Stamatakis@bath.ac.uk and Zimmer, Johannes, E-mail: zimmer@maths.bath.ac.uk},

abstractNote = {Nonlinear diffusion ∂{sub t}ρ = Δ(Φ(ρ)) is considered for a class of nonlinearities Φ. It is shown that for suitable choices of Φ, an associated Lyapunov functional can be interpreted as thermodynamic entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role.},

doi = {10.1063/1.4960748},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 57,

place = {United States},

year = {Mon Aug 15 00:00:00 EDT 2016},

month = {Mon Aug 15 00:00:00 EDT 2016}

}

DOI: 10.1063/1.4960748

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