# Nonequilibrium Entropy in a Shock

## Abstract

In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of equilibrium theory to nonequililbrium processes is usually made through the assumption of *local thermodynamic equilibrium* (LTE). However, gas kinetic theory provides a perfectly general formulation of a nonequilibrium entropy in terms of the probability distribution function (PDF) solutions of the Boltzmann equation. In this paper I will evaluate the Boltzmann entropy for the PDF that underlies the Navier–Stokes equations and also for the PDF of the Mott–Smith shock solution. I will show that both monotonically increase in the shock. As a result, I will propose a new nonequilibrium thermodynamic entropy and show that it is also monotone and closely approximates the Boltzmann entropy.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1398924

- Report Number(s):
- LA-UR-17-24413

Journal ID: ISSN 1099-4300; ENTRFG

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Entropy

- Additional Journal Information:
- Journal Volume: 19; Journal Issue: 7; Journal ID: ISSN 1099-4300

- Publisher:
- MDPI

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; shocks; nonequilibrium thermodynamics; entropy

### Citation Formats

```
Margolin, Len G. Nonequilibrium Entropy in a Shock. United States: N. p., 2017.
Web. doi:10.3390/e19070368.
```

```
Margolin, Len G. Nonequilibrium Entropy in a Shock. United States. doi:10.3390/e19070368.
```

```
Margolin, Len G. Wed .
"Nonequilibrium Entropy in a Shock". United States. doi:10.3390/e19070368. https://www.osti.gov/servlets/purl/1398924.
```

```
@article{osti_1398924,
```

title = {Nonequilibrium Entropy in a Shock},

author = {Margolin, Len G.},

abstractNote = {In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of equilibrium theory to nonequililbrium processes is usually made through the assumption of local thermodynamic equilibrium (LTE). However, gas kinetic theory provides a perfectly general formulation of a nonequilibrium entropy in terms of the probability distribution function (PDF) solutions of the Boltzmann equation. In this paper I will evaluate the Boltzmann entropy for the PDF that underlies the Navier–Stokes equations and also for the PDF of the Mott–Smith shock solution. I will show that both monotonically increase in the shock. As a result, I will propose a new nonequilibrium thermodynamic entropy and show that it is also monotone and closely approximates the Boltzmann entropy.},

doi = {10.3390/e19070368},

journal = {Entropy},

number = 7,

volume = 19,

place = {United States},

year = {2017},

month = {7}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

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##
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##
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