# Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics

## Abstract

We show that the equations which describe irreversible evolution of a system can be derived from a variational principle. We suggest a Lagrangian, which depends on the properties of the normal and the so-called “mirror-image” system. The Lagrangian is symmetric in time and therefore compatible with microscopic reversibility. The evolution equations in the normal and mirror-imaged systems are decoupled and describe therefore independent irreversible evolution of each of the systems. The second law of thermodynamics follows from a symmetry of the Lagrangian. Entropy increase in the normal system is balanced by the entropy decrease in the mirror-image system, such that there exists an “integral of evolution” which is a constant. The derivation relies on the property of local equilibrium, which states that the local relations between the thermodynamic quantities in non-equilibrium are the same as in equilibrium.

- Authors:

- School of Applied Sciences, RMIT University, GPO Box 2476, Melbourne, Victoria 3001 (Australia)

- Publication Date:

- OSTI Identifier:
- 22415860

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 142; Journal Issue: 20; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; BALANCES; ENTROPY; EQUATIONS; EQUILIBRIUM; EVOLUTION; IMAGES; INTEGRALS; LAGRANGIAN FUNCTION; MIRRORS; SYMMETRY; THERMODYNAMICS; VARIATIONAL METHODS

### Citation Formats

```
Glavatskiy, K. S.
```*Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics*. United States: N. p., 2015.
Web. doi:10.1063/1.4921558.

```
Glavatskiy, K. S.
```*Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics*. United States. doi:10.1063/1.4921558.

```
Glavatskiy, K. S. Thu .
"Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics". United States. doi:10.1063/1.4921558.
```

```
@article{osti_22415860,
```

title = {Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics},

author = {Glavatskiy, K. S.},

abstractNote = {We show that the equations which describe irreversible evolution of a system can be derived from a variational principle. We suggest a Lagrangian, which depends on the properties of the normal and the so-called “mirror-image” system. The Lagrangian is symmetric in time and therefore compatible with microscopic reversibility. The evolution equations in the normal and mirror-imaged systems are decoupled and describe therefore independent irreversible evolution of each of the systems. The second law of thermodynamics follows from a symmetry of the Lagrangian. Entropy increase in the normal system is balanced by the entropy decrease in the mirror-image system, such that there exists an “integral of evolution” which is a constant. The derivation relies on the property of local equilibrium, which states that the local relations between the thermodynamic quantities in non-equilibrium are the same as in equilibrium.},

doi = {10.1063/1.4921558},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 20,

volume = 142,

place = {United States},

year = {2015},

month = {5}

}