# Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality

## Abstract

In this paper we revisit the notion of the “minus logarithm of stationary probability” as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a detrministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1340861

- Report Number(s):
- PNNL-SA-119608

Journal ID: ISSN 1099-4300

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Entropy; Journal Volume: 18; Journal Issue: 8

- Country of Publication:
- United States

- Language:
- English

- Subject:
- nonequilibium steady states; stochastic nonequilibrium thermodynamics; generalized potentials; entropy

### Citation Formats

```
Thompson, Lowell, and Qian, Hong.
```*Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality*. United States: N. p., 2016.
Web. doi:10.3390/e18080309.

```
Thompson, Lowell, & Qian, Hong.
```*Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality*. United States. doi:10.3390/e18080309.

```
Thompson, Lowell, and Qian, Hong. Mon .
"Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality". United States.
doi:10.3390/e18080309.
```

```
@article{osti_1340861,
```

title = {Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality},

author = {Thompson, Lowell and Qian, Hong},

abstractNote = {In this paper we revisit the notion of the “minus logarithm of stationary probability” as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a detrministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems.},

doi = {10.3390/e18080309},

journal = {Entropy},

number = 8,

volume = 18,

place = {United States},

year = {Mon Aug 01 00:00:00 EDT 2016},

month = {Mon Aug 01 00:00:00 EDT 2016}

}