# Storage of Energy in Constrained Non-Equilibrium Systems

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## Abstract

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**99**, 042118 (2019)). We evaluate this hypothesis using an ideal gas system with three methods of energy delivery: from a uniformly distributed energy source, from an external heat flow through the surface, and from an external matter flow. By introducing internal constraints into the system, we determine $\mathcal{T}$ with and without constraints and find that $\mathcal{T}$ is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows $U={U}_{0}\ast f\left({J}_{U}\right)$. In this context, we discuss natural variables for NESS, define the embedded energy (an analog of Helmholtz free energy for NESS), and provide its interpretation.

## 1. Introduction

## 2. Models and Results

#### 2.1. Energy Source

#### 2.2. Heat Flow

#### 2.3. Matter Flow

#### 2.4. Energy Density as Function of Heat Flow

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Schemes of (

**a**) unconstrained and (

**b**) constrained ideal gas model under an external energy supply. The two diathermal walls of area A and temperature ${T}_{0}$ are positioned at $x=\pm L$. An external energy is supplied homogeneously to the bulk with a density $\lambda $. The heat flux $2\overrightarrow{J}$ leaves the system through boundaries. In (

**b**), the vertical plane at $x={x}_{1}$ represents the internal constraint, which is a diathermal wall.

**Figure 2.**Schemes of (

**a**) unconstrained, (

**b**) and (

**c**) constrained ideal gas systems with an external heat flow. Two diathermal walls at temperatures ${T}_{1}$ and ${T}_{0}$ are placed at $x=0$ and L, respectively. In (

**b**,

**c**), the black surface inside the system represents the constraint, which is an adiabatic wall. In (

**b**), the constraint has a height h and extends from $(0,h/2)$ to $(L/2,h/2)$ to $(L/2,-h/2)$ to $(L,-h/2)$. In (

**c**), the constraint has a slope k and it stretches from $(0,-kL/2)$ to $(L,kL/2)$. The red arrows denote the heat flux.

**Figure 3.**Contour plots of temperature profiles: (

**a**) results of a vertical constraint; and (

**b**) results of a linear constraint. In both figures, the temperatures at the boundaries are ${T}_{1}=10,{T}_{0}=2$. The size of the system is $L=10$ and $H=60$. For the vertical constraint, the height of the wall is $h=10$. For the linear constraint, the slope of the wall is $k=1$.

**Figure 4.**Plots of total energy storage per volume $\Delta {U}_{tot}/V=\Delta ({U}_{1}+{U}_{2})/V$, total out-going heat flow per area ${J}_{tot}/A=({J}_{{U}_{1}}+{J}_{{U}_{2}})/A$ and their ratio ${\mathcal{T}}_{1|2}=\Delta {U}_{tot}/{J}_{tot}=\Delta ({U}_{1}+{U}_{2})/({J}_{{U}_{1}}+{J}_{{U}_{2}})$: results for vertical constraints (

**a**,

**c**,

**e**); and results for linear constraints (

**b**,

**d**,

**f**). Each panel is evaluated for six different system sizes of a fixed $L=10$ and $H=60,80,100,200,400$ and 600.

**Figure 5.**Schemes of (

**a**) unconstrained and (

**b**) constrained Poiseuille flow. The system is bounded by two plates with a fixed temperature ${T}_{0}$ and area A that are placed at $y=\pm h$. A constant pressure gradient is applied across the system. In (

**b**), the system is divided by an adiabatic slip wall placed at $y={y}_{1}$.

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**MDPI and ACS Style**

Zhang, Y.; Giżyński, K.; Maciołek, A.; Hołyst, R. Storage of Energy in Constrained Non-Equilibrium Systems. *Entropy* **2020**, *22*, 557.
https://doi.org/10.3390/e22050557

**AMA Style**

Zhang Y, Giżyński K, Maciołek A, Hołyst R. Storage of Energy in Constrained Non-Equilibrium Systems. *Entropy*. 2020; 22(5):557.
https://doi.org/10.3390/e22050557

**Chicago/Turabian Style**

Zhang, Yirui, Konrad Giżyński, Anna Maciołek, and Robert Hołyst. 2020. "Storage of Energy in Constrained Non-Equilibrium Systems" *Entropy* 22, no. 5: 557.
https://doi.org/10.3390/e22050557