Storage of Energy in Constrained Non-Equilibrium Systems
We study a quantity <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> defined as the energy U, stored in non-equilibrium steady states (NESS) over its value i...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-05-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/22/5/557 |
| _version_ | 1797567774407524352 |
|---|---|
| author | Yirui Zhang Konrad Giżyński Anna Maciołek Robert Hołyst |
| author_facet | Yirui Zhang Konrad Giżyński Anna Maciołek Robert Hołyst |
| author_sort | Yirui Zhang |
| collection | DOAJ |
| description | We study a quantity <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium <inline-formula> <math display="inline"> <semantics> <msub> <mi>U</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo>−</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> divided by the heat flow <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mi>U</mi> </msub> </semantics> </math> </inline-formula> going out of the system. A recent study suggests that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is minimized in steady states (Phys.Rev.E.<b>99</b>, 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> with and without constraints and find that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows <inline-formula> <math display="inline"> <semantics> <mrow> <mi>U</mi> <mo>=</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>∗</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mi>U</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. 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. |
| first_indexed | 2024-03-10T19:47:17Z |
| format | Article |
| id | doaj.art-0ebe396e9a464d389e92db31d3fe25ea |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T19:47:17Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-0ebe396e9a464d389e92db31d3fe25ea2023-11-20T00:39:43ZengMDPI AGEntropy1099-43002020-05-0122555710.3390/e22050557Storage of Energy in Constrained Non-Equilibrium SystemsYirui Zhang0Konrad Giżyński1Anna Maciołek2Robert Hołyst3Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, PolandWe study a quantity <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium <inline-formula> <math display="inline"> <semantics> <msub> <mi>U</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo>−</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> divided by the heat flow <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mi>U</mi> </msub> </semantics> </math> </inline-formula> going out of the system. A recent study suggests that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is minimized in steady states (Phys.Rev.E.<b>99</b>, 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> with and without constraints and find that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows <inline-formula> <math display="inline"> <semantics> <mrow> <mi>U</mi> <mo>=</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>∗</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mi>U</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. 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.https://www.mdpi.com/1099-4300/22/5/557non-equilibrium stationary statesenergy fluxesinternal energyideal gasheat transfer |
| spellingShingle | Yirui Zhang Konrad Giżyński Anna Maciołek Robert Hołyst Storage of Energy in Constrained Non-Equilibrium Systems Entropy non-equilibrium stationary states energy fluxes internal energy ideal gas heat transfer |
| title | Storage of Energy in Constrained Non-Equilibrium Systems |
| title_full | Storage of Energy in Constrained Non-Equilibrium Systems |
| title_fullStr | Storage of Energy in Constrained Non-Equilibrium Systems |
| title_full_unstemmed | Storage of Energy in Constrained Non-Equilibrium Systems |
| title_short | Storage of Energy in Constrained Non-Equilibrium Systems |
| title_sort | storage of energy in constrained non equilibrium systems |
| topic | non-equilibrium stationary states energy fluxes internal energy ideal gas heat transfer |
| url | https://www.mdpi.com/1099-4300/22/5/557 |
| work_keys_str_mv | AT yiruizhang storageofenergyinconstrainednonequilibriumsystems AT konradgizynski storageofenergyinconstrainednonequilibriumsystems AT annamaciołek storageofenergyinconstrainednonequilibriumsystems AT roberthołyst storageofenergyinconstrainednonequilibriumsystems |