Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems †
Abstract
:1. Introduction
2. Kinematics
2.1. The Components
2.2. The Mixture
2.3. The Diffusion Flux
3. The Energy–Momentum Tensor
3.1. Free and Interacting Components
3.2. (3+1)-Split
3.3. Additivity
3.4. (3+1)-Components of the Mixture
4. Entanglement of Energy and Momentum Balances
5. The Spin Tensor
5.1. (3+1)-Split
5.2. Additivity
5.3. (3+1)-Components of the Mixture
5.4. Spin Balance Equation
6. Thermodynamics of Interacting Components
6.1. The Entropy Identity
6.2. Exploitation of the Entropy Identity
6.2.1. Entropy Density, Gibbs and Gibbs–Duhem Equations
6.2.2. Entropy Flux, -Supply and -Production
6.3. Fields of Lagrange Multipliers
6.4. Multi-Temperature Relaxation and the Partial Temperatures
- diffusion: ;
- by diffusion modified chemical reaction: ;
- heat conduction: ;
- multi-component modified internal friction: ;
- multi-component interaction (this term vanishes in equilibrium and for 1-component systems: see Section 7): ;
- multi-temperature relaxation: ; and
- four terms describing entropy production by the spin .
6.5. The 4-Entropy
6.6. Equilibrium
6.6.1. Equilibrium Conditions
6.6.2. Killing Relation of the 4-Temperature
6.6.3. The Gradient of the 4-Temperature
7. Special Case: 1-Component System
7.1. Entropy Flux, -Supply and -Density
7.2. Equilibrium and Reversibility
8. Thermodynamics of a Mixture
8.1. Additivity of 4-Entropies
8.1.1. Entropy Density and -Flux
8.1.2. Entropy Supply and Production Density
8.2. Partial and Mixture Temperatures
8.3. Multi-Temperature Relaxation Equilibrium
8.3.1. Entropy and Entropy Flux Densities
8.3.2. Entropy Production and -Supply
8.4. Total Equilibrium
8.5. (3+1)-Entropy-Components and Spin
- The entropy density in Equation (154) of an -component depends on the spin density and on the spin density vector , whereas the entropy density of the mixture in Equation (223) depends on the four spin quantities in Equations (72) and (73). In 1-component systems, the entropy density in Equation (207) depends only on the spin vector . In equilibrium, the entropy density is for all cases independent of the spin, Equations (176) and (243).
- The entropy flux density in Equation (139) of an -component depends on the couple stress and on the spin stress , whereas the entropy flux density of the mixture in Equation (235) depends on the four spin quantities in Equations (72) and (73). In 1-component systems, the entropy flux density in Equation (208) depends only on the spin vector . In equilibrium, the entropy flux density in Equations (176) and (243) vanishes and induces .
- The entropy supply of an -component in Equation (156) is as well independent of the spin as for the mixture in Equation (239) and for a 1-component system in Equation (208). The entropy supply vanishes in equilibrium, and a connection between the force density and the angular momentum density is established, Equations (178) and (244).
- The entropy production density in Equations (157) and (240) does not depend on the spin density for an -component and for the mixture, but a dependence upon the three other (3+1)-spin-components exists. In 1-component systems, the entropy production density in Equation (209) depends on the spin stress and on the couple stress . The entropy production density vanishes in equilibrium, and a connection between the viscosity tensor and the spin stress and the couple stress is established, Equations (179) and (245).
9. Balances, Constitutive Equations and the 2nd Law
10. Special Case: General Relativity Theory
10.1. Extended Belinfante/Rosenfeld Procedure
10.2. Example: 2-Component Plain-Ghost Mixture
10.2.1. The Plain Component
10.2.2. The Ghost Component
10.2.3. The Plain-Ghost Mixture
10.3. “Dark Matter” as a Ghost Component?
11. Summary
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Appendices
Appendix A.1. Rest Mass Densities
Appendix A.2. Example: Uniform Component Velocities
Appendix A.3. Stoichiometric Equations
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Muschik, W. Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems. Entropy 2019, 21, 1034. https://doi.org/10.3390/e21111034
Muschik W. Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems. Entropy. 2019; 21(11):1034. https://doi.org/10.3390/e21111034
Chicago/Turabian StyleMuschik, Wolfgang. 2019. "Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems" Entropy 21, no. 11: 1034. https://doi.org/10.3390/e21111034
APA StyleMuschik, W. (2019). Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems. Entropy, 21(11), 1034. https://doi.org/10.3390/e21111034