Symmetry of the linearized Boltzmann equation: Entropy production and Onsager-Casimir relation
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1 Symmetry of the linearized Boltzmann equation: Entropy production and Onsager-Casimir relation Shigeru TAKATA ( 髙田滋 ) Department of Mechanical Engineering and Science, (also Advanced Research Institute of Fluid Science and Engineering) Kyoto University takata@aero.mbox.media.kyoto-u.ac.jp 1
2 Contents What is the Onsagar-Casimir relation? Definition of entropy, its flow and production Expression of entropy production Weakly disturbed system Some consideration on the disturbance from the surroundings Case 1: conventional-type reciprocity Case 2: non-conventional reciprocity Note: The discussed reciprocity holds for arbitrary Knudsen number. (NOT restricted to the fluid dynamic regime) 2
3 Onsagar-Casimir relation (#1) (Non-equilibrium thermodynamics) In non-equilibrium systems, the entropy is produced. The driving force that makes the system out of equilibrium is called the thermodynamic force or affinity. Entropy production Thermodynamic force (e.g. pressure grad., temperature grad.) Conjugate thermodynamic flux (e.g. mass flux, heat flux) 3
4 Onsagar-Casimir relation (#2) (Non-equilibrium thermodynamics) Entropy production Thermodynamic force (e.g. pressure grad., temperature grad.) Conjugate thermodynamic flux (e.g. mass flux, heat flux) For weakly non-equilibrium systems, i.e., when X i is small enough, J i may be expressed by a linear combination of {X i } kinetic coefficient The matrix of kinetic coefficients is known to be symmetric, which is called the Onsager-Casimir relation: 4
5 Onsagar-Casimir relation (#3) (Non-equilibrium thermodynamics) The matrix of kinetic coefficients is known to be symmetric, which is called the Onsager-Casimir relation: In what sense? It was empirically known and explained for bulk or near continuum systems. There are, however, numerical evidences suggesting that the relation holds for the entire range of the Knudsen number. Goal of the present talk Develop the theory of the Onsagar-Casimir relation for the entire range of the Knudsen number. Clarify the situation where the relation holds or not. 5
6 Contents What is the Onsagar-Casimir relation? Definition of entropy, its flow and production Expression of entropy production Weakly disturbed system Some consideration on the disturbance from the surroundings Case 1: conventional-type reciprocity Case 2: non-conventional reciprocity Note: The discussed reciprocity holds for arbitrary Knudsen number. (NOT restricted to the fluid dynamic regime) 6
7 (*) is confined in a finite region is a solution of (*) 7
8 Real boundary 1. Non-negative 2. Uniqueness condition 3. Detailed balance Real boundary Imaginary boundary Imaginary boundary 1. Non-negative 2. Extended detailed balance Most typically: Specular- and periodic-type boundary condition Key feature: No information is lost on the imaginary boundary 8
9 Entropy, its flow and production in non-equilibrium gases entropy Entropy production Entropy flow 9
10 Entropy production Entropy production (#1) Production in the internal gas (A) (B) (A)= ( + + ) Mass conservation (B)= + + (=0) Observation Green terms: definite moment (flux) of f Blue term: indefinite moment (NOT PREFERABLE) 10
11 Entropy production (#2) Entropy balance at the boundary Condensed phase or Simple solid (steady case; conservation law across the boundary) (due to Waldmann, Kuscer) Production at the surface interaction between the gas and the real boundary 11
12 Production in the internal gas Entropy production (#3) Production at the surface interaction between the gas and the real boundary sum 12
13 Contents What is the Onsagar-Casimir relation? Definition of entropy, its flow and production Expression of entropy production Weakly disturbed system Some consideration on the disturbance from the surroundings Case 1: conventional-type reciprocity Case 2: non-conventional reciprocity Note: The discussed reciprocity holds for arbitrary Knudsen number. (NOT restricted to the fluid dynamic regime) 13
14 Weakly disturbed system =0 14
15 (additional restriction) (2-1) (2-2) otherwise 15
16 Symmetric relation (general version) (1) Finite domain (2) Infinite domain Symmetric relation 16
17 Contents What is the Onsagar-Casimir relation? Definition of entropy, its flow and production Expression of entropy production Weakly disturbed system Some consideration on the disturbance from the surroundings Case 1: conventional-type reciprocity Case 2: non-conventional reciprocity Note: The discussed reciprocity holds for arbitrary Knudsen number. (NOT restricted to the fluid dynamic regime) 17
18 (#1) Thermodynamic forces Conjugate thermodynamic fluxes 18
19 (#2) Thermodynamic forces Conjugate thermodynamic fluxes 19
20 (#3) 20
21 (#1) Case (2-1) Thermodynamic forces Conjugate thermodynamic fluxes 21
22 (#2) Case (2-1) Conjugate thermodynamic fluxes 22
23 (#3) Case (2-1) Conjugate thermodynamic fluxes 23
24 Contents What is the Onsagar-Casimir relation? Definition of entropy, its flow and production Expression of entropy production Weakly disturbed system Some consideration on the disturbance from the surroundings Case 1: conventional-type reciprocity Case 2: non-conventional reciprocity Note: The discussed reciprocity holds for arbitrary Knudsen number. (NOT restricted to the fluid dynamic regime) 24
25 (#1) Case (2-2) general case Decomposition: Thermodynamic forces Conjugate thermodynamic fluxes??? Reciprocity is NOT recovered 25
26 Symmetric relation (general version) (1) Finite domain (2) Infinite domain Symmetric relation 26
27 (#2) Case (2-2) general case Thermodynamic forces Conjugate thermodynamic fluxes 27
28 (#3) Case (2-2) general case Conjugate thermodynamic fluxes Thermodynamic forces 28
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CHAPTER 5 Entropy and the Second and Third Laws of Thermodynamics Key Points Entropy, S, is a state function that predicts the direction of natural, or spontaneous, change. Entropy increases for a spontaneous