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Entropy 2011, 13(9), 1611-1647; doi:10.3390/e13091611
2. Uniqueness of Bekenstein-Hawking Entropy
2.1. Ordinary Thermodynamics in Axiomatic Formulation
2.1.1. Adiabatic Process and Composition
2.1.2. Basic Properties of State Variables
2.1.3. Entropy in Ordinary Thermodynamics
2.2. Black Hole Thermodynamics
2.2.1. Thermal Equilibrium of Schwarzschild Black Hole
2.2.2. Thermal Stability of Schwarzschild Black Hole
2.2.3. Scaling Behavior of State Variables
- Extensive variable: These variables, X (e.g., , and ), are scaled as, .
- Intensive variable: These variables, Y (e.g., and ), are scaled as, .
- Thermodynamic energy: These energies, Z (e.g., and ), are scaled as, .
2.2.4. Adiabatic Process and Composition
2.2.5. Basic Properties of Bekenstein-Hawking Entropy
- Extensivity of formulated in Definition 3.
- Additivity of in Equation (18).
- Entropy principle shown in Fact 1. (See comments given below.)
- Uniqueness of . (Proof is given in the next subsection.)
- Concavity of about internal energy and system size . (Detail is given below.)
- Monotone increasing nature of about . (Detail is given below.)
2.3. Uniqueness Theorem of Bekenstein-Hawking Entropy
2.3.1. Statement of Theorem
2.3.2. Proof of Theorem 1 (preparations)
- Step 1:
- Show the positivity of .
- Step 2:
- Show that the relation, , holds with arbitrary X, where .
- Step 3:
- Show that the relation, , holds with arbitrary Y.
- Step 4:
- Show the extensivity of η.
2.3.3. Step 1 of the Proof
2.3.4. Step 2 of the Proof
2.3.5. Step 3 of the Proof
2.3.6. Step 4 of the Proof
3. Conditions Justifying Boltzmann formula
3.1. Statements of Theorems and a Corollary without Proof
- Condition A : Arbitrary j-particle interaction, , becomes negative for sufficiently large distribution of j particles. That is, there exists a constant , such that
- Condition B : The potential Φ is bounded below. That is, there exists a constant , such that
- Result 1 : The “large system limit” of the density of ground state energy exists,
- Result 2 : When does not diverge to , the “thermodynamic limit” of the logarithmic density of number of states exists,
- Presupposition C : The j-particle interactions for disappear, and the total interaction potential Φ is a sum of two-particle interactions, .
- Presupposition D : Introduce a differential quantity, , of potential Φ defined as
3.2. Proof of Dobrushin Theorem
4. Conclusion: Suggestion on Universal Property of Quantum Gravity
References and Notes
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- The author could not obtain the original paper  of Dobrushin. But this theorem is found in Ruelle’s book  as Proposition 3.2.4. The Dobrushin theorem in Ruelle’s book is proven only for classical systems. However, we are interested in quantum system in this paper. Therefore, the statement and proof of Dobrushin theorem in this paper are the extended version by this author so as to match with quantum system under consideration.
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A. Construction of Free Energy of Schwarzschild Black Hole
B. Sketch of Proof of Ruelle-Tasaki Theorem
- Step 1 :
- Introduce a basic technique used in the proof. (Mini-max principle is used.)
- Step 2 :
- Show the Result 1 of Ruelle-Tasaki theorem.
- Step 3 :
- Show the Result 2 of Ruelle-Tasaki theorem. This step consists of three substeps:
- Substep 3-1 :
- Show the existence of unique thermodynamic limit, . (Proposition 2 is used.)
- Substep 3-2 :
- Show the concavity of about ε and ρ.
- Substep 3-3 :
- Show the monotone increasing nature of about ε.
B.2. Step 1 of the Proof: Basic Technique
B.3. Step 2 of the Proof: Result 1
- Let be a cubic region of edge length , where is a constant, in which identical particles exist. Let denote the volume of this cube, . The Hamiltonian of this system, , is expressed as that in Equation (66). Require that the Conditions A and B are satisfied.
- Let be a cubic region of edge length , and denote its volume, . Then, make eight copies of the cube (including particles), and place them inside as shown in Figure 7 so as to share the eight vertices of with the eight copies of . By this construction of larger cube , the distance between smaller cubes is longer than or equal to . In the larger cube , there exist particles. The Hamiltonian of this system, , is expressed as that in Equation (70),
- Let be a cubic region of edge length , and denote its volume. Repeat the procedure (ii) and construct the larger system in including particles with Hamiltonian, , where . Then, repeating again the same procedure n times, the n-th cube of edge length is constructed, which includes particles with Hamiltonian, , where . For sufficiently large n, we obtain a large system. Obviously, the Inequalities (77) and (79) can be applied to this large system with appropriate modifications.
B.4. Step 3 of the Proof: Result 2
B.4.1. Substep 3-1
B.4.2. Substep 3-2
B.4.3. Substep 3-3
C. Proof of Proposition 2
D. Preparations for Proposition 2
E. Proof of Corollary 1
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