/Math/MathML"> Δ x Δ y Δ z for the N given particles, (3.18) gives (3.15) in the case δ Q = 0 . In the case of δ Q 0 , (3.15) holds due to energy conservation law.

Now we derive the relation ρ ¯ ( J ) from (3.15) for the equilibrium process with δ Q = 0 . For clearness, we use the the static mass density ρ ¯ = N m ¯ V to

replace V. Substituting (3.7), (3.8) and (3.9) into (3.15), we get dimensionless differential equation

1 ρ ¯ d ρ ¯ d J = 3 [ ( J + σ ) 2 μ ¯ σ 2 ] ( J + σ ) ( J 2 + B J 2 μ σ ) (3.20)

The solution is given by

ρ ¯ = ϱ 0 ( J + σ ) 3 μ ¯ 3 μ ¯ + 1 ( J 2 + J B 2 σ μ ) 3 2 ( 1 μ ¯ 3 μ ¯ + 1 ) ( 2 J + B A 2 J + B + A ) α 2 A (3.21)

where parameters are defined by

B = ( 2 + 3 μ ¯ ) σ 2 μ , A = 8 σ μ + B 2 , α = 3 ( 1 + 4 μ ¯ ) ( 2 μ 3 μ ¯ σ ) (3.22)

(3.21) shows how the internal potentials w n influence the mass density. (3.21) reduces to (3.10) if μ = μ ¯ = 0 .

4. Discussion and Conclusion

1) The above calculations show that functions of state consistent with relativity should include the influences of gravity. The energy-momentum tensor and geodesics connect the macro concepts with micro movements of particles.

2) In the case of ideal gas with w ¯ = 0 at low temperature, (3.11)-(3.12) give the equation of state for the adiabatic monatomic gas

P = ˙ ρ J = ˙ P 0 ρ 5 3 , ( J 1 or k T m ¯ ) (4.1)

which is identical to the empirical law in thermodynamics. When J 1 , we have P 1 3 ρ , so the adiabatic index is not a constant for large range of

temperature due to the relativistic effect. These results show the validity of (3.11)-(3.13) and the consistence with normal thermodynamics.

3) By (3.11), letting J or m ¯ 0 , we get the Stefan-Boltzmann’s law ρ T 4 . This means that the above results automatically include photons, and the Stefan-Boltzmann’s law is also valid for the ultra-relativistic particles.

4) In general relativity, all processes occur automatically, and ϱ 0 is independent of any practical process. Of course, ϱ 0 is related to the property of particles. Furthermore, equation of state (3.11)-(3.12) provides a singularity-free stellar structure in thermal equilibrium [18] .

5) In the case w ¯ > 0 , the motion of the particles will slightly deviate from the geodesic. By (3.21) we find, ρ ¯ = 0 leads to J = 2 σ μ B μ , which means the

zero temperature can not reach. The physical reason for such conclusion is unclear.

Acknowledgements

The author is grateful to Prof. Ta-Tsien, Prof. Tie-Hu Qin and Prof. Ji-Zong Li for encouragement.

Conflicts of Interest

The authors declare no conflicts of interest.

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