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

Now we derive the relation $\stackrel{¯}{\rho }\left(J\right)$ from (3.15) for the equilibrium process with $\delta Q=0$ . For clearness, we use the the static mass density $\stackrel{¯}{\rho }=\frac{N\stackrel{¯}{m}}{V}$ to

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

$\frac{1}{\stackrel{¯}{\rho }}\frac{\text{d}\stackrel{¯}{\rho }}{\text{d}J}=\frac{3\left[{\left(J+\sigma \right)}^{2}-\stackrel{¯}{\mu }{\sigma }^{2}\right]}{\left(J+\sigma \right)\left({J}^{2}+BJ-2\mu \sigma \right)}$ (3.20)

The solution is given by

$\stackrel{¯}{\rho }={\varrho }_{0}{\left(J+\sigma \right)}^{\frac{3\stackrel{¯}{\mu }}{3\stackrel{¯}{\mu }+1}}{\left({J}^{2}+JB-2\sigma \mu \right)}^{\frac{3}{2}\left(1-\frac{\stackrel{¯}{\mu }}{3\stackrel{¯}{\mu }+1}\right)}{\left(\frac{2J+B-A}{2J+B+A}\right)}^{\frac{\alpha }{2A}}$ (3.21)

where parameters are defined by

$B=\left(2+3\stackrel{¯}{\mu }\right)\sigma -2\mu ,\text{ }A=\sqrt{8\sigma \mu +{B}^{2}},\text{ }\alpha =3\left(1+4\stackrel{¯}{\mu }\right)\left(2\mu -3\stackrel{¯}{\mu }\sigma \right)$ (3.22)

(3.21) shows how the internal potentials ${w}_{n}$ influence the mass density. (3.21) reduces to (3.10) if $\mu =\stackrel{¯}{\mu }=0$ .

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 $\stackrel{¯}{w}=0$ at low temperature, (3.11)-(3.12) give the equation of state for the adiabatic monatomic gas

$P\stackrel{˙}{=}\rho J\stackrel{˙}{=}{P}_{0}{\rho }^{\frac{5}{3}},\text{ }\left(J\ll 1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{ }kT\ll \stackrel{¯}{m}\right)$ (4.1)

which is identical to the empirical law in thermodynamics. When $J\gg 1$ , we have $P\to \frac{1}{3}\rho$ , 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 $J\to \infty$ or $\stackrel{¯}{m}\to 0$ , we get the Stefan-Boltzmann’s law $\rho \propto {T}^{4}$ . 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 ${\varrho }_{0}$ is independent of any practical process. Of course, ${\varrho }_{0}$ is related to the property of particles. Furthermore, equation of state (3.11)-(3.12) provides a singularity-free stellar structure in thermal equilibrium  .

5) In the case $\stackrel{¯}{w}>0$ , the motion of the particles will slightly deviate from the geodesic. By (3.21) we find, $\stackrel{¯}{\rho }=0$ leads to $J=\frac{2\sigma \mu }{B}\approx \mu$ , 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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