Thermodynamics of ideal gas in doubly special relativity

We study thermodynamics of an ideal gas in doubly special relativity. A new type of special functions (which we call “incomplete modified Bessel functions”) emerge. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that doubly special relativity thermodynamics is nonperturbative in the special relativity and massless limits. A stiffer equation of state is found.

Figure 1

The single-particle partition function is plotted vs $m_0$ (left) and $\frac{1}{\kappa }$ (right). The plots for the unmodified measure are in solid lines, while those for modified measure are in dashed lines. The plots in massless (left) and SR (right) cases are shown in red. The leading-order behaviors are plotted in blue. The numerical plots for Eqs. (2.27, 2.12) are shown in green (solid and dashed, respectively) for comparison. Different values for the parameters in natural units are as follows: $V={10}^{35},N={10}^{25},T=0.01,a_0=1,a_1=0.2,a_2=a_3=\dots =0$, and $\kappa =1$ (left), $m_0=0.01$ (right).

Figure 2

The plots for the unmodified measure are in solid lines, while those for modified measure are in dashed lines. The SR plots are shown in red. The figures in the left column are for Case I, while those in the right column are for Case III. The DSR plots for Case I are plotted in green. For Case III, the leading order behaviors are plotted in blue. The numerical plots for Case III are also shown in green for comparison. Different values for the parameters in natural units are as follows: $V={10}^{35},N={10}^{25},\kappa =1,m_0=1(\mathrm{\text{Case}}\mathrm{I}),0.01(\mathrm{\text{Case}}\mathrm{III}),a_0=1,a_1=0.2,a_2=a_3=\dots =0$.