Transport coefficients of hot and dense hadron gas in a magnetic field: A relaxation time approach

We estimate various transport coefficients of hot and dense hadronic matter in the presence of magnetic field. The estimation is done through solutions of the relativistic Boltzmann transport equation in the relaxation time approximation. We have investigated the temperature and the baryon chemical potential dependence of these transport coefficients. Explicit calculations are done for the hadronic matter in the ambit of hadron resonance gas model. We estimate thermal conductivity, electrical conductivity, and the shear viscosity of hadronic matter in the presence of a uniform magnetic field. Magnetic field, in general, makes the transport coefficients anisotropic. It is also observed that all the transport coefficients perpendicular to the magnetic field are smaller compared to their isotropic counterpart.

Figure 1

Variation of normalized thermal conductivity ($k_0/T^2$) with temperature ($T$) for different values of magnetic field ($B$) at finite baryon chemical potential. Red solid line represents $B=0$ case, blue dotted line and brown dashed line represent $eB=0.01{\mathrm{GeV}}^2$ and $eB=0.03{\mathrm{GeV}}^2$, respectively. In the presence of magnetic field, $k_0/T^2$ decreases. At low temperature, decrease in $k_0/T^2$ due to magnetic field is significant. But at higher temperature, the effect of magnetic field on $k_0/T^2$ is not significant.

Figure 2

Left plot: variation of $\omega /n_B$ with temperature ($T$) for different values of ${\mu }_B$ at zero magnetic field. Right plot: variation of $\omega /n_B$ only for baryons denoted as ${\omega }_B/n_B$, with temperature ($T$) for different values of ${\mu }_B$ for a vanishing magnetic field. With increasing temperature and ${\mu }_B$, $\omega /n_B$ of the hadron resonance gas decreases. However, baryonic contribution to $\omega /n_B$ i.e., ${\omega }_B/n_B$ increases with temperature.

Figure 3

Left plot: variation of normalized thermal conductivity ($k_0/T^2$) with temperature ($T$) for different values of ${\mu }_B$ at zero magnetic field. Right plot: variation of normalized thermal conductivity ($k_0/T^2$) with temperature ($T$) for different values of ${\mu }_B$ for a nonvanishing magnetic field. With increasing temperature and ${\mu }_B$, $k_0/T^2$ decreases. In the presence of magnetic field, $k_0/T^2$ decreases.

Figure 4

Left plot: variation of Hall-type thermal conductivity $k_1/T^2$ with temperature for nonvanishing magnetic field and different values of ${\mu }_B$. Right plot: variation of $k_1/T^2$ with temperature for different values of magnetic fields. With ${\mu }_B$ $k_1/T^2$ decreases. But for a fixed value of ${\mu }_B$ variation of $k_1/T^2$ is nonmonotonic with magnetic field. At low temperature, $k_1/T^2$ decreases with magnetic field, but at higher temperature $k_1/T^2$ increases with magnetic field.

Figure 5

Left plot: variation of $k_2/T^2$ with temperature ($T$) for various values of baryon chemical potential at finite magnetic field. Right plot: variation of $k_2/T^2$ with temperature ($T$) for various values of magnetic field. For a fixed value of magnetic field and ${\mu }_B$, $k_2/T^2$ decreases with temperature. With increasing magnetic field, generically $k_2/T^2$ increases within the range of $T$, ${\mu }_B$, and $B$ considered in this investigation. But with increasing ${\mu }_B$, $k_2/T^2$ decreases.

Figure 6

Left plot: variation of ${\sigma }_2/T$ with temperature for a nonvanishing magnetic field and for different values of ${\mu }_B$. With increasing ${\mu }_B$, ${\sigma }_2/T$ decreases. Right plot: variation of ${\sigma }_2/T$ with temperature for nonvanishing values of magnetic field at finite ${\mu }_B$. With magnetic field, ${\sigma }_2/T$ increases.

Figure 7

Variation of ${\eta }_1/T^3$ with temperature for vanishing magnetic field but with different values of ${\mu }_B$. At zero magnetic field, ${\eta }_1={\eta }_2$. From this figure, it is clear that ${\eta }_1/T^3$ decreases with temperature and increases with ${\mu }_B$.

Figure 8

Left plot: variation of ${\eta }_1/T^3$ with temperature for vanishing ${\mu }_B$ but with different values of magnetic field. Right plot: variation of ${\eta }_2/T^3$ with temperature for vanishing ${\mu }_B$ but with different values of magnetic field. For vanishing magnetic field, ${\eta }_1={\eta }_2$, which can be seen in this figure. Also, for nonvanishing magnetic field, ${\eta }_1\ne {\eta }_2$ can be seen in these plots. With increasing magnetic field, both ${\eta }_1/T^3$ and ${\eta }_2/T^3$ decrease. This decrease is very prominent at low temperature. At high temperature, the effect of magnetic field is not significant.

Figure 9

Left plot: variation of ${\eta }_1/T^3$ with temperature for different values of ${\mu }_B$ in the presence of magnetic field. Right plot: variation of ${\eta }_2/T^3$ with temperature for different values of ${\mu }_B$ in the presence of magnetic field. Behavior of ${\eta }_1/T^3$ and ${\eta }_2/T^3$ is similar apart from their numerical values. ${\eta }_2$ is larger than ${\eta }_1$ as can be seen from their analytical expressions. For higher ${\mu }_B$, value of ${\eta }_1/T^3$ and ${\eta }_2/T^3$ is higher. Variation of both ${\eta }_1/T^3$ and ${\eta }_2/T^3$ with temperature shows nonmonotonic behavior with a peak structure.

Figure 10

Left plot: variation of ${\eta }_3/T^3$ with temperature for a nonvanishing magnetic field and different values of ${\mu }_B$. Right plot: variation of ${\eta }_4/T^3$ with temperature for a nonvanishing magnetic field and different values of ${\mu }_B$. Behavior of ${\eta }_3/T^3$ and ${\eta }_4/T^3$ is similar with ${\mu }_B$ and temperature apart from their numerical values. With increasing ${\mu }_B$, Hall-type shear viscosities ${\eta }_3/T^3$ and ${\eta }_4/T^3$ increase.

Figure 11

Left plot: variation of ${\eta }_3/T^3$ with temperature for different values of nonvanishing magnetic field and finite ${\mu }_B$. Right plot: variation of ${\eta }_4/T^3$ with temperature for different values of nonvanishing magnetic field and finite ${\mu }_B$. Variation of ${\eta }_3/T^3$ and ${\eta }_4/T^3$ with temperature and magnetic field is similar apart from their numerical values. For low temperature, ${\eta }_3/T^3$ and ${\eta }_4/T^3$ decrease with magnetic field. On the other hand, for high temperature, both ${\eta }_3/T^3$ and ${\eta }_4/T^3$ increase with magnetic field. Variation of ${\eta }_3/T^3$ and ${\eta }_4/T^3$ with temperature is nonmonotonic with a peak.