Magneto-Seebeck coefficient and Nernst coefficient of a hot and dense hadron gas

We discuss the thermoelectric effect of hot and dense hadron gas within the framework of the hadron resonance gas model. Using the relativistic Boltzmann equation within the relaxation time approximation we estimate the Seebeck coefficient of the hot and dense hadronic medium with a gradient in temperature and baryon chemical potential. The hadronic medium in this calculation is modeled by the hadron resonance gas (HRG) model with hadrons and their resonances up to a mass cutoff $\mathrm{\Lambda }\sim 2.6\mathrm{GeV}$. We also extend the formalism of the thermoelectric effect for a nonvanishing magnetic field. The presence of magnetic field also leads to a Hall type thermoelectric coefficient (Nernst coefficient) for the hot and dense hadronic matter apart from a magneto-Seebeck coefficient. We find that generically in the presence of a magnetic field the Seebeck coefficient decreases while the Nernst coefficient increases with the magnetic field. At higher temperature and/or baryon chemical potential these coefficients approach to their values at vanishing magnetic field.

Figure 1

Variation of Seebeck coefficient ($S$) for vanishing magnetic field with temperature ($T$) and baryon chemical potential (${\mu }_B$). With increasing baryon chemical potential (${\mu }_B$), Seebeck coefficient ($S$) increases. On the other hand with increasing temperature Seebeck coefficient decreases for values of ${\mu }_B$ ($\lesssim 300\mathrm{MeV}$), but for higher values of baryon chemical potential Seebeck coefficient increases with temperature.

Figure 2

Left plot: variation of normalised electrical conductivity (${\sigma }_{el}/T$) with temperature ($T$) and baryon chemical potential (${\mu }_B$) for vanishing magnetic field. With increasing $T$ and ${\mu }_B$ normalized electrical conductivity decreases. Right plot: variation of ${\mathcal{I}}_{31}/T^2$ with temperature and baryon chemical potential for vanishing magnetic field. With increasing $T$ and ${\mu }_B$, ${\mathcal{I}}_{31}/T^2$ increases.

Figure 3

Variation of the magneto-Seebeck coefficient ($S_B$) with temperature ($T$) for different values of baryon chemical potential (${\mu }_B$). With increasing baryon chemical potential (${\mu }_B$) the magneto-Seebeck coefficient ($S_B$) increases. On the other hand with increasing temperature the magneto-Seebeck coefficient shows nonmonotonic behavior for ${\mu }_B\lesssim 300\mathrm{MeV}$ and for higher values of ${\mu }_B$ the magneto-Seebeck coefficient increases.

Figure 4

Left plot: variation of normalized electrical conductivity ${\sigma }_{el}/T$ with temperature for different values of baryon chemical potential for nonvanishing magnetic field. For low temperature range ${\sigma }_{el}/T$ increases with ${\mu }_B$, otherwise it decreases with baryon chemical potential. With temperature ${\sigma }_{el}/T$ shows nonmonotonic variation with a peak structure. Right plot: variation of normalized Hall conductivity ${\sigma }_H/T$ with temperature for different values of baryon chemical potential.

Figure 5

Left plot: variation of ${\mathcal{I}}_{31}^B/T^2$ with temperature for different values of baryon chemical potential for nonvanishing magnetic field. With temperature as well as with baryon chemical potential ${\mathcal{I}}_{31}^B/T^2$ increase. Right plot: variation of ${\mathcal{I}}_{42}^B/T^2$ with temperature for different values of baryon chemical potential for nonvanishing magnetic field. With increasing temperature as well as with ${\mu }_B$, ${\mathcal{I}}_{42}^B/T^2$ decreases. In ${\mathcal{I}}_{31}^B/T^2$ mesonic contributions vanishes due to exact and opposite contributions of the particles and the antiparticles. On the other hand in ${\mathcal{I}}_{42}^B/T^2$ mesonic contributions are dominant relative to the baryonic contributions.

Figure 6

Left plot: variation of magneto-Seebeck coefficient ($S_B$) with temperature for different values of magnetic field. With increasing magnetic field $S_B$ decreases, however in the high temperature range the effect of magnetic field on $S_B$ is less significant. Right plot: variation of magneto-Seebeck coefficient $S_B$ with baryon chemical potential for different values of magnetic field. From this plot it is clear that with increasing ${\mu }_B$ magneto-Seebeck coefficient ($S_B$) increases.

Figure 7

Left plot: variation of normalized electrical conductivity, ${\sigma }_{el}/T\equiv {\sum }_a{q}_a^2{L}_{1_a}/T$ with temperature for nonvanishing value of the magnetic field and baryon chemical potential. With magnetic field electrical conductivity generically decrease and with temperature ${\sigma }_{el}/T$ shows nonmonotonic behavior. In the high temperature range the effect of magnetic field is less significant on ${\sigma }_{el}/T$. Right plot: variation of normalized Hall conductivity ${\sigma }_H/T\equiv {\sum }_a{q}_a^2{L}_{2_a}/T$ with temperature for nonvanishing magnetic field. With increasing magnetic field generically Hall conductivity increases for the range of magnetic field considered here.

Figure 8

Left plot: variation of ${\mathcal{I}}_{31}^B/T^2$ with temperature at finite baryon chemical potential for nonvanishing values of magnetic field. With increasing magnetic field ${\mathcal{I}}_{31}^B/T^2$ increase with respect to its value for vanishing magnetic field. Right plot: variation of ${\mathcal{I}}_{42}^B/T^2$ with temperature at finite baryon chemical potential for nonvanishing values of magnetic field. With magnetic field ${\mathcal{I}}_{42}^B/T^2$ first increases, then it decreases for higher values of magnetic field. Also for higher values of magnetic field it shows a nonmonotonic variation with temperature for the range of temperature considered here.

Figure 9

Left plot: variation of normalised Nernst coefficient ($NB$) with temperature and magnetic field. With temperature $NB$ decrease and $NB$ increases with increasing magnetic field. However in the high temperature range the effect of magnetic field on $NB$ is less significant. Right plot: Variation of $NB$ with baryon chemical potential ${\mu }_B$ for nonvanishing magnetic field. With baryon chemical potential $NB$ decreases.