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We investigate the thermoelectric effect for baryon rich plasma produced in heavy ion collision experiments. We estimate the associated Seebeck coefficient for the hadronic matter. Using kinetic theory within relaxation time approximation we calculate the Seebeck coefficient of a hadronic medium with a temperature gradient. The calculation is performed for hadronic matter modeled by the hadron resonance gas model with hadrons and resonance states up to a cutoff in the mass as 2.25 GeV. We argue that the thermoelectric current produced by such effect can produce a magnetic field in heavy ion collision experiments.

Transport coefficients of strongly interacting matter under extreme conditions of temperature, density, and/or magnetic fields have been one of the most challenging interests in the field of strong interaction physics. In the context of relativistic heavy ion collision experiments (RHIC), these are important input parameters that enter in the dissipative relativistic hydrodynamics as well as transport simulations, that are being used to describe the evolution of the matter subsequent to a heavy ion collision. Indeed, a small shear viscosity to entropy ratio (

In the present work, we investigate another related coefficient relevant for high density heavy ion collision, namely, the thermoelectric behavior of the strongly interacting matter in heavy-ion collisions. The phenomenon in which a temperature gradient in a conducting material is converted to electrical current and vice versa is known as thermoelectric effect, which is also known as the Seebeck effect. The Seebeck effect in a conductor is a manifestation of the fact that when there exists a temperature gradient, the charge carriers would diffuse toward the region of lower temperature. This diffusion continues till the electric field generated by the motion of charge carriers becomes strong enough to stop this motion. The Seebeck coefficient is defined as the electric field produced in a conducting medium due to a temperature gradient when the electrical current is set to zero

In the present investigation, we study the Seebeck effect for hot and dense hadronic matter. It may be noted that in the usual condensed matter systems, the thermoelectric effect requires only a temperature gradient, as the ions in the lattice are stationary. On the other hand, e.g., in an electron-positron plasma, just having a temperature gradient is not enough to lead to any thermoelectric current. This will be similar in quark gluon plasma (QGP) with zero baryon density. However, the situation is different at finite baryon chemical potential, when the number of baryons and antibaryons are different. In the presence of a temperature gradient, there will be net thermoelectric current driven by the temperature gradient as there will be unequal number of positive and negative charge carriers. For the heavy-ion collisions at Facility for Antiproton and Ion Research (FAIR) at Darmstadt

The hadronic phase of the strongly interacting medium created in heavy ion collisions are well described in terms of the hadron resonance gas (HRG) model, at chemical freeze-out

This paper is organized as follows, in Sec.

We consider here the linearized Boltzmann equation in relaxation time approximation. For a linear problem or weak external fields, the Boltzmann equation can be interpreted as a linear expansion of the distribution function around the equilibrium distribution function, hence

Using Eqs.

Electric current density is defined as,

Note that due to the presence of the external force, the velocity will not be isotropic in general. But since the external force is considered to be small, the change in the velocity can be ignored. For later calculations, it is convenient to rewrite the momentum integration in Eqs.

The Seebeck coefficient

From Eq.

In this context, it may be relevant to note that, similar to the expression for electrical conductivity, the Seebeck coefficient in Drude picture has been estimated in Ref.

The central quantity in hadron resonance gas model is the thermodynamic potential which is given by

Let us note that, while the individual Seebeck coefficient is independent of the relaxation time, the Seebeck coefficient of the medium is not. In the following we therefore estimate the same. The relaxation time is defined as

Thus the thermal average relaxation time can be expressed by averaging the relaxation time over

The energy averaged relaxation time

As mentioned earlier for the hadron resonance gas model we shall include all the hadrons and resonances up to a mass cutoff

Variation of the electrical conductivity of

Next we discuss the Seebeck coefficient for a single species as given by Eq.

It is important to mention that the Seebeck coefficient of single species is independent of the relaxation time. We might mention here that the Seebeck coefficient has been estimated in condensed matter systems in the Drude limit. In this limit the Seebeck coefficient is also independent of relaxation time as is shown explicitly in Ref.

We next show the variation of the total Seebeck coefficient for the hadronic medium of Eq.

Behavior of Seebeck coefficient (S) of hadron resonance gas as a function of temperature and baryon chemical potential. We have used temperature range 80 to 160 MeV, because degrees of freedom of HRG model are hadrons and resonances. We have also taken the range of baryon chemical potential from 60 to 250 MeV. In the left plot we have shown the variation of Seebeck coefficient of hadron resonance gas with temperature for different values of chemical potential. In the right plot we have shown the variation of Seebeck coefficient with baryon chemical potential for various temperature. In this calculation we have taken into account all the hadrons and resonances having mass up to 2.25 GeV.

To understand the behavior of the Seebeck coefficient with baryon chemical potential, let us first note that for the temperature (

Left plot: Variation of

In a similar way one can understand the temperature dependence of Seebeck coefficient for the system of hadron resonance gas from the behavior of proton Seeback coefficient which is dominant in the sum given in Eq.

Left Plot: Variation of proton Seebeck coefficient as a function of temperature at

It might be relevant here to note that the thermoelectric current so produced due to the temperature gradient can generate a magnetic field in the heavy ion collision experiments. One can estimate an order of magnitude of the magnetic field produced due to this thermoelectric current. The magnitude of electric current density produced by the temperature gradient can be expressed as,

It may be seen in Fig.

In this work, we have attempted to study the thermoelectric effect of a thermalized hadronic medium with a temperature gradient. We have estimated the corresponding Seebeck coefficient of hot hadronic matter within hadron resonance gas model (HRG). Thermoelectric effect necessarily requires a temperature gradient which is achievable in heavy ion collision experiment due to the temperature difference in the central and peripheral part of the fireball produced in these collisions. One of the important outcomes of this calculation is that, for a baryon free plasma, contributions in total Seebeck coefficient due to the mesonic degrees of freedom cancel out. This happens because of the fact that each meson particle comes with its antiparticle with opposite charge leading to cancellation of the corresponding Seebeck coefficient for the charged mesons. However, in a baryon rich plasma, the contributions to the total Seebeck coefficient due to the baryons do not cancel out. The total Seebeck coefficient of thermalized hadron resonance gas increases with increasing temperature for fixed baryon chemical potential and increases with baryon chemical potential for fixed temperature. It is important to note that the formalism we are using is a nonrelativistic one. It will be important to study the thermoelectric effect in a relativistic formalism, particularly for QGP medium. Electrical current produced due to the temperature gradient can be a source of magnetic field. According to our crude estimate the strength of the transient magnetic field so generated can be

The idea discussed in the present investigation arose during a visit of one of the author’s (H. M.) to the research group of Prof. Ajit M. Srivastava at Institute of Physics Bhubaneswar. The authors would like to thank Ajit. M. Srivastava for suggesting the problem and members of his research group for subsequent extensive discussions. The authors would also like to thank Sabyasachi Ghosh, Abhishek Atreya, Aman Abhishek, Chowdhury Aminul Islam, Rajarshi Ray for many discussions during working group activities at WHEPP 2017, IISER Bhopal. We thank Guruprasad Kadam for useful comments on the manuscript.