_{1}

^{*}

**We compute the shear viscosity of superfluid Bose and Fermi gases on the base of Boltzmann equation and rela****xation times. We show that, in the low temperature limit, the shear viscosities of Bose and Fermi gases are pro****portional to T^{-1}**

*e*^{vp0/T }and**T**

^{-4}, respectively. For the superfluid Bose gas at low temperature limit, only splitting processes contribute to the shear viscosity.A Bose-Einestein Condensation (BEC) whose particles interact via dipole-dipole forces constitutes an example of a superfluid with anisotropic inter-particle interactions. The action has been predicted to lead to BECs with unusual stability properties [

In this article, we explore the shear viscosity of a superfluid dipolar gas. Recent experimental development in trapping and cooling of polar molecules [14,15] has shown that the dominant interactions are dipole-dipole interactions. We calculate the shear viscosity by using the Boltzmann equation. It should be noted that the Boltzmann equation is not appropriate for long-range potential like coulomb potential, , whereas it is suitable for weak, short-ranged potentials like contact and probably dipole-dipole potentials [

The rest of the article is laid out as follows: Section 2 will discuss the basic equations relating the shear viscosity to the phonon collision operator. The transition probability is derived in Section 3, and then g(p) is obtained for the calculation of the viscosity. The discussion of the result is presented in Section 4.

Shear viscosity can be defined simply as below,

Viscosity is related to stress tensor, which is the deviation from equilibrium of the stress-energy tensor for a fluid with pressure and energy density

where

the ellipsis part of is related to bulk viscosity and thermal conductivity. is the fluid velocity at a given position and time. At low temperature, a linear dispersion relation for dipolar Bose gas is the phonons:

The stress-energy tensor and the viscosity can be calculated by using kinetic theory [

where is the distribution function of the phonons with speed, momenta, and energy We work in units where. The full distribution function is given by

where is the Bose-Einstein distribution and is a small departure from equilibrium. We write the deviation from equilibrium as

where

and

By substituting Equation (7) in to Equation (5) and Equation (2), we find

On the basis of symmetry consideration and by using the definition of [see Equation (3)], we can write [Equation (2)] in the following form

Then, by contracting the tensor on the left-hand side with respect to the pairs of indices (and) we can determine the shear viscosity in terms of the function g(p),

The process now is to evaluate from Equation (11), we need to find a form for g(p) in two cases: dipolar Bose and Fermi dilute gas at low temperature limit. For the former case, we follow the approach of Eckern [

To calculate the first case, we can use the Boltzmann equation given in the absence of external forces by

the left-hand side can be written as [

where only the contribution relevant for shear viscosity was retained in the linear response approximation, leading order in the small deviation from equilibrium. Thermal gradients and bulk flows would give an additional term on the right-hand side of Equation (13).

The collision operator should contain any possible collision terms that are typically considered: 1) binary collisions (2↔2) in which the number of particles is conserved, and 2) splitting processes (1↔2) in which the number of particles is not conserved.

Eckern [

We show in the following that even with this scattering amplitude, the binary collisions do not contribute to the collision operator and the dominated ones are the splitting processes at low temperature limit.

The collision integral is written as [

where the scattering amplitudes are [

and the Fourier transform of the dipole-dipole interaction:

Manuel et al. [

By using the definition of collision integral in Equation (13) and Equation (14) and using the ansatz in Equation (7) for and the linear dispersion relation, we get

by inserting Equation (16) in Equation (11), we get finally

The explicit expression for the shear viscosity of a superfluid dipolar Bose gases is given by Equation (17). The temperature dependence of the shear viscosity coefficient in normal dipolar Bose gas is proportional to [

In the case of the phonon interaction for the unitarity gas in the superfluid Fermi gas, the phonon cross section has been calculated by Rupak and Schafer [

with. Equations (18), (19) and (11) finally gives

We have calculated the shear viscosity arising from dipolar interaction in a dipolar Bose and Fermi gas system. At low temperatures, the linear dispersion relation for dipolar gas is phonon, and the viscosity is dominated by phonons in superfluid dipolar gas. For dipolar Bose gas, the basis of the calculation is the linearized Boltzmann equation. By using small-angle processes, we obtain that at low temperatures, only splitting processes are dominating in the collision integrals. However, this result is contrary to the case of dilute Fermi gases. Rupak and Schafer [

We obtain for dipolar Bose gas and the temperature dependence of the shear viscosity as (see Equation (17)).

Experimental results [

At the leading order in the polynomial expansion and variational calculation, the temperature dependence of shear viscosity in superfluid Fermi gas in the unitarity limit is observed as T^{−5} by Rupak and Schafer [^{−4} on the basis of damping relaxation time.