Transport Properties of Superfluid Dipolar Fermi Gas at Low Temperatures ()
1. Introduction
Fermi gases of dipolar particles present a different physical picture. Being magnetically polarized, these particles interact via anisotropic dipole-dipole forces. The potential dipole-dipole interaction is partially repulsive and partially attractive. These features of the dipole-dipole interaction have very important consequences not only for the scattering properties of the particles in ultra cold gas, but also for the stability of the system and for the variety of its properties. The recent success in observing quantum degeneracy in ultra cold atomic Fermi gas [1] -[4] stimulates a search for gaseous Fermi systems with an achievable temperature of super fluid phase transition, which is generally very low.
Fermionic superfluidity was obtained in 2003 and allowed for the exploration of BEC-BCS crossover physics, a theoretical scenario bridging the gap between the Bardeen-Cooper-Schrieffer mean-field theory describing the behavior of weakly attracting fermions (scattering length a small and negative), and the strongly attractive regime (a small and positive) where the system behaves as a Bose-Einstein condensate of tightly bound dimers. Remarkably, although the theoretical foundations of this crossover physics had been laid in the early 1980s by the pioneering works of Nozières, Schmitt-Rink and Leggett [5] [6] , its first experimental confirmation was only made possible by the possibility of tuning interactions in ultra-cold atom vapors using Fano-Feshbach resonances [7] -[8] .
Braby et al. [9] have computed the thermal conductivity and sound attenuation length of a dilute atomic Fermi gas in the framework of kinetic theory. Above the critical temperature for superfluidity, 
, the quasiparticles are fermions, whereas below
, the dominant excitations are phonons. They have calculated the thermal conductivity in both cases. They have found that at unitarity the thermal conductivity 
in the normal phase scales as
.In the superfluid phase they have found
.
The shear viscosity and the second viscosity of superfluid 
have been formulated in the scope of Boltzmann equation approach [10] [11] . The value of the shear viscosity decreases as 
with temperature; and the second viscosity of superfluid 
is temperature independent in the limit of low temperatures, and in the near transition temperature behaves as inverse of the gap temperature. Theoretical study show that a gas of spin-polarized atomic 
becomes superfluid at densities and temperatures comparable with those at which the Bose-Einstein experiments are performed [12] .
The shear viscosity tensor of the 
-phase of superfluid 
is calculated at low temperatures and melting pressure, by using the Boltzmann equation approach. They obtained that the elements of the shear viscosities
, 
and 
are proportional to 
[13] [14] . Shahzamanian [15] calculated the components of the diffusive thermal conductivity of superfluid 3He-A using approximate collision integrals at low temperatures. The results are 
and
.
Transport coefficients of the superfluid p-wave Fermi gas with weak interaction were calculated using the Boltzmann equation approach and the quasiparticle relaxation rate [16] . They have found that the superfluid Fermi gases under their considerations are similar to
.
Diffusive thermal conductivity tensor of p-wave superfluid was calculated using the Boltzmann equation approach at low temperatures [17] . They have shown that their results are in good agreement with 
-phases of superfluid 
at low temperatures [18] .
Here we present a calculation of the transport properties of a superfluid dipolar Fermi gas, based on the derivations of Abrikosov and Khalatnikov [19] (shear viscosity and diffusive thermal conductivity). We used a sufficiently high magnetic field so that all quasiparticles with spin up will go to superfluid state. Important collision processes are the binary scattering of quasiparticles, the decay processes in which one Bogoliubov quasi particle decays into three and coalescence processes in which three Bogoliubov quasi particles coalesce to produce one. We obtained the transition probabilities at low temperature. By using the Boltzmann equation, we calculated shear viscosity and diffusive thermal conductivity tensors.
2. Transition Probabilities
The interaction between quasiparticles in the superfluid dipolar Fermi gas was found by performing a Bogoliubov transformation on the normal state interaction:
(1)
The Hamiltonian interaction contains the terms
, 
, 
, 
and
. The last two processes are not allowed because in each process the total energy should be conserved.
At low temperatures Bogoliubov coefficients can be written as 
and
, because more particles are gathered in the nodes of gap, that gap in direction 
on the Fermi surface is
, where 
is the maximum gap and 
is the angle between the quasiparticle momentum and gap axis
, which is supposed to be in the direction of z-axis, in this region of temperature 
[20] . Using the bogoliubov transformation, transition probabilities in polarized Fermi gas are:
(2)
The interparticle interaction in polarized dipolar gases that the dipoles are polarized along the z-axis reads,
(3)
where 
is magnetic dipole, and 
is the angle between the momentum 
and the z-axis [21] . Choosing 
along the z-axis, 
is the angle between 
and
, and 
is the angle between the plane of 
with
. The relations 
and 
according to the momentum conservation are [22]
(4)
(5)
At low temperatures more particles are contributing in contacts about Fermi momentum [19] . By replacing Equations (3)-(5) into (2) and considering the fact that the size of momentums are equal to Fermi momentum and by neglecting the potential effects of trap in Fermi momentum, transition probability is obtained in the form below:
(6)
3. Collision Integral
The Boltzmann transport equation reads
(7)
where 
is the collision integral. The full distribution function is given by
. where 
and 
is a small departure from equilibrium. We wrote the deviation from equilibrium as:
(8)
by using (8) in (7) we got [23]
(9)
The collision integral is form [23]
(10)
here
,
,
.
We expressed 
as a series of spherical harmonics in the polar angles
,
. Thus
(11)
by inserting (11) in (10) and integrate over y we have
(12)
where
(13)
We divided 
in to even and odd functions of
, and in the same way separated
. Since
, only the even part of 
contributes to the even part of
, and similarly for the odd part [24] .
(14)
(15)
that an even function of 
used for calculating of shear viscosity and odd function used for thermal conductivity [24] .
4. Shear Viscosity
The coefficients shear viscosity describes the response of the momentum current density to a transverse velocity field
(16)
is expressed in terms of the bogoliubov qusipartcle distribution function 
by
(17)
to solve the linearized Boltzmann equation, it is suitable to define 
as
(18)
By substituting (18) in (9) and then in (17) and compared with (16), we got [24]
(19)
By solving the even part of 
in (15) and with regard to the fact that system is polarized, we have [24] ,
(20)
in this equation
, 
and 
are
(21)
(22)
(23)
and 
is Euler,s constant, 
is a digamma function and
,
(24)
only replaces with
, because 
and 
are nearly zero.
For obtaining 
and
, we used (6) and note that 
is small for superfluid case and its maximum value is 
[20] , where maximum gap, 
, due to strong coupling effects is equal to 
[25] . Then, we obtained:
, (25)
(26)
(27)
After substituting 
, 
, and 
in (20) and taking the angular integrations, we have
(28)
(29)
(30)
(31)
(32)
5. Thermal Conductivity
The diffusive thermal conductivity tensor 
is defined by
(33)
where 
is the diffusive heat current. It is noted that, in addition to heat transfer by a random diffusive process of thermal excitations described by Equations (33), there is a convective contribution to the heat current in a superfluid, even in the absence of mass flow, owing to the possibility of normal-superfluid counter flow.
At low temperatures, this contribution is negligible [24] -[26] and the measurement of the components of thermal conductivity at these temperatures is more meaningful than the measurement of the thermal conductivity components in the vicinity of
.
The diffusive heat current may be written in terms of the polarized quasi-particle distribution functions:
(34)
is defined at below[26]
(35)
By using (35) in (34) and compared with (33) and using odd function of 
in (15) for polarized gas we got [24]
(36)
By using Sykes and Brooker procedure we defined 
as bellows
(37)
and 
are introduced as below
(38)
(39)
where 
and 
are defined by (24).
By using (6) in (38), (39) we took the following explicit forms
(40)
(41)
By using the Sykes and Brooker procedure [24] , we calculated the diffiusive thermal conductivity tensor. After substituting
,
, and 
in (37), and taking the angular integrations, we got:
(42)
(43)
(44)
6. Discussion and Concluding Remarks
As a result, in a superfluid dipolar Fermi gases we calculated the transition probabilities for the cases where the Bogoliubov quasiparticles are presented in coalescence, decay and binary processes and obtained that binary processes are dominated in calculation of transition probabilities, and other processes are nearly zero. Then, by using collision integral in the Boltzmann equation contained binary processes, the components of shear viscosity and thermal conductivity were calculated. We obtained
;
; and 
with temperatures dependences
;
; and 
respectively (see Equations (28)-(32)); and we have found the components of the diffusive thermal conductivity, 
with temperature dependency 
and 
which is proportional to 
and other components which are zero. Temperature dependency of shear viscosity and thermal conductivity along the z-axis (
and 
respectively) were showing that the polarization is along the z-axis.
Then, we had to compare our results in transport coefficients with [13] [14] . (They found the shear viscosities 
and 
of A1-phase of superfluid 3He are proportional to (T/Tc)-2.). Two major differences were shown: a) Transition probabilities in superfluid dipolar Fermi gas are the function of temperature, but in superfluid 3He-A1 are constant. b) 3He-A1 have two kind of the quasiparticles, one for Bogoliubov quasiparticle, and another for normal quasiparticle. The wave function of the system in A1 phases of 3He is 
that interacts with Bogoliubov quasipaticle 
and normal quasiparticle
, but in superfluid dipolar Fermi gas only 
contributes to Bogoliubov quasiparticles 
in interaction.
It is noted that, in superfluid Fermi gas in p-wave state
, 
were contributed to calculations [16] [17] but in superfluid dipolar Fermi gas only 
with temperature dependency contributes to calculations.
Also, in [15] the components of thermal conductivity tensor of superfluid 3He-A were calculated using approximate collision integrals at low temperatures, and the thermal coefficients 
and 
have been obtained with temperature dependences 
and
, respectively. This is due to the fact that 
was taken to be a constant parameter with respect to temperatures.
In [18] , 
, 
and 
were contributed in calculations and the wave function 
interacted with Bogoliubov and normal quasiparticles, but in superfluid dipolar gas only 
contributed to calculation and only 
interacted with Bogoliubov quasiparticles. These differences in transition probabilities of 
and superfluid Fermi gas in p-wave state with superfluid dipolar Fermi gas, have shown themselves in transport coefficients.