hall and normal component of electrical conductivity in ... · for normal component for hall...
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Hall and Normal component of electrical conductivityin presence of magnetic field
Sabyasachi Ghosh (IIT Bhilai)
Jayanta DeyJayanta Dey Sarthak SatapathySarthak Satapathy Prashant MurmuPrashant Murmu
Drude’s model in solid state physics
Resistivity Matrix:
Electrical Conductivity Matrix:
Classical Hall Effect:
Quantum Hall Effect:
Now, let us go to Relativistic decription of Drude’s model:
First, without magnetic field case
Microscopic Kinetic Theory of Ellectrical current density
Macroscopic (Ohm’s law)
Now, let us go to Relativistic decription of Drude’s model:
With magnetic field case
Relativistic Boltzmann Equation in presence of magnetic field in RTA method:
Unknown constants:
Hall and Normal conductivity of relativistic fluid :
Massless Analytic expression (B=0) :
Massless Analytic expression at finite B :
For Normal component For Hall component
Graphical representation of massless expression at finite B :
Phys. Rev. E 84, 011905 (2011)Phys. Rev. E 84, 011905 (2011)
Massless results at finite B :
Interesting imilarity found in Bio-direction :
Shear viscosity is another important transport coefficients like electrical conductivity
Elli
ptic
flo
w
Roy, Chaudhuri & Mohanty PRC (2012)
Hydrodynamical Simulations
Shear viscosity inversely measure the interaction strength of the matter
Strongly Interacting Matter
Expecting : W
eakly-
interacting
Momentum transfer
Distant between quarks
Ru
nn
ing
co
up
ling
co
nst
ant
of
QC
DQuarks are
asymptoticallyfree
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● Muronga, Phys. Rev. C 69, 044901 (2004).
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● Itakura, Morimatsu, Otomo, Phys. Rev. D 77, 014014 (2008).
● S. Sarkar, R. Sharma, Phys.Rev. D96 (2017) 094025 (2016)
● S. Banik, R. Nandi, D. Bandyopadhyay (2011)
● S. Pal, Phys. Lett. B 684, 211 (2010).
● M. Kurian, S. Mitra, S. Ghosh, V. Chandra, Eur. Phys. J. C79 (2019) 134.
● R. Lang, N. Kaiser, and W. Weise Eur. Phys. J. A 48, 109 (2012).
● S. Mitra, U. Gangopadhyaya, S. Sarkar, Phys. Rev. D 91, 094012 (2015).
● N. Demir and S. A. Bass, Phys. Rev. Lett. 102, 172302 (2009).
● G. P. Kadam, H. Mishra, Nucl.Phys. A 934 (2014) 133;
Phys. Rev. C 92, 035203 (2015).
● ................List is not small......................
● Dobado and Santalla, Phys. Rev. D 65, 096011 (2002).
● Muronga, Phys. Rev. C 69, 044901 (2004).
● Fernandez-Fraile and Gomez Nicola, Eur. Phys. J. C 62, 37 (2009).
● Itakura, Morimatsu, Otomo, Phys. Rev. D 77, 014014 (2008).
● S. Sarkar, R. Sharma, Phys.Rev. D96 (2017) 094025 (2016)
● S. Banik, R. Nandi, D. Bandyopadhyay (2011)
● S. Pal, Phys. Lett. B 684, 211 (2010).
● M. Kurian, S. Mitra, S. Ghosh, V. Chandra, Eur. Phys. J. C79 (2019) 134.
● R. Lang, N. Kaiser, and W. Weise Eur. Phys. J. A 48, 109 (2012).
● S. Mitra, U. Gangopadhyaya, S. Sarkar, Phys. Rev. D 91, 094012 (2015).
● N. Demir and S. A. Bass, Phys. Rev. Lett. 102, 172302 (2009).
● G. P. Kadam, H. Mishra, Nucl.Phys. A 934 (2014) 133;
Phys. Rev. C 92, 035203 (2015).
● ................List is not small......................
● Arnold, Moore, Yaffe JHEP (2003) 05051.● Arnold, Moore, Yaffe JHEP (2003) 05051.
List of microscopic calculations of Shear viscosity/entropy density
KSS bound=1/(4pi)
HTL > 2HTL > 2
Weakly interacting GasWeakly interacting Gas
Strongly Interacting
Macro Micro
Relativistic Boltzmann Equation
Kinetic Theory (Relaxation T ime Approximation)
Viscous (shear only) stress tensors at finite B :
Macro Micro
Relativistic Boltzmann Equation in presence of magnetic field in RTA method:
Normal component
Hall component
Get back B=0 results:
Shear viscosity components in presence of magnetic field:
R. K
ub
o,
J. Ph
ys. So
c. Jpn
.
12, 570 (1957).R
. Ku
bo
,
J. Ph
ys. So
c. Jpn
.
12, 570 (1957).
Dynamical structure
Green-Kubo Relation of Transport coefficients
Static Limit
Shear ViscosityShear Viscosity
Bulk ViscosityBulk Viscosity
Thermal Conductivity
Thermal ConductivityTr
ansp
ort
Co
effi
cien
ts
Th
erm
al C
ore
lato
rs
Energy-momentum tensor & conserved current
Dynamical structure
Op
erat
ors
Mo
men
tum
Tra
nsp
ort
Hea
t E
ner
gy
Tran
spo
rt
Velocity gradientVelocity gradient
Temperature gradientTemperature gradient
Classical Picture Picture of QFT at finte temperature
Th
erm
al C
ore
lato
rs
Energy-momentum tensor
Lagrangian densityFermion Field
Boson Field
Tran
spor
t Coe
ffic
ient
s
1/20, 1/2, 1/(6T)Speed of Sound Enthalpy per baryon
Fermion Self-energyin Medium
Boson Self-energyin Medium
Degeneracy Factor Thermal Width
Pauli Blocked/Bose enhanced
probabilities
Landau Cuts
q_0 ->0
qve
c