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

● 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......................

● 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