Hall and Normal component of electrical conductivity in ... · For Normal component For Hall...
Transcript of 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
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Drude’s model in solid state physics
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Resistivity Matrix:
Electrical Conductivity Matrix:
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Classical Hall Effect:
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Quantum Hall Effect:
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Now, let us go to Relativistic decription of Drude’s model:
First, without magnetic field case
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Microscopic Kinetic Theory of Ellectrical current density
Macroscopic (Ohm’s law)
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Now, let us go to Relativistic decription of Drude’s model:
With magnetic field case
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Relativistic Boltzmann Equation in presence of magnetic field in RTA method:
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Unknown constants:
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Hall and Normal conductivity of relativistic fluid :
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Massless Analytic expression (B=0) :
Massless Analytic expression at finite B :
For Normal component For Hall component
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Graphical representation of massless expression at finite B :
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Phys. Rev. E 84, 011905 (2011)Phys. Rev. E 84, 011905 (2011)
Massless results at finite B :
Interesting imilarity found in Bio-direction :
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Shear viscosity is another important transport coefficients like electrical conductivity
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Roy, Chaudhuri & Mohanty PRC (2012)
Hydrodynamical Simulations
Shear viscosity inversely measure the interaction strength of the matter
Strongly Interacting Matter
Expecting : W
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Momentum transfer
Distant between quarks
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● 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
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Macro Micro
Relativistic Boltzmann Equation
Kinetic Theory (Relaxation T ime Approximation)
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Viscous (shear only) stress tensors at finite B :
Macro Micro
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Relativistic Boltzmann Equation in presence of magnetic field in RTA method:
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Normal component
Hall component
Get back B=0 results:
Shear viscosity components in presence of magnetic field:
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Dynamical structure
Green-Kubo Relation of Transport coefficients
Static Limit
Shear ViscosityShear Viscosity
Bulk ViscosityBulk Viscosity
Thermal Conductivity
Thermal ConductivityTr
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Energy-momentum tensor & conserved current
Dynamical structure
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Velocity gradientVelocity gradient
Temperature gradientTemperature gradient
Classical Picture Picture of QFT at finte temperature
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Th
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Energy-momentum tensor
Lagrangian densityFermion Field
Boson Field
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Fermion Self-energyin Medium
Boson Self-energyin Medium
Degeneracy Factor Thermal Width
Pauli Blocked/Bose enhanced
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Landau Cuts
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