Introduction
Plasma physics was considered to be a purely classical field. With time, there has been a curiosity on plasma systems where quantum effects in degenerate plasma are important. When we compare the de Broglie wavelength of the charge carriers with the inter-particle distance than there is a significant overlap of the solitary wave functions. This problem is solved by Fermi-Dirac statistics which is in contrast to the laboratory and space plasmas which obey Maxwell-Boltzmann statistics. The fermionic behaviour becomes significant for dense plasmas. This kind of plasma exists in astrophysical objects such as in white dwarfs and the atmosphere of neutron stars or in intense laser-solid density plasma interaction experiments, etc.
Degenerate Plasma
Degenerate plasma usually exists at high densities. They have subatomic particles with half-integral spin. A gas in which all the energy states below a critical value are filled with the increase in density is called a fully degenerate, or zero-temperature fermion gas. Such particles as electrons, protons, neutrons, and neutrinos are all fermions and obey Fermi-Dirac statistics.
Degeneracy Pressure
The process of gradually filling in the higher-energy states increases the pressure of the fermion gas which is as termed as degeneracy pressure. They depend only on the fermion number density, but not on its temperature. The degenerate pressure arises due to the combined effect of Pauli’s exclusion principle and Heisenberg’s uncertainty principle.
Solitons In Degenerate Plasma
A soliton is a self-reinforcing solitary wave that maintains its shape while it travels at a constant speed. They are wave packets that can be propagated as a travelling wave in nonlinear systems. They do not obey the superposition principle and does not dissipate. The soliton phenomenon was first described by John Scott Russell who observed a solitary wave in Scotland. The non-linear propagation of these waves in degenerate dense plasma has been theoretically investigated by Hass, Misra, Samanta and many more. For this, the K-dV equation has been derived by using the reductive perturbation method and by taking into account the effect of different plasma parameters in plasma fluid. The derived K-dV equation is examined to identify the basic properties of solitary structures. The effect of pressures on electrons, ions, and positrons modify the basic features of solitary waves that are found to exist in nature.
History
In 1873, Johannes Diderik van der Waals developed a new model of dense gases and fluids. This work opened the way to understanding matter in non-ideal states which are not described by the classical ideal gas law. After that, there has been a great deal of interest in understanding the basic properties of matter. There are many interstellar compact objects where matters support themselves against gravitational collapse by cold degenerate fermion/electron pressure. They are of two categories:
• The first category is close to a dense solid which is surrounded by degenerate electrons, and possibly other heavy particles or dust. The example of this kind of star is a white dwarf which is supported by the pressure of degenerate electrons.
• The second category is close to a giant atomic nucleus which is a mixture of interacting nucleons and electrons and other heavy elementary particles or dust. The example of this kind of stars is a neutron star which is supported by the pressure due to a combination of nucleon degeneracy and nuclear interactions.
These unique states of matter occur by significant compression of the interstellar medium.
The degenerate plasma number density in such a compact object is so high of the order of 1030 cm−3 in dwarfs and 1036 cm−3 in neutron stars. The equation of state for degenerate electrons in such interstellar compact objects was mathematically explained by
Chandrasekhar for two limits, namely non-relativistic and ultra-relativistic limits. He built the concept first on the mass density, gravitational equilibrium, pressure density and the equations were solved for the electron-proton system. The degenerate electron equation of state of Chandrasekhar is Pe ∝ N5/3 for non-relativistic limit and Pe ∝ N4/3 for ultra-relativistic limit, where Pe is the degenerate electron pressure and Ne is the degenerate electron number density. After that many scientists use these limits to make their theoretical investigations. So, these interstellar compact objects provide us cosmic laboratories for studying the properties of the matter as well as solitons and instabilities at degenerate state for which quantum, as well as relativistic effects, become important Further, these effects on linear and non-linear propagation of electrostatic and electromagnetic waves have been investigated by using the quantum hydrodynamic (QHD) model which is an extension of the classical fluid model in plasma, and by using the quantum magneto-hydrodynamic (QMHD) model which involve half spin and one-fluid MHD equations.
Explanation In Research papers
The behaviour and characteristics of solitary waves of different kinds with special conditions are discussed below :
1) Planar and Non-planar Solitary Waves in a Four-Component Relativistic
Degenerate Dense Plasma
The nonlinear propagation of electrostatic perturbation modes in an un-magnetized, collisionless, relativistic, degenerate plasma, which contain both non-relativistic and ultra-relativistic degenerate electrons, non-relativistic degenerate ions, and arbitrarily charged static heavy ions has been investigated theoretically. The Korteweg-de Vries (K-dV) equation has been derived.
Their solitary wave solution is obtained and numerically analyzed in case of both planar and non-planar geometry. It has been observed that the ion-acoustic (IA) and modified ion-acoustic (mIA) solitary waves have been significantly changed. IA waves are low-frequency electrostatic waves. The modified ion-acoustic (mIA) waves are nothing but IA-type waves in the presence of static heavy-ion. We have found that:
• The phase speed with negatively (positively) charged heavy ions for the ultra-relativistic case is higher (lower) than that for the nonrelativistic case.
• With the decrease in time the amplitude of the solitary wave in both geometries increase.
• In spherical case, the amplitude is always higher than cylindrical geometries for K-dV solitons which indicates that the density compression can be more effectively obtained in a spherical.
This work is applicable for the matter under extreme conditions like IA and mIA solitary waves propagation in the interior of interstellar stellar for polytropes, hadronic matter, quark-gluon plasma and proton-neutron stars where planar or non-planar geometry comes.
2) Solitary waves in an ultra-relativistic degenerate dense plasma
Solitary waves in an ultra-relativistic degenerate dense plasma have been investigated by the reductive perturbation method. The modified Korteweg–de Vries equation has been derived and its numerical solutions have been analyzed to identify the basic features of spherical electrostatic solitary structures that may form in such a degenerate dense plasma. It has been shown here that the amplitude, width, and speed increase with the increase of the plasma number density.
This investigation will help understand the basic features of the localized electrostatic disturbances in compact astrophysical objects, for example, white dwarf stars which have spherical shapes.
3) Standing electromagnetic solitons in degenerate relativistic plasmas
The existence of standing high frequency electromagnetic (EM) solitons in a fully degenerate under dense electron plasma is studied by applying relativistic hydrodynamics and Maxwell equations. The possibility of existence and stability of solitons in underdense plasma occurs when ω ≤ Ωe. Soliton exists for the entire range of physically allowed electron densities, i,e for n0 = 1024 cm−3 and higher. They are found in both relativistic and nonrelativistic degenerate plasmas. The intensity of the solitons can be small for ω → Ωe and becomes relativistically strong for ω → ωc.
This model is generalized for underdense plasma. These results are used to understand X-ray pulses emerging from compact astrophysical objects. It also tells us about the interaction of intense laser pulses and dense degenerate plasma.
4) Ion-acoustic solitary waves in dense pair-ion plasma containing degenerate electrons and positrons
The nonlinear propagation of ion-acoustic solitary waves in a collisionless dense electron-positron–ion plasma is probed in this. The electrons and positrons follow the Thomas–Fermi density distribution and the ions are described by the hydrodynamic equations. An energy balance-like equation involving a Sagdeev-type pseudo-potential is derived in this.
Analytical and numerical calculations reveal that both subsonic and supersonic ion-acoustic solitary waves may exist for low values of the positron-to-electron number density ratio. However, any increase in the positron-to-electron number density ratio allows the propagation of subsonic solitons only. The pseudopotential and pulse excitation character depends upon the density ratio and Mach number.
The method helps understand the excitation of nonlinear ion-acoustic solitary waves in a degenerate plasma such as in super-dense white dwarfs.
Conclusion
If we summarized the above explanation we can say that solitons in degenerate plasma are present in relativistic, ultra-relativistic and non-relativistic regimes. They have high density and a strong relation occurs between degenerate plasma and solitons. The degenerate plasma supports compressive or rarefactive solitary structures. Without the study of solitons in degenerate plasma, we are unable to understand the astrophysical aspects of nature completely.
Besides, its applications to diverse systems ranging from nanoscale electronic devices and dense astrophysics environments to intense laser-solid density plasma interaction experiments, this is a significant step in justifying the attention to quantum plasmas with the development of efficient macroscopic models.
References
1) Melrose, D., 2008. Quantum Plasmadynamics. Berlin: Springer.
2) Masood, W. and Eliasson, B., 2011. Electrostatic solitary waves in a quantum plasma with relativistically degenerate electrons. Physics of Plasmas, 18(3), p.034503.
3) Hossein, M., Nahar, L. and Mamun, A., 2014. Planar and Nonplanar Solitary Waves in a Four-Component Relativistic Degenerate Dense Plasma. Journal of Astrophysics, 2014, pp.1-8.
4) Mamun, A. and Shukla, P., 2010. Solitary waves in an ultrarelativistic degenerate dense plasma. Physics of Plasmas, 17(10), p.104504
5) Mikaberidze, G. and Berezhiani, V., 2015. Standing electromagnetic solitons in degenerate relativistic plasmas. Physics Letters A, 379(42), pp.2730-2734.
6) Abdelsalam, U., Moslem, W. and Shukla, P., 2008. Ion-acoustic solitary waves in a dense pair-ion plasma containing degenerate electrons and positrons. Physics Letters A, 372(22), pp.4057-4061
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