Steady Magnetohydrodynamic Equations for Quantum Plasmas

doi:10.4236/jmp.2012.312233

Paper Menu >>

Journal Menu >>

Journal of Modern Physics, 2012, 3, 1856-1857

http://dx.doi.org/10.4236/jmp.2012.312233 Published Online December 2012 (http://www.SciRP.org/journal/jmp)

Steady Magnetohydrodynamic Equations for

Quantum Plasmas

Muhammad Asif, Umar Bashir

Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan

Email: [email protected], [email protected]

Received September 30, 2012; revised October 30, 2012; accepted November 13, 2012

ABSTRACT

Steady Magnetohydrodynamic (MHD) Equations of force, density and energy for quantum plasmas have been derived.

These equations constitu te our Steady Magnetohydrodynamic model for quantum plasmas. All the quantum effects are

contained in the last term of quantum force equation and in the last three terms of quantum Energy Equation, so-called

Bohm potential and may be valuable for the description of quantum phenomena like tunneling.

Keywords: Steady Magnetohydrodynamic; Quantum Plasmas; Inhomogeneous Magnetic Field

1. Introduction

In classical plasma physics, fluid models are ubiquitous,

with their application ranging from astrophysics to con-

trolled nuclear fusion [1-3]. Magnetohydrodynamics, for

instance, provides one of the most useful fluid models,

focusing on the global properties of the plasma. The

purpose of work [4] is to produce a quantum counterpart

of magnetohydrodynamics, starting with the quantum

hydrodynamic model for charged particle systems. This

may provide yet another approach to the study of the

ways in which quantum physics can modify classical

plasma physics. However, it must be noted that the

quantum hydrodynamic model for charged particle sys-

tems was built for non-magnetized systems only [4]. In

order to arrive at a quantum modified magnetohydrody-

namics, work [4] sugg ests what the appropriate extension

would be of the quantum hydrodynamic model in cases

of nonzero magnetic field.

In this paper, we pr esent the Steady Magnetohydrody-

namic (MHD) Equations of force, density and energy for

quantum plasmas. These equations constitute our Steady

Magnetohydrodynamic model for quantum plasmas. All

the quantum effects are contained in the las t term of qu a n-

tum force equation and in th e last three terms of quantum

Energy Equation, so-called Bohm potential and may be

valuable for the description of quantum phenomena like

tunneling.

2. Steady Magnetohydrodynamic Equations

The Steady Magnetohydrodynamic (MHD) equations [5]

describe the plasma as a conducting fluid with conduc-

tivity which experiences electric and magnetic forces.

The fluid is specified by a mass density ρ, a flow velocity

V, and a pr essure P, which are all functions of space and

time. Here, we shall simply draw on the ordinary equa-

tions of hydrodynamics. The equation of motion has an

additional force per unit volume J due to the net mag-

netic force on the plasma particles. On fluid scales







 the plasma is overall charge neutral, so that the

net electric force is neglibly small. The MHD equations

represent the conservation of momentum, the conserva-

tion of mass and some equation of state (i.e., an energy

relationship). The simplified Steady Magnetohydrody-

namic Equations [5] are





VV PJB





 (1)









 (2)





0PV



 (3 )

There are a number of assumptions behind the above

equations: the energy equation is the adiabatic equation

of state with



being the ratio of specific heats (5





for a monatomic gas); there are other possible choices for

an energy equation such as isothermal , etc. Oh-

mic heating J.E has been ignored, i.e., assumed to be less

important than the plasma’s thermal energy; radiative or

other heat losses are also neglected. We have also as-

sumed that the pressure is isotropic which is not gener-

ally true when the magnetic field can impose different

kinds of motion on the particles parallel and perpendicu-

lar to the field direction. Any possible heat flux has been

ignored, and there is no relative flow between the differ-







M. ASIF, U. BASHIR 1857

ent species in the plasma, so that the one fluid description

is valid. While the conductivity is usually large due to the

mobile electrons, the absence of collisions has the oppo-

site impact on other transport processes such as viscosity,

which w e h av e also therefore neglected.

3. Steady Quantum Force Balance Equation

The Steady force balance equation for quantum plasma is

obtaine d by putti ng the pressure P gi ven by [4,6] as

koQ

Pp p p



pp pn

(4)

Since the kinetic and osmotic pressures measures the

kinetic and osmotic velocity dispersions, we can write

from [4,6]

(5)

Equation (5) depend only on the density, we obtain

from [4,6]



Ppn













 (6)

Using Equations (5) and (6), the Steady force balance

Equation (1) for quantum plasma is obtained as



VV pJB



  













0VV



 

(7)

The derived Steady quantum force Equation (7) con-

stitute our quantu m hydrodynamic model for magnetized

systems. All the quantum effects are contained in the last

term of Equation (7), the so-called Bohm potential and

may be valuable for the description of quantum phe-

nomena like tunneling.

4. Steady Quantum Density Equation

The density Equation (2), for quantum plasmas is



 (8)

This is Steady density equation, which does not con-

tained the quantum effects.

5. Steady Quantum Energy Equation

Using Equations (5) and (6), the Steady energy Equation

(3), for quantum plasmas is obtained as





ie ie

mm mm













 













 





(9)

The derived Steady Energy Equation (9) constitu te our

quantum hydrodynamic model for magnetized systems.

All the quantum effects are contained in the last three

terms of Equation (9), the so-called Bohm potential and

may be valuable for the description of quantum phe-

nomena like tunneling.

In comparison with standard fluid models for charged

particle systems, the Bohm potential is th e only quantum

contribution. In order to have a deeper understanding of

the importance of quantum effects, we compare Equa-

tions (7)-(9) with the rescaling of ideal quantum magne-

tohydrodynamic Equation (46) of [4], and obtained a non-

dimensional parameter 22



, measuring the

relevance of quantum effects. Where A

V is a Alfvén

velocity and c



is the cyclotron velocity. While for

ordinary plasmas H is n egligible, for dense astrophysical

plasmas, H can be of the order unity or more. Hence, in

dense astrophysical plasmas [7] such as the atmosphere

of neutron stars or the interior of massive white dwarfs,

quantum corrections to magnetohydrodynamics can be

experimentally important.

6. Conclusion

Steady Magnetohydrodynamic (MHD) Equations of force ,

density and energy for quantum plasmas have been de-

rived. The Steady force, density and Energy equations

constitute our reduced quan tum hydrodynamic model for

magnetized systems. All the quantum effects are con-

tained in the last term of force Equation (7) and in the

last two terms of Energy Equation (9), so-called Bohm

potential and may be valuable for the description of

quantum phenomena like tunneling.

REFERENCES

[1] D. R. Nicholson, “Introduction to Plasma Theory,” Wiley,

New York, 1983.

[2] J. A. Bittencourt, “Fundamentals of Plasma Physics,” Na-

tional Institute for Space Research, São José dos Campos,

1995.

[3] M. Asif, X. Gao, J. Li, et al., “Plasma Density Behavior

with New Graphite Li mite rs in the He fei T oka mak-7,” Phys-

ics of Plasmas, Vol. 12, No. 8, 2005, Article ID: 082502.

doi:10.1063/1.1995627

[4] F. Haas, “A Magnetohydrodynamic Model for Quantum

Plasmas,” Physics of Plasmas, Vol. 12, No. 6, 2005, Arti-

cle ID: 062117.

[5] S. J. Schwartz, C. J. Owen and D. Burgess, “Astrophysi-

cal Plasmas Astronomy Unit, Queen Mary,” University of

London, London, 2004, p. 35.

[6] F. Haas, “An Introduction to Quantum Plasmas,” Brazil-

ian Journal of Physics, Vol. 41, No. 4, 2011, p. 349.

doi:10.1007/s13538-011-0043-0

[7] G. Chabrier, F. Douchin and A. Y. Potekhin, “Dense

Astrophysical Plasmas,” Journal of Physics: Condensed

Matter, Vol. 14, No. 40, 2002, p. 9133.

doi:10.1088/0953-8984/14/40/307