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

ko

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]

2

ie

mm

2

2

Ppn

(6)

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

Equation (1) for quantum plasma is obtained as

2

ie

2

2

VV pJB

mm

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

2

22

22

ie ie

Vp

V

pV

mm mm

2

0

(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

c

A

H

mV

, 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.

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tional Institute for Space Research, São José dos Campos,

1995.

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

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doi:10.1063/1.1995627

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

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

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