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This study aims to estimate the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy in curvilinear coordinates. We used the results to calculate the quantum binary and triplet distribution functions in curvilinear coordinates. The analytical form of the quantum distribution functions was obtained for dusty plasma in Saturn’s rings model. We use particles-in-cell (PIC) simulations to find a visualization of dusty three-component plasma phase space in curvilinear coordinates. Our results were compared with others.

The quantum distribution functions are of great interest for understanding the properties of dusty plasma. In statistical mechanics, we can get the thermodynamic functions such as the internal energy, the osmotic pressure, and the excess free energy by using the distribution functions for the plasma particles. The importance of quantum distribution function in statistical physics is to give the particles number density in the phase space at time t.

Many authors have calculated the quantum distribution functions. Hussein and Hassan [

The calculation of the distribution functions in general form by using curvilinear coordinates makes it easier for the researchers to find the form of the distribution function in the spherical or cylindrical coordinates or any type of coordinates that facilitates the study of the physical problem. Błaszak and Domanski [

Dusty Plasmas plays an important role in experimental physics and in many astrophysical situations [

Dusty plasma was originally important in the field of astrophysics. Examples of astronomical dusty plasmas include planetary ring systems (rings of Saturn). The rings of Saturn are the most important models in space dusty plasma study [

Saturn’s ring system extends up to 175,000 miles (282,000 kilometres) from the planet, yet the vertical height is typically about 30 feet (10 meters) in the main rings [

Particle-in-cell (PIC) simulations are a useful tool in modelling plasma. The electron velocity distribution function and the plasma potential are found by particle-in-cell (PIC) simulations [

This work is aimed to calculate the quantum binary and triplet distribution functions of a dusty plasma in curvilinear coordinates. In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. Commonly used curvilinear coordinate systems include rectangular, spherical, and cylindrical coordinate systems. The calculation is based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [

The cluster expansion method consists of writing the binary distribution function as a power series in the density. The coefficients of different powers of the density involve integrals of the order of that power. These coefficients are then expressed as a sum of a product of integrals. The power series of the binary distribution function in the density convergence badly at high densities, many attempts have been made to overcome such difficulty.

Consider 3 Dimensions space with coordinates X = ( x 1 , x 2 , x 3 ) . A point p in 3d space can be defined using Cartesian coordinates or it can also be defined by its curvilinear coordinates ξ = ( ξ 1 , ξ 2 , ξ 3 ) . The relation between the coordinates is then given by the invertible transformation functions:

ξ s = ξ s ( x 1 , x 2 , x 3 ) , s = 1 , 2 , 3

x i = x i ( ξ 1 , ξ 2 , ξ 3 ) , i = 1 , 2 , 3 (1)

The surfaces ξ 1 = constant, ξ 2 = constant, ξ 3 = constant are called the coordinate surfaces. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates. The momentum operator in quantum mechanics is the gradiant operator p = ( ℏ / i ) ∇ . By defining the Jacobi matrix as:

j α β = ∂ x α ∂ ξ β , α , β = 1 , 2 , 3. (2)

The Jacobian of the transformation is the determinant of the Jacobi matrix

J ( ξ ) = det [ j α β ] (3)

Define the natural basis vectors:

h i = ∂ r ∂ ξ α , α = 1 , 2 , 3

r = ( x 1 , x 2 , x 3 ) (4)

The reduced s-particle density operators defined by Bogoliubov [

F s ( ξ 1 , P ξ 1 , ξ 2 , P ξ 2 , ⋯ , ξ N , P ξ N ) = V s T r ξ s + 1 ⋯ ξ N Q − 1 exp ( − β H N ) ; (5)

where H N is the Hamiltonian of our system given by

H N = − ℏ 2 2 m ∇ 2 + V ( ξ ) (6)

where V is the potential of the system and Q is the configuration integral is given by

Q = T r ξ s + 1 ⋯ ξ N e − β H N , (7)

The solution of N-particle schrödinger (time-dependent) equation with this Hamiltonian are given by | Ψ ( 1 ) 〉 ⋯ | Ψ ( M ) 〉 and from a complete orthonormal basis

〈 Ψ ( 1 ) | p n | Ψ ( 2 ) 〉 = ℏ i ∫ d ξ 1 ∫ d ξ 2 ∫ d ξ 3 J Ψ 1 ∗ ∂ J Ψ ( 2 ) ∂ ξ n

∑ k = 1 M | Ψ ( k ) 〉 〈 Ψ ( k ) | = 1 (8)

Define the N-particle density operator

ρ ^ = ∑ k = 1 M w k | Ψ ( k ) 〉 〈 Ψ ( k ) | (9)

where w_{k} are positive real probabilities

∑ k = 1 M w k = 1 (10)

0 ≤ w k ≤ 1

The density operator ρ ^ follow the Von Neumann equation

i ℏ ∂ ∂ t ρ ^ − [ H ^ , ρ ^ ] = 0 (11)

In order to derive the quantum BBGKY-hierarchy, we introduce the reduced s-particle density operator as

F ^ 1 ⋯ s = C s N T r s + 1 ⋯ N ρ ^ (12)

and T r 1 ⋯ s F ^ 1 ⋯ s = C s N = N ! ( N − s ) ! .

The equation of motion for the reduced density operator obeys directly the Von Neuman equation. Now by substituting from Equation (12) into Equation (11) we get

i ℏ ∂ ∂ t F ^ 1 ⋯ s − [ H ^ 1 ⋯ s , F ^ 1 ⋯ s ] = T r s + 1 ∑ i = 1 s [ v ^ i , s + 1 , F ^ 1 ⋯ s + 1 ] (13)

where v ^ i , j is the potential between particles i,j and H ^ 1 ⋯ s is the s-particle Hamiltonian operator. The above equation constitutes the quantum generalization of the (BBGKY) hierarchy.

We assume that the momentum of the electron lies between ξ 1 and ξ 1 + d ξ 1 is P ξ 1 . Also the momentum of the positron lies between ξ 2 and ξ 2 + d ξ 2 is P ξ 2 and the momentum of the dust (ion) lies between ξ 3 and ξ 3 + d ξ 3 is P ξ 3 . In this section, we shall find the binary distribution function of a quantum dusty in curvilinear coordinates. Firstly, define the quantum N particle distrbution function as follows:

F N q u ( ξ N , P ξ N ) = N ! exp [ − β ∑ i = 1 N P ξ i 2 2 m − β U ( ξ N ) ] ∫ d P ξ N exp [ − β ∑ i = 1 N P ξ i 2 2 m ] ∫ d ξ N exp [ − β U ( ξ N ) ] (14)

The one particle distribution function is obtained by reducing F N q u ( ξ N , P ξ N ) by integrating over N − 1 positions and momenta then

F 1 q u ( ξ 1 , P ξ 1 ) = 1 ( N − 1 ) ! ∬ d ξ N d P ξ N F N q u ( ξ N , P ξ N ) (15)

If there are no external fields

F 1 q u ( ξ 1 , P ξ 1 ) = α exp [ − β 2 m P ξ 1 2 ] (16)

where the value of α can be found by normalization:

∬ d ξ 1 d P ξ 1 F 1 q u ( ξ 1 , P ξ 1 ) = N (17)

By substituting from Equation (16) into Equation (17) we get

α − 1 = V N ( 2 π m β ) 3 / 2 (18)

Then the one particle distribution function is given by

F 1 q u ( ξ 1 , P ξ 1 ) = N V ( β 2 π m ) 3 / 2 exp [ − β 2 m P ξ 1 2 ] (19)

By putting s = 1 into Equation (13) we can get the first equation of quantum BBGKY as

i ℏ ∂ ∂ t F ^ 1 − [ H ^ 1 , F ^ 1 ] = T r s + 1 ∑ i = 1 s [ v ^ 12 , F ^ 12 ] (20)

Then we have the binary distribution function by substituting from Equation (19) into Equation (20):

F 12 q u ( ξ 1 , P ξ 1 , ξ 2 , P ξ 2 ) = F 1 q u ( ξ 1 , P ξ 1 ) F 2 q u ( ξ 2 , P ξ 2 ) + g ( ξ 1 , ξ 2 ) (21)

where g ( ξ 1 , ξ 2 ) is the correlation function.

Let us now study the model of dusty three component plasma i.e. the neutral system of point-like particles of positive and negative charges such as electrons, positrons and dust particles like ions. This model is an important model in both laboratory physics and space physics and has many applications [^{th} species and Z d is the charge state of dust [

Substituting Equations (19) and (21) into (13) for s = 1 , 2 we obtain

F 12 q u = N 2 8 V 2 ( β π ) 3 [ exp [ − β P ξ e 2 m e ] m e 3 + 2 exp [ P ξ e 2 2 m e + P ξ i 2 2 m i ] ( m e m i ) 3 / 2 ] − e 1 e 2 K T ξ 12 e − κ ξ 12 − e 1 e 2 p ξ 1 ⋅ p ξ 2 e − κ ξ 12 ( m c ) 2 K T ξ 12 + ∑ s = 1 3 q s 2 m s 1 ξ s + ⋯ (22)

The quantum triplet distribution function F 123 q u defined in such a way that F 123 q u ( ξ 1 , P ξ 1 , ξ 2 , P ξ 2 , ξ 3 , P ξ 3 , t ) d ξ 1 d ξ 2 d ξ 3 d P ξ 1 d P ξ 2 d P ξ 3 is the probability of finding a particle of the type 1^{th} in the volume element d ξ 1 surrounding ξ 1 ,with momentum in range P ξ 1 → P ξ 1 + d P ξ 1 , a particle of type 2^{th} in the volume element d ξ 2 surrounding ξ 2 , with momentum in range P ξ 2 → P ξ 2 + d P ξ 2 and a particle of the type 3^{th} in the volume element d ξ 3 surrounding ξ 3 , with momentum in range P ξ 3 → P ξ 3 + d P ξ 3 respectively at time t.

The quantum triplet distribution function F 123 q u ( ξ 1 , P ξ 1 , ξ 2 , P ξ 2 , ξ 3 , P ξ 3 ) is defined by the calculation of the interaction between three charged particles seems rather involved, because the force on particle 1 at time t would depend on the position and momentum of particles 2 and 3 at a retarded time. For simplification the quantum triplet distribution function F 123 q u can be written as

F 123 q u ( 123 ) = F 1 q u ( 1 ) F 1 q u ( 2 ) F 1 q u ( 3 ) + F 1 q u ( 1 ) g ( 23 ) + F 1 q u ( 2 ) g ( 13 ) + F 1 q u ( 3 ) g ( 12 ) + h ( 123 ) (23)

where h ( 123 ) is the correlation function between particles 1, 2 and 3.

Substituting Equations (19) and (22) into (13) for s = 1 , 2 , 3 we obtain

F 123 q u ( 123 ) = N 3 16 2 V 3 ( β π ) 9 2 [ exp [ P ξ 1 2 2 m 1 + P ξ 2 2 2 m 2 + P ξ 3 2 2 m 3 ] ( m 1 m 2 m 3 ) 3 / 2 ] + N V ( β 2 π m ) 3 / 2 { exp [ − β 2 m P ξ 1 2 ] g ( 23 ) + exp [ − β 2 m P ξ 2 2 ] g ( 13 ) + exp [ − β 2 m P ξ 3 2 ] g ( 12 ) } + h ( 123 ) (24)

For three component plasma, we can use the two particle correlation function g ( 12 ) which is given by

g ( 12 ) = g 1 ( ξ 12 ) [ 1 + p ξ 1 ⋅ p ξ 2 ( m c ) 2 + ( ξ 12 ⋅ p ξ 1 ) ( ξ 12 ⋅ p ξ 2 ) 2 ( m c ) 2 ξ 12 2 ] (25)

where

g 1 ( ξ 12 ) = − e 1 e 2 K T ξ 12 e − κ ξ 12 (26)

g 1 ( r 12 ) is the Debye-Hückel solution and the three particle correlation function G ( 1,2,3 ) which is given by

h ( 123 ) = g 1 ( ξ 12 ) g 1 ( ξ 13 ) g 1 ( ξ 23 ) [ 1 + p ξ 1 ⋅ p ξ 2 ( m c ) 2 + p ξ 1 ⋅ p ξ 3 ( m c ) 2 + p ξ 2 ⋅ p ξ 3 ( m c ) 2 + ∑ i , j = 1 , i ≠ j 3 ( ξ i j ⋅ p ξ i ) ( ξ i j ⋅ p ξ j ) 2 ( m c ) 2 ξ i j 2 + ⋯ ] (27)

Whatever particles 1, 2, 3 are, we can write the quantum triplet distribution function F 123 q u ( 123 ) as

F 123 q u ( 123 ) = N 3 16 2 V 3 ( β π ) 9 2 [ exp [ P ξ 1 2 2 m 1 + P ξ 2 2 2 m 2 + P ξ 3 2 2 m 3 ] ( m 1 m 2 m 3 ) 3 / 2 ] + N V ( β 2 π m ) 3 / 2 { exp [ − β 2 m P ξ 1 2 ] ( g 1 ( ξ 23 ) [ 1 + p ξ 2 ⋅ p ξ 3 ( m c ) 2 + ( ξ 23 ⋅ p ξ 2 ) ( ξ 23 ⋅ p ξ 3 ) 2 ( m c ) 2 ξ 23 2 ] ) + exp [ − β 2 m P ξ 2 2 ] ( g 1 ( ξ 13 ) [ 1 + p ξ 1 ⋅ p ξ 3 ( m c ) 2 + ( ξ 13 ⋅ p ξ 1 ) ( ξ 13 ⋅ p ξ 3 ) 2 ( m c ) 2 ξ 13 2 ] )

+ exp [ − β 2 m P ξ 3 2 ] ( g 1 ( ξ 12 ) [ 1 + p ξ 1 ⋅ p ξ 2 ( m c ) 2 + ( ξ 12 ⋅ p ξ 1 ) ( ξ 12 ⋅ p ξ 2 ) 2 ( m c ) 2 ξ 12 2 ] ) } + g 1 ( ξ 12 ) g 1 ( ξ 13 ) g 1 ( ξ 23 ) [ 1 + p ξ 1 ⋅ p ξ 2 ( m c ) 2 + p ξ 1 ⋅ p ξ 3 ( m c ) 2 + p ξ 2 ⋅ p ξ 3 ( m c ) 2 + ∑ i , j = 1 , i ≠ j 3 ( ξ i j ⋅ p ξ i ) ( ξ i j ⋅ p ξ j ) 2 ( m c ) 2 ξ i j 2 ] (28)

An alternative statistical description of planetary rings was formulated in a series of papers by Hämeen-Anttila [

The biggest advantage is that the balance equations for mass, stress, and scale hight of the ring, are given analytically as partial differential equations. Thus, the theory can be applied to investigate the dynamical evolution of a planetary ring. The disk’s self-gravity potential ϕ d i s k couples to the surface mass density through Poisson’s equation [

∇ 2 ϕ d i s k = 4 π a σ δ (29)

where σ is the surface mass density, a is gravitation parameter and δ is the Dirac delta function.

1 h 1 h 2 h 3 [ ∂ ∂ ξ 1 ( h 2 h 3 h 1 ∂ ϕ d i s k ∂ ξ 1 ) + ∂ ∂ ξ 2 ( h 1 h 3 h 2 ∂ ϕ d i s k ∂ ξ 2 ) + ∂ ∂ ξ 3 ( h 2 h 1 h 3 ∂ ϕ d i s k ∂ ξ 3 ) ] = 4 π a σ δ (30)

where h 1 , h 2 h 3 is the natural basis vectors. The Kinetic theory describes the evolution of the local velocity distribution function of an ensemble of particles in terms of the Boltzmann equation or a suitable generalization of it, like Enskog’s theory of hard sphere gases. The kinetic equation can be derived from Liouville’s theorem, appearing as the leading equation in a hierarchy of equations describing n-particle distribution functions in phase space and neglecting correlations between particle pairs. Then the one particle distribution function in Equation (19) in in the local vertical gravity field can be given by

F 1 q u ( ξ 1 , P ξ 1 ) = N V ( β 2 π m ) 3 / 2 exp [ − β 2 m P ξ 1 2 − e ϕ ] (31)

Kinetic theory allows us to incorporate the full complexity of the dynamics of a planetary ring in a statistical description, such as the effects of the motion of ring particles on curved orbits between inelastic collisions, their finite size, the anisotropy of the velocity dispersion, and in principle also coagulation and fragmentation of the ring particles.

The collision motion of an ensemble of identical particles in the plane, in the frame of reference rotating with angular velocity ω can be described by the Boltzmann equation [

∂ f ∂ t + p ξ 1 m h 1 ∂ f ∂ ξ 1 + ( ω + p ξ 2 m h 2 ) ∂ f ∂ ξ 2 + ( 2 ω m p ξ 2 + p ξ 2 2 m h 2 − ∂ ϕ 1 h 1 ∂ ξ 1 ) ∂ f ∂ p ξ 1 = ( κ 2 2 m h 1 ω p ξ 1 + p ξ 1 p ξ 2 m 2 h 1 h 2 + ∂ ϕ 1 h 2 ∂ ξ 2 ) ∂ f ∂ p ξ 2 + ( ∂ f ∂ t ) c l (32)

where ξ 1 , ξ 2 , ξ 3 was defined by Equation (1) and ( ∂ f ∂ t ) c l is the collision integral

which takes into account effects due to the discrete-point nature of the gravitational charges, or collision effects (including diffusion in space and velocity), and defines the change of the distribution function f(r, v, t) arising from ordinary interparticle collisions (in a plasma this term represents the change of f arising from collisions with particles at distances shorter than a Debye length). The Boltzmann form for the collision integral is based on an assumption that the duration of a collision is much less than the time between collisions instantaneous collisions are considered [

( ∂ f ∂ t ) c l = − v c ( f − f 0 ) (33)

where f is the actual distribution function of particles and f_{0} is the steady-state equilibrium distribution function (Shu and Stewart, 1985) [

f 0 = σ 0 2 π c 2 exp ( − p ξ 2 2 m 2 c 2 ) (34)

The gravitational potential of a rotating oblate planet

ϕ ( ξ , β ) = − a M 0 2 [ 1 + ( R ξ ) 2 ( 1 2 − 3 2 sin 2 β ) I 2 ] (35)

where r is the distance from the center of the planet to the point at which the potential is sought, β is the planetocentric latitude of the point, R is the radius of the planet, and M 0 is its mass. For saturn I 2 = 0.017 , M 0 = 5.7 × 10 29 g and R = 6 × 10 7 m for more exact values of the parameters see Ref. [

In this work we obtained the quantum binary and triplet distribution functions of dusty plasma; the calculation is based on curvilinear coordinates and the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We consider only the thermal equilibrium plasma. The model under consideration is the three-component dusty plasma i.e. neutral system of point like particles of positive and negative charges (electrons and positrons) interspersed with dust particles (ions) [

The first derivation of reactive quantum Boltzmann equations by Olmstead and Curtiss [

Initial quantum curvilinear coordinates phase-space ( ξ , V ξ ) for dusty plasma electrons (gray color), positrons (red color) and dust (orange color) was given in

We observe from the phase space that the speed of dust particles is much lower than the speed of electrons and positrons. The density of dust particles varies due to the large volume of dust particles.

The quantum binary distribution function for dusty plasma was given in

Model of Saturn planet rings in the radius interval (92,000 km, 139,350 km) was given in

The authors declare no conflicts of interest regarding the publication of this paper.

Hussein, N.A. El R., Ahmed, A.El-R.H. and Sayed, E.G. (2019) Quantum Distribution Functions for Dusty Plasma in Saturn’s Rings by Using Curvilinear Coordinates. International Journal of Astronomy and Astrophysics, 9, 115-132. https://doi.org/10.4236/ijaa.2019.92009