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Conservation equation in phase space for a guiding-centre plasma in a constant magnetic field
Volume 73A, number 2
PHYSICS LETTERS
3 September 1979
CONSERVATION EQUATION IN PHASE SPACE FOR A GUIDING-CENTRE PLASMA IN A CONSTANT MAGNETIC FIELD M. PSIMOPOULOS Department of Physics, Imperial College, London SW7, UK
Received 28 May 1979
The guiding-centre model is used to derive in a 4N-dimensional space the probability conservation equation for a plasma in a very strong constant magnetic field. The derived equation shows that the ensemble flow in this space is incompressible.
The guiding-centre model [1,2] has been originally introduced to describe the so called “anomalous” (1/B) diffusion of a plasma transverse to a very strong magnetic field [3]. According to this model when the magnetic field is sufficiently strong it is assumed that the position of each electron coincides with the position of its guiding centre and that the motion of each electron is described by the velocity of its guiding centre. Quantitatively the validity of the above model is based on the inequality ‘r ~ d, where ‘r represents the Larmor radius and d the interparticle distance. In the present communication a general equation based on the guiding-centre model is derived to describe the evolution of a plasma in the presence of a very strong constant magnetic field B. In order to simplify notations the usual model of a plasma consisting of a gas of electrons moving in a uniform continuous positively charged background [4] is adopted. The extension to a twocomponent plasma is simple. Assuming that the magnetic field is in the z-direction the components v~,va,, / = 1,2, ,N of the electron guiding-centre velocities depend on the electron positions through the relations:
(x1,
,XN) and the z-components u1Z, V~ of the particles. In order to derive an equation for the evolution of such a system we introduce the familiar notion of a microcanonical ensemble of systems which is a large set of systems each of constant energy E, constant volume V and constant number of particles N. Each system of the ensemble can be represented by a point in a 4N-dimensional orthogonal cartesian system whose coordinates correspond to the positions x1,..., XN and the z-components Viz, ,VNz of the velocities of the particles. We call this space the B-space as it is introduced to describe the behaviour of a guiding. centre plasma in a magnetic field and in contrast to the r-space used in usual statistical systems [5]. The whole ensemble forms a cloud of points which considered as a fluid streaming in B-space can be described by the continuity equation [6] ...
apN -~
+
div ( VPN) = 0,
(2)
...
~.
JX
=_.~
3
~
B
~
Ix.—x.1 ~‘ ‘
~.
‘
JY
B z=I ix•—x.i~’
=.~
~
______
where .
A
.1
4
PN(x1,...,xN,01z,.,L~Nz,tj~1 ...U.4NUV1Z... VNZ = l~/ir, (3) w being the number of systems of the ensemble con-
/
(1)
tamed in the volume element dx 1 dxNdulz dv~ and ir being the total number of systems of the en...
and therefore cannot be considered as independent variables. In this case the state of the system at any time t is completely determined by the positions
semble; ~N is the N-particle specific distribution func-
tion and represents the probability density that the 107
Volume 73A, number 2
PHYSICS LETFERS
system is in a volume element dx
1 d~Ndu12 ... dvNZ of B-space. Vis the average “velocity” of the cloud within the element dx1 ... dxNdvlZ .., du~zand obvi.
ously coincides with the velocity of one point of the element due to the way B-space has been constructed, V = {u~, ,UN, dvi~/dt, ...
...
,
dv~,z/dt}.
(4)
3 September 1979
av~/ax1+
apN N ~ at /=1’ / ax1
or explicitly as:
~PN i=ikaxi ~ lao1
~/z
aPN
apN
dt
aIdviz av,z~~)jo~
(5)
(10)
Eq. (11) implies that the flow in B-space is incompressible and eq. (5) can be simply written as: dpp,r/dt = 0,
+
0.
Substituting eqs. (9) and (10) into eq. (8) we get au1/ax1 = 0. (11)
Introducing eq. (4) into eq. (2) we get:
—+~.
=
avjy/ayj
at
(12)
N
E NB
i,/=1 j5I~j
X
apN
~1
3 au
We have
+ j=1 V~
zi_zi 3 m i*j 11 Ix,—x11
dt
(6)
a
and
(7)
We consider next the quantity:
au1
—
av1~ av1~ av1~
a
=
a
(8)
a
+
The z-velocity component ~ of the particle/ is independent of z1. This can be best understood if we consider the fact that two systems of the ensemble can have different for their/th particle even if all the other coordinates of their representative points in the B-space coincide. We write
at~~/az,=o.
au1.,
a))1
108
differences between eq. (12a) and the classical Liouville equation. The first is physical and is related to the fact that whereas Liouville’s equation can describe the present system only for time steps dt’~w~, WQ being the Larmor frequency, during which the motion of the electrons is linear, eq. (12a) can well describe the system for time steps much larger than The second difference is structural and related to the ~
fact that the coefficients of apN/aXJ and apN/ayl in eq. (12a) are functions of the positions rather than independent velocity variables. The integration of eq. (12a) and subsequent derivation of the kinetic equations describing the evolution of the one-particle distribution functions will be given elsewhere. [1] J.B. Taylor and B. McNamara, Phys. Fluids 14 (1971)
1492. [2] D. Montgomery, Physica 82 (1976) 111.
3ec B
(x1 —x~)(y1 Yi) 1
=
hence
scribing evolution of a guiding-centre one-compoEq. (12a)the can be considered as a general equation de-
(9)
From eqs. (1) we have
au1~= ax1
(12a)
equation in r-space. There are however two essential
dv1~ 0.
ZI aP~\,T i~j ~—x;I / —zi —=0. 1~
+ ~m ~
nent plasma and corresponds to the classical Liouville
hence dv 1~/dtis independent of
aPN~
—
B
/i ~. 1
~
(x1 —x1)(y15—v1) ~ —x11
(31
D. Bohm, in: The characteristics of electrical discharge
in magnetic fields, eds. A. Gutlirie and R. Wakening (McGraw-Hill, 1949). [41 R. Balescu, Statistical mechanics of charged particles [51 R. (Wiley-Interscience, Balescu, Equilibrium 196 and 3). non-equilibrium statistical mechanics (Wiley-Interscience, 1975). [61 L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, 1959).
PHYSICS LETTERS
3 September 1979
CONSERVATION EQUATION IN PHASE SPACE FOR A GUIDING-CENTRE PLASMA IN A CONSTANT MAGNETIC FIELD M. PSIMOPOULOS Department of Physics, Imperial College, London SW7, UK
Received 28 May 1979
The guiding-centre model is used to derive in a 4N-dimensional space the probability conservation equation for a plasma in a very strong constant magnetic field. The derived equation shows that the ensemble flow in this space is incompressible.
The guiding-centre model [1,2] has been originally introduced to describe the so called “anomalous” (1/B) diffusion of a plasma transverse to a very strong magnetic field [3]. According to this model when the magnetic field is sufficiently strong it is assumed that the position of each electron coincides with the position of its guiding centre and that the motion of each electron is described by the velocity of its guiding centre. Quantitatively the validity of the above model is based on the inequality ‘r ~ d, where ‘r represents the Larmor radius and d the interparticle distance. In the present communication a general equation based on the guiding-centre model is derived to describe the evolution of a plasma in the presence of a very strong constant magnetic field B. In order to simplify notations the usual model of a plasma consisting of a gas of electrons moving in a uniform continuous positively charged background [4] is adopted. The extension to a twocomponent plasma is simple. Assuming that the magnetic field is in the z-direction the components v~,va,, / = 1,2, ,N of the electron guiding-centre velocities depend on the electron positions through the relations:
(x1,
,XN) and the z-components u1Z, V~ of the particles. In order to derive an equation for the evolution of such a system we introduce the familiar notion of a microcanonical ensemble of systems which is a large set of systems each of constant energy E, constant volume V and constant number of particles N. Each system of the ensemble can be represented by a point in a 4N-dimensional orthogonal cartesian system whose coordinates correspond to the positions x1,..., XN and the z-components Viz, ,VNz of the velocities of the particles. We call this space the B-space as it is introduced to describe the behaviour of a guiding. centre plasma in a magnetic field and in contrast to the r-space used in usual statistical systems [5]. The whole ensemble forms a cloud of points which considered as a fluid streaming in B-space can be described by the continuity equation [6] ...
apN -~
+
div ( VPN) = 0,
(2)
...
~.
JX
=_.~
3
~
B
~
Ix.—x.1 ~‘ ‘
~.
‘
JY
B z=I ix•—x.i~’
=.~
~
______
where .
A
.1
4
PN(x1,...,xN,01z,.,L~Nz,tj~1 ...U.4NUV1Z... VNZ = l~/ir, (3) w being the number of systems of the ensemble con-
/
(1)
tamed in the volume element dx 1 dxNdulz dv~ and ir being the total number of systems of the en...
and therefore cannot be considered as independent variables. In this case the state of the system at any time t is completely determined by the positions
semble; ~N is the N-particle specific distribution func-
tion and represents the probability density that the 107
Volume 73A, number 2
PHYSICS LETFERS
system is in a volume element dx
1 d~Ndu12 ... dvNZ of B-space. Vis the average “velocity” of the cloud within the element dx1 ... dxNdvlZ .., du~zand obvi.
ously coincides with the velocity of one point of the element due to the way B-space has been constructed, V = {u~, ,UN, dvi~/dt, ...
...
,
dv~,z/dt}.
(4)
3 September 1979
av~/ax1+
apN N ~ at /=1’ / ax1
or explicitly as:
~PN i=ikaxi ~ lao1
~/z
aPN
apN
dt
aIdviz av,z~~)jo~
(5)
(10)
Eq. (11) implies that the flow in B-space is incompressible and eq. (5) can be simply written as: dpp,r/dt = 0,
+
0.
Substituting eqs. (9) and (10) into eq. (8) we get au1/ax1 = 0. (11)
Introducing eq. (4) into eq. (2) we get:
—+~.
=
avjy/ayj
at
(12)
N
E NB
i,/=1 j5I~j
X
apN
~1
3 au
We have
+ j=1 V~
zi_zi 3 m i*j 11 Ix,—x11
dt
(6)
a
and
(7)
We consider next the quantity:
au1
—
av1~ av1~ av1~
a
=
a
(8)
a
+
The z-velocity component ~ of the particle/ is independent of z1. This can be best understood if we consider the fact that two systems of the ensemble can have different for their/th particle even if all the other coordinates of their representative points in the B-space coincide. We write
at~~/az,=o.
au1.,
a))1
108
differences between eq. (12a) and the classical Liouville equation. The first is physical and is related to the fact that whereas Liouville’s equation can describe the present system only for time steps dt’~w~, WQ being the Larmor frequency, during which the motion of the electrons is linear, eq. (12a) can well describe the system for time steps much larger than The second difference is structural and related to the ~
fact that the coefficients of apN/aXJ and apN/ayl in eq. (12a) are functions of the positions rather than independent velocity variables. The integration of eq. (12a) and subsequent derivation of the kinetic equations describing the evolution of the one-particle distribution functions will be given elsewhere. [1] J.B. Taylor and B. McNamara, Phys. Fluids 14 (1971)
1492. [2] D. Montgomery, Physica 82 (1976) 111.
3ec B
(x1 —x~)(y1 Yi) 1
=
hence
scribing evolution of a guiding-centre one-compoEq. (12a)the can be considered as a general equation de-
(9)
From eqs. (1) we have
au1~= ax1
(12a)
equation in r-space. There are however two essential
dv1~ 0.
ZI aP~\,T i~j ~—x;I / —zi —=0. 1~
+ ~m ~
nent plasma and corresponds to the classical Liouville
hence dv 1~/dtis independent of
aPN~
—
B
/i ~. 1
~
(x1 —x1)(y15—v1) ~ —x11
(31
D. Bohm, in: The characteristics of electrical discharge
in magnetic fields, eds. A. Gutlirie and R. Wakening (McGraw-Hill, 1949). [41 R. Balescu, Statistical mechanics of charged particles [51 R. (Wiley-Interscience, Balescu, Equilibrium 196 and 3). non-equilibrium statistical mechanics (Wiley-Interscience, 1975). [61 L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, 1959).