ABSTRACT (original article): We consider a reduced dynamics for the first four fluid moments of the one-dimensional Vlasov–Poisson equation, namely, fluid density, fluid velocity, pressure, and heat flux. This dynamics depends on an equation of state to close the system. This equation of state (closure) connects the fifth-order moment—related to the kurtosis in velocity of the Vlasov distribution—with the first four moments. By solving the Jacobi identity, we derive an equation of state, which ensures that the resulting reduced fluid model is Hamiltonian. We show that this Hamiltonian closure allows symmetric homogeneous equilibria of the reduced fluid model to be stable. CITATION (original article): C. Chandre and B. A. Shadwick, “Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation. Phys. Plasmas 29, 102101 (2022). https://doi.org/10.1063/5.0102418

Introduction

The purpose of this notebook is to check some equations relating moments of the Vlasov distribution
f(x,v)
, namely
ρ∫f(x,v)dv
,
u
1
ρ
vf(x,v)dv
and
S
n

1
n+1
ρ

n
(v-u)
f(x,v)dv
such that the reduced fluid model is Hamiltonian, i.e., its corresponding Poisson bracket satisfies the Jacobi identity.
The
parametric equations
for the fluid moments
S
n
are given as:
S
2
(
Γ
2
,
Γ
3
)
3
Γ
2
+
Γ
2
(κ-
Γ
2
)
2
Γ
3
​
S
3
(
Γ
2
,
Γ
3
)
Γ
2
Γ
3
(κ-
Γ
2
)3
2
Γ
2
+(κ-2
Γ
2
)
2
Γ
3

​
S
4
(
Γ
2
,
Γ
3
)
9κ
5
5
Γ
2
+6
3
Γ
2
2
(κ-
Γ
2
)
2
Γ
3
+
Γ
2
(κ-
Γ
2
)(
2
κ
-3
Γ
2
(κ-
Γ
2
))
4
Γ
3
​
S
5
(
Γ
2
,
Γ
3
)9
5
κΓ
2
(κ-
Γ
2
)
Γ
3
+10
3
Γ
2
3
(κ-
Γ
2
)
3
Γ
3
+
Γ
2
(κ-
Γ
2
)(κ-2
Γ
2
)
2
κ
-2
κΓ
2
+2
2
Γ
2

5
Γ
3
​​These parametric equations have to
satisfy the identities
derived from the Jacobi identity:
S
3

1
2
∂
S
2
∂
Γ
2
∂
S
2
∂
Γ
3
​
S
4

9
5
2
S
2
+
1
5
∂
S
2
∂
Γ
2
∂
S
3
∂
Γ
3
+
1
5
∂
S
2
∂
Γ
3
∂
S
3
∂
Γ
2
​
S
5
4
S
2
S
3
+
1
3
∂
S
3
∂
Γ
2
∂
S
3
∂
Γ
3
​We also
verify some identities
on the Poisson bracket:
2
∂
S
i
∂
Γ
n
∂
Γ
k

β
kl
∂
S
j
∂
Γ
l

∂
γ
ij
∂
Γ
n
where

β

0
1
1
0

. We then
plot the surfaces
defined parametrically by
(
S
2
(
Γ
2
,
Γ
3
),
S
3
(
Γ
2
,
Γ
3
),
S
4
(
Γ
2
,
Γ
3
))
and
(
S
2
(
Γ
2
,
Γ
3
),
S
3
(
Γ
2
,
Γ
3
),
S
5
(
Γ
2
,
Γ
3
))
.

Parametric expressions of
S
n
(
Γ
2
,
Γ
3
)

In[]:=
Clear["Global‘*"];​​
S
2
[x_,y_]:=
3
x
+x(κ-x)
2
y
;​​
S
3
[x_,y_]:=xy(κ-x)(3
2
x
+
2
y
(κ-2x));​​
S
4
[x_,y_]:=9κ
5
x
/5+6
3
x
2
(κ-x)
2
y
+x(κ-x)(
2
κ
-3x(κ-x))
4
y
;​​
S
5
[x_,y_]:=9κ
5
x
(κ-x)y+10x^3(κ-x)^3y^3+x(κ-x)(κ-2x)(κ^2-2κx+2x^2)y^5;

Verification of the parametric expressions for
S
n
(
Γ
2
,
Γ
3
)

In[]:=
Simplify4
S
3
[x,y]-2
∂
{x}
S
2
[x,y]
∂
{y}
S
2
[x,y]​​Simplify5
S
4
[x,y]-9
2
S
2
[x,y]
-
∂
{y}
S
2
[x,y]
∂
{x}
S
3
[x,y]-
∂
{x}
S
2
[x,y]
∂
{y}
S
3
[x,y]​​Simplify6
S
5
[x,y]-24
S
2
[x,y]
S
3
[x,y]-2
∂
{x}
S
3
[x,y]
∂
{y}
S
3
[x,y]
Out[]=
0
Out[]=
0
Out[]=
0

Verification of the constraints on the closure functions

In[]:=
β={{4
S
3
[x,y],5
S
4
[x,y]-9
2
S
2
[x,y]
},{5
S
4
[x,y]-9
2
S
2
[x,y]
,6
S
5
[x,y]-24
S
2
[x,y]
S
3
[x,y]}};​​

β
={{0,1},{1,0}};​​γ={{2
S
3
[x,y],2
S
4
[x,y]-3
2
S
2
[x,y]
},{3
S
4
[x,y]-6
2
S
2
[x,y]
,3
S
5
[x,y]-12
S
2
[x,y]
S
3
[x,y]}};​​var={x,y};​​SimplifyTable
2
∑
k=1
2
∑
l=1

β
k,l
∂
{var〚l〛}
S
2
[x,y]
∂
{var〚k〛},{var〚n〛}
S
2
[x,y]-
∂
{var〚n〛}
γ1,1,{n,1,2}​​SimplifyTable
2
∑
k=1
2
∑
l=1

β
k,l
∂
{var〚l〛}
S
3
[x,y]
∂
{var〚k〛},{var〚n〛}
S
2
[x,y]-
∂
{var〚n〛}
γ1,2,{n,1,2}​​SimplifyTable
2
∑
k=1
2
∑
l=1

β
k,l
∂
{var〚l〛}
S
2
[x,y]
∂
{var〚k〛},{var〚n〛}
S
3
[x,y]-
∂
{var〚n〛}
γ2,1,{n,1,2}​​SimplifyTable
2
∑
k=1
2
∑
l=1

β
k,l
∂
{var〚l〛}
S
3
[x,y]
∂
{var〚k〛},{var〚n〛}
S
3
[x,y]-
∂
{var〚n〛}
γ2,2,{n,1,2}
Out[]=
{0,0}
Out[]=
{0,0}
Out[]=
{0,0}
Out[]=
{0,0}

Surfaces
S
n
(
Γ
2
,
Γ
3
)

κ=1;​​umin=0;​​umax=κ;​​vmax=5;​​vmin=-vmax;​​S2max=1.5;​​S3max=1;​​S4max=3;​​S5max=5;​​ParametricPlot3D[{
S
2
[u,v],
S
3
[u,v],
S
4
[u,v]},{u,umin,umax},{v,vmin,vmax},Exclusions{
S
2
[u,v]≤0,
S
4
[u,v]≤0},BoxRatios{1,1,1},AxesLabel{Style[
S
2
,FontSize30],Style[
S
3
,FontSize30],Style[
S
4
,FontSize30]},BaseStyle{FontSize18},MeshNone,PlotRange{{0,S2max},{-S3max,S3max},{0,S4max}},PlotStyleOpacity[0.5],PlotPoints50,ColorFunction(ColorData["BrassTones"][#5]&)]​​ParametricPlot3D[{
S
2
[u,v],
S
3
[u,v],
S
5
[u,v]},{u,umin,umax},{v,vmin,vmax},Exclusions{
S
2
[u,v]≤0,
S
4
[u,v]≤0},BoxRatios{1,1,1},AxesLabel{Style[
S
2
,FontSize30],Style[
S
3
,FontSize30],Style[
S
5
,FontSize30]},BaseStyle{FontSize18},MeshNone,PlotRange{{0,S2max},{-S3max,S3max},{-S5max,S5max}},PlotStyleOpacity[0.6],PlotPoints50,ColorFunction(ColorData["BrassTones"][#5]&)]​​Quit[];
Out[]=
Out[]=

CITE THIS NOTEBOOK

Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation​
by Cristel Chandre and Bradley Allan Shadwick
Wolfram Community, STAFF PICKS, July 3, 2024
​https://community.wolfram.com/groups/-/m/t/3204625