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\noindent{\title Quantum Plasma Model with Hydrodynamical}
\vskip 0truemm
\noindent{\title Phase Transition}
\vskip 12 truemm
\noindent {{\auth By Geoffrey L. Sewell}\footnote *{Partially
supported by European Capital and Mobility Contract No. CHRX-CT-
0007}}
\vskip 5truemm
\noindent Department of Physics, Queen Mary and Westfield
College,
\hfil\break
Mile End Road, London E1 4NS
\vskip 26truemm
\noindent {\abs {\titabs Abstract.}
We derive the electro-hydrodynamics of the Jellium plasma model
from its many-particle Schr\"odinger equation, subject to certain
general initial and regularity conditions, and prove that it
undergoes a transition from deterministic to stochastic flow when
a certain parameter, representing the non-uniformity in the
initial density and drift velocity profiles, reaches a certain
critical value. Thus, the model exhibits a phase transition far
from equilibrium.}
\vskip 12truemm
\noindent {\subt 1 Introduction}
\medskip\noindent
The quantum Jellium model is a system of electrons, interacting
via Coulomb forces both with one another and with a uniform,
positively charged, neutralising background. It is thus a model
of a many-particle system with realistic interactions. At the
level of mathematical physics, it has been proved to enjoy 'good'
thermodynamic [1,2] and hydrodynamic [3] properties. In fact,
apart from Davies's [4] derivation of Fourier's law of heat
conduction for a certain model of interacting atoms, the passage
from quantum mechanics to Eulerian hydrodynamics in [3]
represents, to the best of our knowledge, the only rigorous
quantum statistical derivation of a macroscopic continuum
mechanics. It is, however, based on the assumption of regularity
conditions, which exclude the possibility of hydrodynamical phase
transitions.
\vskip 0.2cm\noindent
The object of the present article is to provide a further quantum
mechanical treatment of the hydrodynamics of the Jellium model,
in which certain regularity assumptions of [3] are weakened in
such a way as to admit non-equilibrium phase transitions. In
fact, we show that, under the new assumptions, the model exhibits
a transition from a deterministic to a stochastic hydrodynamics
when a certain parameter, representing the non-uniformity of the
initial density and velocity profiles, attains a critical value.
The method by which we obtain this result is based on two
main steps. Firstly, we derive a Vlasov equation for the large
scale dynamics of the model from its many-particle Schr\"odinger
equation, subject to specified initial and regularity conditions.
We then show that the Vlasov dynamics reduces to a deterministic
Eulerian hydrodynamics if the initial density and velocity
profiles lie below a certain non-uniformity threshold, and that
otherwise the flow becomes stochastic. Specifically, in the
latter case, the flow corresponds to a {\it statistical mixture}
of different streams, and thus the local density and drift
velocity have macroscopic dispersions.
\vskip 0.2cm\noindent
We present our treatment as follows. In ${\S}$2, we extend
the scheme of Refs. [3,5] so as to derive a Vlasov equation,
governing the large-scale dynamics of the Jellium model, from its
many-particle Schr\"odinger equation, subject to rather
general initial conditions. To be more precise, we provide a
treatment here of both the Jellium model itself and a regularised
version of it, obtained by introducing a short distance cut-off
in the Coulomb potential. For the regular model, our derivation
of the Vlasov dynamics is based exclusively on the Schr\"odinger
equation and the initial conditions. For the true Jellium model,
on the other hand, certain supplementary regularity assumptions
are also required. The essential idea behind these is that the
repulsive character of the inter-electronic forces keeps the
electrons apart and thereby tames the singularity in the Coulomb
potential. The precise form of the assumptions is specified by
the conditions (R.1-4). We remark here that we have not avoided
repeating much of the formalism of Ref. [3] in this Section,
because it is essential for our present purposes to reset it
in the context of our new (weakened) regularity conditions
\vskip 0.2cm\noindent
In ${\S}$3, we note that the Vlasov equation is just the
Liouville equation for a certain {\it Lagrangian}
hydrodynamics, governing the evolution of the time (t)-dependent
position, $X_{t}(x),$ of a 'fluid particle'
located initially at the point $x.$ We then show that the Vlasov
dynamics reduces to a deterministic Eulerian hydrodynamics if and
only if the function $X_{t}$ is invertible: otherwise it
corresponds a stochastic flow, in the sense described above.
\vskip 0.2cm\noindent
In ${\S}$4, we analyse the conditions for deterministic
versus stochastic flow in both the true Jellium model and the
regularised one, in the situation where the initial density and
velocity profiles depend on only one spatial coordinate and so
are essentially one-dimensional. For this case, we are able to
show explicitly that both models undergo transitions from
deterministic to stochastic flow when the initial conditions
attain certain non-uniformity thresholds.
\vskip 0.2cm\noindent
We conclude, in ${\S}$5, with some brief further comments
on the results obtained here and on their possible relevance to
the theory of turbulence.
\vfill\eject
\noindent {\subt 2. Basis of the Vlasov Dynamics}
\medskip\noindent
The Jellium model, ${\Sigma}^{(N,L)},$ consists of $N$ electrons
in a cube, $K^{(L)},$ of side $L,$ with uniform neutralising
positive charge background. We assume periodic boundary
conditions. Our objective will be to obtain a quantum theoretical
derivation of the hydrodynamics of ${\Sigma}^{(N,L)}$ in a limit
where $N$ and $L$ tend to infinity and the mean particle density,
$${\overline n}=N/L^{3}\eqno(2.1)$$
remains fixed and finite.
\vskip 0.3cm \noindent
We denote the position vectors and momenta of the electrons by
$X_{1},.. \ .,X_{N}$ and $P_{1},.. \ .,P_{N},$ respectively.
Thus, $P_{j}=-i{\hbar}{\nabla}_{j}^{(L)},$ where
${\nabla}^{(L)}$
is the gradient operator in $K^{(L)}.$ At the microscopic
level, the pure states of the system are given by the normalised,
antisymmetric wave-functions ${\Psi}^{(N)}(X_{1},.. \ .,X_{N}),$
and the Hamiltonian takes the form
$$H^{(N,L)}={-{\hbar}^{2}\over 2m}{\sum}_{j=1}^{N}
{\Delta}_{j}^{(L)}+e^{2}{\sum}_{j,k(>j)=1}^{N}U^{(L)}(X_{j}-X_
{k})
\eqno(2.2)$$
where $-e,m$ are the electronic charge and mass, respectively,
${\Delta}^{(L)}$ is the Laplacian for $K^{(L)},$ and $U^{(L)}(X)$
is the difference between ${\vert}X{\vert}^{-1},$ periodicised
w.r.t. $K^{(L)},$ and its space average over that cube, i.e.
$$U^{(L)}(X)={4{\pi}\over L^{3}}{\sum}^{(L)}
{{\exp}(iQ.X)\over Q^{2}}\eqno(2.3)$$
the superscript $(L)$ over ${\Sigma}$ signifying that summation
is taken over the non-zero vectors $Q=(2{\pi}/L)(n_{1},n_{2},
n_{3}),$ with the $n$'s integers. The time-dependent
Schr\"odinger equation for ${\Sigma}^{(N,L)},$ with $T$ the time
variable, is
$$i{\hbar}{{\partial}{\Psi}_{T}^{(N)}\over {\partial}T}=
H^{(N,L)}{\Psi}_{T}^{(N)}\eqno(2.4)$$
We shall assume the following initial kinetic and potential
energy bounds for ${\Sigma}^{(N,L)}$-more precisely, for the
family of systems ${\lbrace}{\Sigma}^{(N,L)}{\rbrace},$ with
$N,L$ satisfying (2.1).
\vskip 0.2cm\noindent
$(I.1)^{(L)}$ The expectation value of the total kinetic energy
per particle, for the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant $B/2m,$ i.e.
$$({\Psi}_{0}^{(N)},P_{1}^{2}{\Psi}_{0}^{(N)})**j)=1}^{N}U(x_{j}-x_{k})\eqno(2.10)$$
$$p_{j}=-i{\hbar}_{N}{\nabla}_{j}\eqno(2.11)$$
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}}=
{{\hbar}\over m{\omega}}({{\overline n}\over
N})^{2/3}\eqno(2.12)$$
is a dimensionless effective 'Planck constant', ${\nabla}$ is the
gradient operator for $K,$ and
$$U(x)=U_{c}(x):={\sum}_{q}^{(1)}{\exp}(iq.x)/q^{2}\eqno(2.13)$$
the superscript $(1)$ over ${\Sigma}$ signifying that summation
is taken over the non-zero vectors $2{\pi}(n_{1},n_{2},n_{3}),$
with the $n$'s integers. Our reason for introducing the symbol
$U_{c}$ here is that we want to employ the Hamiltonian given by
(2.10) both for the model ${\Sigma}^{(N)},$ with $U=U_{c},$ and
for a modified version of this, where $U$ is a 'smoothed out'
Coulomb potential. We note that it follows from (2.13) that
$${\Delta}U_{c}(x)=1-{\delta}(x)\eqno(2.14)$$
where ${\Delta}$ is the Laplacian and ${\delta}$ the Dirac
distribution for $K.$
\vskip 0.2cm\noindent
{\ssubt Note.} Two key features of the rescaled description, as
given by (2.9)-(2.14), are that
\vskip 0.2cm\noindent
(a) the effective, dimensionless Planck constant, ${\hbar}_{N},$
governing the quantum behaviour of the model, tends to zero as
$N{\rightarrow}{\infty};$ and
\vskip 0.2cm\noindent
(b) the pair interaction potential scales as $N^{-1}.$
\vskip 0.2cm\noindent
The properties (a) and (b) are generally the hallmarks of a
classical and of a mean field theory, respectively, in the limit
$N{\rightarrow}{\infty}.$ In fact, as we shall presently show,
the model does indeed reduce to a classical mean field theoretic
one, governed by Vlasov dynamics, in this limit.
\vskip 0.2cm\noindent
We formulate the dynamics of ${\Sigma}^{(N)}$ in
terms of its characteristic functions,
$${\mu}_{t}^{(N,n)}({\xi}_{1},.. \ .,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})=({\psi}_{t}^{(N)},{\Pi}_{j=1}^{n}
({\exp}(i{\xi}_{j}.p_{j}/2){\exp}(i{\eta}_{j}.x_{j})
{\exp}(i{\xi}_{j}.p_{j}/2)){\psi}_{t}^{(N)})
\eqno(2.15)$$
where the ${\xi}'$s and ${\eta}'$s run over the ranges ${\bf
R}^{3}$ and $(2{\pi}{\bf Z})^{3},$ respectively. These functions
are in one-to-one correspondence with the reduced density
matrices for the state ${\psi}_{t}^{(N)};$ and, in particular,
the n-particle spatial density function
$${\rho}_{t}^{(N,n)}(x_{1},.. \ .,x_{n})={\int}
dx_{n+1}.. \ .dx_{N}{\vert}{\psi}_{t}(x_{1},x_{2},.. \ .,x_{N})
{\vert}^{2}\eqno(2.16)$$
is the Fourier transform of ${\mu}_{t}^{(N,n)}$ w.r.t. the
${\eta}'$s, when the ${\xi}'$s are held at the value zero.
\vskip 0.2cm\noindent
The initial condition for ${\Sigma}^{(N)},$ corresponding to
$(I.1)^{(L)}$ for ${\Sigma}^{(N,L)},$ is
$$({\psi}_{0}^{(N)},p_{1}^{2}{\psi}_{0}^{(N)})
0),$ there is a constant,
$B_{\tau}(<{\infty}),$ such that}
$${\int}dy{\rho}_{t}^{(N,1)}(x){\vert}{\nabla}U(x-y){\vert}
0){\vert} \ (1+u_{0}^{\prime}(x)t)=0 \ for
\ some \ x{\in}[0,1]{\rbrace}\eqno(4.6)$$
it follows immediately from (4.4)-(4.6) that $J_{t}$ is
strictly positive, and hence that $X_{t}$ is invertible, if
$0{\le}t0,$ then the regularity
condition (2.39) is satisfied.}
\vskip 0.3cm\noindent
{\ssubt Proof.} We note first that, by (2.13) and (4.2),
$$F(x)={\sum}_{n=1}^{\infty}i{\exp}
(2{\pi}inx)/(2{\pi}n)$$
which implies that $F$ is square integrable, hence absolutely
integrable, over $[0,1].$ Thus, since, by
(3.2), (4.2) and (4.3), the l.h.s. of (2.39) is equal to
$${\int}_{0}^{1}dy{\sigma}_{0}(y)J_{t}(y){\vert}F
(X_{t}(x)-X_{t}(y)){\vert}$$
$${\equiv}{\int}_{0}^{1}dy{\sigma}_{0}(X_{t}^{-1}(y))
{\vert}F(X_{t}(x)-y){\vert}$$
by the invertibility of $X_{t},$ it follows that the condition
(2.39) is satisfied.
\vskip 0.3cm\noindent
{\ssubt Proof of Prop. 4.2.} Since $U=U_{c}$ here, it follows
from (2.13), (2.14), (4.1) and (4.3) that
$$J_{t}(x)=1+u_{0}^{\prime}(x)t+{\int}_{0}^{t}ds(t-s)
J_{s}(x)(-1+{\int}_{0}^{1}dy{\sigma}_{0}(y)
{\delta}(X_{s}(x)-X_{s}(y)))\eqno(4.9)$$
where now ${\delta}$ is the Dirac distribution on $[0,1],$
subject to periodic boundary conditions.
\vskip 0.2cm\noindent
Let us first suppose that $X_{t}$ is invertible, i.e. that
$J_{t}$ is strictly positive, over a time interval
$0{\le}t<{\tau}_{0},$ for some positive ${\tau}_{0}.$ In this
case,
$$J_{s}(x){\delta}(X_{s}(x)-X_{s}(y)){\equiv}{\delta}(x-y) \
{\forall}s{\in}[0,{\tau}_{0})$$
and therefore (4.9) reduces to
$$J_{t}(x)=1+u_{0}^{\prime}(x)t+{\int}_{0}^{t}ds(t-s)
({\sigma}_{0}(x)-J_{s}(x))\eqno(4.10)$$
i.e.
$$({d^{2}\over dt^{2}}+1)J_{t}(x)={\sigma}_{0}(x);
\ with \ J_{0}(x)=1; \ {\dot J}_{0}(x)
=u_{0}^{\prime}(x)\eqno(4.11)$$
where ${\dot J}_{t}=dJ_{t}/dt.$ Hence,
$$J_{t}(x)={\sigma}_{0}(x)+(1-{\sigma}_{0}(x))
{\cos}(t)+u_{0}^{\prime}(x){\sin}(t)\eqno(4.12)$$
In view of the non-negativity of ${\sigma}_{0},$ this equation
implies that $J_{t}$ is strictly positive for all $t{\ge}0$
if and only if the condition (4.7) is fulfilled. Otherwise,
$J_{t}$ changes sign at some point $x{\in}[0,1]$ when $t$ reaches
the value ${\tau}$ specified in the statement of the Proposition.
\vskip 0.2cm\noindent
We may thus summarise these results as follows.
\vskip 0.2cm\noindent
(a) If (4.7) is satisfied, then the function $X_{t}$ given by
substituting the formula (4.12) for $J_{t}$ into the r.h.s. of
(4.1)$^{\prime}$ is invertible and satisfies both the Newtonian
mean field equation (4.10) and, by Lemma (4.3), the regularity
condition (2.39), for all $t{\ge}0.$ Hence, by (R.4), it is the
unique solution of the Newtonian mean field equation, and
persists for all positive $t.$ Hence, by Prop. 3.1, the model
exhibits the Eulerian hydrodynamics given by (3.5)-(3.7) at all
times.
\vskip 0.2cm\noindent
(b) If (4.7) is violated, then, by the same argument, the
system exhibits this deterministic hydrodynamics for times
$t{\in}[0,{\tau}),$ with ${\tau}$ as specified in Prop. 4.2.
\vskip 0.2cm\noindent
(c) If (4.7) is violated, then there must be a transition to
stochastic flow at $t={\tau},$ since an assumption to the
contrary becomes invalid when $t$ passes through that
value.
\vskip 0.2cm\noindent
The results (a)-(c) establish the Proposition.
\medskip\noindent
{\subt 5. Concluding Remarks.}
\medskip\noindent
We have shown here that the quantum dynamics of the Jellium model
leads to a hydrodynamics, which supports both deterministic and
stochastic flows, and exhibits phase transitions between them. This
hydrodynamics is therefore richer than that of the deterministic flow
given by the Euler-cum-Maxwell equations. Furthermore, since the flow
in the stochastic phase corresponds to a statistical mixture
of different streams, one might envisage that this carries
a germ of turbulence.
\vskip 0.2cm\noindent
As regards possible ramifications of the present work, we note
that the above hydrodynamical properties of the model stemmed from
its Vlasov dynamics, which is simply the Liouville probabilistic
version of its Lagrangian hydrodynamics (cf. ${\S}3$). This suggests
that, more generally, a natural way of formulating the theory of
stochastic flows, even of turbulence, might be via a probabilistic
treatment of Lagrangian hydrodynamics. That should presumably have
some connection with Foias's [11] formulation of stochastic
hydrodynamics on the basis of a Liouville equation governing
Navier-Stokes flows. It need not, however, be equivalent to it,
since, as we have seen in ${\S}'s$ 3 and 4, the Eulerian and
Lagrangian pictures of the present plasma model are not equivalent.
\vskip 0.2cm\noindent
Finally, we remark here that the hydrodynamics obtained here is
completely inviscid. The reason for this, as in Ref. [3] (cf.
discussion there at the end of ${\S}$1), is that our macroscopic
description is effected on the largest available length scale,
$L,$ and that, consequently, the viscous forces are 'scaled
away'. Thus, the hydrodynamic picture we have obtained should be
regarded as no more than a skeletal version of that of a real
plasma.
\medskip\noindent
{\subt References.}
\medskip\noindent
1. E. H. Lieb and H. Narnhofer: J. Stat. Phys. {\bf 12}, 291
(1975)
\vskip 0.2cm\noindent
2. Ph. Martin and Ch. Oguey: I. Phys. A {\bf 18}, 1995 (1985)
\vskip 0.2cm\noindent
3. G. L. Sewell: J. Math. Phys. {\bf 26}, 2324 (1985)
\vskip 0.2cm\noindent
4. E. B. Davies: J. Stat. Phys. {\bf 18}, 161 (1978)
\vskip 0.2cm\noindent
5. H. Narnhofer and G. L. Sewell: Commun. Math. Phys. {\bf 79},
9 (1981)
\vskip 0.2cm\noindent
6. D. Ruelle: "Statistical Mechanics", Benjamin, New York (1969)
\vskip 0.2cm\noindent
7. G. L. Sewell: "Quantum Theory of Collective Phenomena",
Clarendon Press, Oxford (1991)
\vskip 0.2cm\noindent
8. H. Spohn: Math. Meth. Appl. Sci. {\bf 3}, 445 (1981)
\vskip 0.2cm\noindent
9. H. Neunzert: Fluid Dyn. Trans. {\bf 9}, 229 (1978)
\vskip 0.2cm\noindent
10. C. Radin: Commun. Math. Phys. {\bf 54}, 69 (1974)
\vskip 0.2cm\noindent
11. C. Foias: Russian Math. Surveys {\bf 29}, 293 (1974)
**