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\bigskip
9/13/94
\bigskip
\centerline{Nonunique Stationary States in Driven Systems; The Weakly Ionized Plasma}
\centerline{by}
\centerline{E. Carlen\footnote{$^1$}{School of Mathematics, Georgia
Institute of Technology, Atlanta, GA 30332--0160}, R.
Esposito\footnote{$^2$}{Dipartimento di Matematica, Universita' di Roma Tor
Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy}, J.L.
Lebowitz\footnote{$^3$}{Department of Mathematics and Physics, Rutgers University, New
Brunswick, NJ 08903}, R. Marra\footnote{$^4$}{Dipartimento di Fisica,
Universita' di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy}
and A. Rokhlenko{$^3$}}
\bigskip
\bigskip
\centerline{\bf Abstract}
We study the electron velocity distribution $f(v, t)$
in a weakly ionized spatially homogeneous plasma in an external
electric field $E$. $f$ satisfies a Boltzmann type equation in which
there are linear terms representing collisions with ions and neutral particles having a specified
Maxwellian distribution as well as a nonlinear electron-electron (e-e)
collision term. Using simplified collision terms we prove that, when e-e collisions dominate, the long time
asymptotic behavior of $f$ is non-unique
for certain ranges of $E$. This provides a rigorous proof of a dynamical
phase transition for a realistic (model) system.
\bigskip
\medskip
\noindent PACS 1994: 05.20.Dd, 52.25.Dg, 52.35.-g, 51.50.+v
\vfill \eject
{\bf Introduction}
The understanding of equilibrium phenomena, including their most interesting
aspect, that of phase transitions, has enormously advanced in recent years.
By contrast, our understanding of nonequilibrium phenomena is
still very incomplete at the present time. In particular there is no
general microscopic theory of
nonequilibrium phase transitions. Theories of such phenomena rely mostly
on the qualitative or numerical solutions of model
hydrodynamical equations. These show how the
great variety and complexity of observed nonequilibrium behavior in fluids,
lasers, electron devices, chemical reactions, etc.\ arises from the
nonlinearities in the macroscopic equations [1]. Despite the sucess of
such phenomenological approaches,
there remains the need for reliable microscopic
theories to provide deeper insight into this important area of far from local
equilibrium phenomena, encompassing enormous ranges of spatio-temporal
scales, in plasmas, living systems and
galaxies. Attempts so far in this direction make use of a variety of
approaches. These include molecular dynamic simulations of
Hamiltonian systems of particles ($\leq 10^6$) interacting
via simple pair potentials (hard spheres or disks, cut-off
Lenard-Jones, potential, etc.), and the study via analytic means and/or computer simulations
of highly simplified (lattice) model systems [3]. Lacking thus far are
proofs of the existence of
a nonequilibrium phase transition due to cooperative effects for any physical
system with realistic
modeling of the dynamics.
In this note we prove such a result for a simplified, but still
recognizable, model of a
weakly ionized plasma in the presence of an external electric field. A
more realistic version of this system has been studied in [4].
It was found there, using various approximations, that
this system may undergo a phase transition, including
hysteresis, as the magnitude of the field $E$ is varied. Here we supply a
rigorous mathematical proof of this phenomena. In
particular we prove that there really exist stationary distributions $f(v)$ in the
vicinity of any one
of the stable fixed points on the hysteresis loop obtained in [4] from the
solution of a pair of nonlinear equations for the hydrodynamical variables;
current and temperature. We show furthermore that when the
system is started near such a stable hydrodynamic state it will remain there
forever. If, on the contrary, it is started near the unstable fixed point,
it will leave the neighborhood after some finite time.
Our formal set up is as follows: We consider a weakly ionized gas in $\IR^3$
in the presence of an externally imposed constant electric field
$E$. The density of the gas, the degree of ionization and the strength of
the field are assumed to be such that: (i) the interactions between the
electrons can be described
by some nonlinear, Boltzmann type collision operator, and (ii) collisions between
the electrons and the heavy components of the plasma, ions and neutrals,
are adequately described by assuming the latter ones to have a spatially
homogeneous time independent Maxwellian
distribution with an a priori given temperature. Under these conditions the
time evolution of the spatially homogeneous electron velocity distribution
function $f(v,t)$ will satisfy a Boltzmann type equation
$$
{\partial f(v,t) \over \partial t} = -E \cdot {\bf \nabla}f + Lf +
\epsilon^{-1}Q(f), \eqno(1)
$$
where $\nabla$ is the gradient with respect to $v$ and the mass and charge
of the electron have been set equal to unity.
The linear term $Lf$ represents collisions with the ions
and neutrals and we have introduced a coupling parameter $\epsilon^{-1}$ in
front of the nonlinear term $Q(f)$ representing e-e interactions.
We refer the reader to refs. [4] for a detailed description of the
different terms in (1). In the present work we
use simplified collision terms: We replace the
linear term by a Fokker-Planck type operator
with a velocity dependent diffusion coefficient, chosen to have a maximum at $|v| = 0$
and approach a constant value when $|v| \to \infty$. This mimics the fact
that the cross section of collisions of the electrons with the ions, is
strongly peaked at low speeds (going to zero as $|v| \to \infty$) while
collisions with neutrals has an approximately constant cross-section. In
accordance with assumption (ii) $L$ tries to
bring $f$ to equilibrium with the ``thermal bath'' represented by the
massive neutrals, i.e.\ to a Maxwellian distribution with zero mean velocity and the
a priori specified temperature of the neutrals (and ions). Choosing units
in which this temperature is unity this equilibrium distribution of the
electron velocities is, $M_n(v) \equiv (2\pi)^{-3/2} \exp[{-v^2 \over 2}]$.
In the absence of the external field $E$, $f(v,t)$ would approach $M_n(v)$,
as $t \to \infty$,
independent of the nature of the collisions between the electrons. For $E
\ne 0$ we have to model these collisions.
We simplify the collision
term here keeping, however, its essential physical features: it conserves
momentum and energy and vanishes when $f$ is {\it any} Maxwellian. Thus
while the electrons are being driven away
from equilibrium with the ions and neutrals by the electric field
their internal collisions try to bring the distribution to a general
Maxwellian $M(v; u, T) = (2\pi)^{-3/2} \exp[-(v-u)^2/2T]$ with the
instantaneous value of momentum $u$ and energy $e = {1 \over 2} u^2
+ {3 \over 2} T$.
The nonequilibrium ``phase transition'' in this system, which we show that $f$ and thus
also the current and energy of the electrons undergo, can be understood as
a transition, when the speed of the electrons increases with the field
from a low energy regime
in which there is a strong coupling to the ions to a high energy
regime where this coupling essentially vanishes. In the absence of
electron-neutral collisions this leads to a runaway situation [5]. With
neutrals present the stationary distribution
would be smooth as $E$ changes in the
absence of the nonlinear coupling induced by the
electron-electron collisions, the frequently used ``electron swarm''
approximation [6]. These collisions produce a cooperative
effect which prevents the distribution from deviating
too much from a Maxwellian. The strength of this effect is controlled by
the
parameter $\epsilon$, which measures the effective mean time between
electron-electron collisions. Our proofs apply to the situation when
$\epsilon$ is ``small enough''.
The precise form of the collision operator $L$ we shall use is
$$Lf(v) =
\nabla\cdot\biggl(D(v)M_n(v)\nabla\biggl({f(v)\over M_n(v)}\biggr)
\biggr)\eqno(2)$$
where
$$D(v) = a\exp(-b|v|^2/2) + c\eqno(3)$$
for some strictly positive constants $a$, $b$ and $c$.
For $Q(f)$ we take
the BGK model of the Boltzmann collision kernel [7],
$$Q(f) = M_f - f\eqno(4) $$
where for any velocity distribution $f$, $M_f$ denotes the Maxwellian
distribution
with the same first and second moments as $f$.
Since $Q(f)$ drives $f$ close towards the Maxwellian manifold $M_f$ we
should have, formally at least, that in the limit $\epsilon \to 0$,
$f$ will equal $M_f$ for all time $t>0$. To keep track of the
evolution of $f$ we would then need only keep track of the moments
$$u(t) \equiv \int_{\IR^3}vf(v,t)\d,\eqno(5)$$
and
$$ e(t) \equiv {1 \over 2}\int_{\IR^3}v^{2}f(v,t)\d\quad .\eqno(6)$$
Using the prescription $f= M_f$, one easily obtains
$$
{du \over dt} = F(u,e) = E - u[c + a \exp(-w) {1+b \over (1+b/\beta)^{5/2}}], \eqno(7a)
$$
$$
{de \over dt} = G(u,e) = Eu -c[2e(1-\beta)+\beta u^2] - {a \exp(-w) \over
(1 + b/\beta)^{{5 \over 2}}} [2e(1-\beta) + u^2(\beta - b {1-\beta \over b+\beta})], \eqno(7b)
$$
\noindent where $\beta^{-1} = T = {2 \over 3} (e - {1 \over 2} u^2)$, and
$w = bu^2/2(1+bT)$.
%$${{\rm d}\over {\rm d}t}\left(\matrix{\alpha(t)\cr
%\beta(t)\cr}\right) = \left(\matrix{F(\alpha(t), \beta(t))\cr
%G(\alpha(t), \beta(t))\cr}\right)\eqno(1.6)$$
\noindent Solving (7) for the stationary values of $u$ and $e$
gives, for certain ranges of $|E|/c$, depending on
$b$ and $c/a$, three such pairs, two stable, and one unstable,
see Fig. 1.
Our primary goal here is to show that this situation
actually does hold for small, but positive, values
of $\epsilon$; i.e. that the
interaction between the hydrodynamic and the non--hydrodynamic
modes does not destroy this picture involving only the hydrodynamic modes.
The following
theorems enable us to do this.
\bigskip
\noindent{\bf Theorem 1.} {\it Let $f$ be a solution of (1) with
$f(v,0) = M_0(v;u_0,T_0)$ some arbitrary Maxwellian.
Then there is a constant $K$ depending only on $a$, $b$, $c$, $|E|$
and $e_0$ such that
$$\|f(v,t) - M_{f(\cdot,t)}(v)\| \le K\epsilon^{1/2},\eqno(8)$$
$\|\ \cdot \ \|$ denoting the $L^1(\IR^3)$--norm.}
\bigskip
\noindent{\bf Theorem 2.} {\it Let $(u^*,e^*)$ be
a stationary point
of {(7)} and $T^*$ the corresponding temperature.
Then for $\epsilon$ small enough, there exists a
unique stationary point in an $\epsilon$-neighborhood of $M(v;u^*,T^*)$, and there
are no other stationary solutions of (1) that do not correspond to
stationary solutions of (7).
Moreover, if $(u^*,e^*)$ is a stable stationary solution of (7), then there
is a neighborhood of $M(v;u^*,T^*)$ that is stable for (1). More
precisely, given any $\delta > 0$,
there is an $\epsilon$
greater than zero such that if
$$\|f(v,0) - M(v,u^*,T^*)\| \le \epsilon\eqno(9)$$
then the $f_\epsilon(v,t)$ which solves (1) with the value of $\epsilon$ satisfies
$$\|f_\epsilon(v,t) - M(v;u^*,T^*)\| \le \delta\eqno(10)$$
for all $t\ge 0$.
%If, however, $(u^*,T^*)$ is not stable, then there exist a $\delta >0$
%so that for every $\epsilon >0$, there is a solution $f_\epsilon(\cdot,t)$
%of (1.1) with Maxwellian initial data satisfying
%$$\|f_\epsilon(v,0) - M(v;u^*,T^*)\|_{L^1(\IR^3)} \le \epsilon\ ,\eqno(12)$$
%but such that for some finite $t>0$
%$$\|f_\epsilon(v,t) - M(v,u^*,T*)(\cdot)\|_{L^1(\IR^3)} \ge \delta\
%.\eqno(13)$$}
Likewise, the unstable stationary solutions of (7) can be shown to
correspond to unstable stationary solutions of (1).\/}
\bigskip
The detailed description of the stationary solutions follows by their almost explicit
construction. In fact one can show that the moments $(u_\epsilon, e_\epsilon)$ of the stationary
solution
$f_\epsilon$ satisfy a closed equation of the form
$$F(u_\epsilon, e_\epsilon)+\epsilon \tilde F_\epsilon(u_\epsilon, e_\epsilon)=0,
\eqno(11a)$$
$$G(u_\epsilon, e_\epsilon)+\epsilon \tilde G_\epsilon(u_\epsilon, e_\epsilon)=0,
\eqno(11b)$$ with suitable $\tilde F_\epsilon$ and $\tilde G_\epsilon$.
An application of the implicit function theorem provides the moments $(u_\epsilon,
e_\epsilon)$ of the stationary solution in an $\epsilon$-neighborhood of $(u^*,e^*)$.
These moments determine $M_{f_\epsilon}$ and hence $f_\epsilon$ itself, using the
stationary equation. This construction is peculiar of the choice of the BGK collision
kernel, but the results should extend to more general collisions.
The stability statements in theorem (2) are consequence of a more general result:
if we compute
$u(t)$ and $e(t)$ for a solution $f(v,t)$ of (1) satisfying
$f(v,0) = M_0(v;u_0,T_0)$ then $u(t)$ and $e(t)$ satisfy
$$
{d \over dt} u(t) = F(u,e) + \epsilon^{1/4} \gamma(t)
$$
$$
~~~~~~~~~~~~~~~~~~~~~~~ \eqno(12)
$$
$$
{d \over dt} e(t) = G(u,e) + \epsilon^{1/4} \eta(t)
$$
where $F$ and $G$ are defined in (7) and $\gamma(t), \eta(t)$ are bounded
uniformly in $t$ with a bound independent of $\epsilon$.
Our proof of Theorems 1 and 2, whose details will be presented elsewhere,
has several ingredients. First we prove by direct calculations and
``interpolation inequalities'' some a priori bounds on the moments and
smoothness of $f(v,t)$. We then prove an entropy production inequality to
get strong bounds on the tendency of the BGK operator to keep $f$
nearly Maxwellian.
To get an idea of how the latter works consider the time evolution of the
system's ``free energy'' $A(t) = [e - T_n s]$
$$
A(t) = \int[{1 \over 2} v^2 + \log f] f(v,t) d^3v = \int \log(f/M_n) fd^3v- {3\over 2} \log
2\pi\eqno(13)
$$
where $T_n$ is the ion and neutral temperature which we have set equal to $1$.
Using the properties of $L$ and $Q$ we find
after some manipulations
$$
{d \over dt} A(t) = E\cdot u - 4\int D(v)M_n(v)[\nabla\sqrt{f/M_n}]^2 d^3v - {1 \over
\epsilon} \int \log (f/M_f)[f - M_f] d^3v. \eqno(14)
$$
Combining (14) with other inequalities and bounds leads
to the inequality
$$
\int \log(f/M_f) f(v,t) d^3v \leq \epsilon K, \eqno(15)
$$
where $K$ is a constant. This shows that $f$ has to stay close to $M_f$
when $\epsilon$ is small.
Our
methods allow us also to take into account an
initial layer, i.e. we need not start with a Maxwellian $f(v,0)$. We can
also extend our results to more realistic collision kernels both linear and
nonlinear. We have not tried to formulate this here since these required
additional conditions and our main interest is to
show rigorously how nonuniqueness of the electron distribution in the limit
$t \to \infty$ can come about from the dynamics of the interactions between
the electrons in a strongly driven nonequilibrium system. Just how strong
the field needs to be depends of course on the model. It is easy to see
that for $E = 0$, $f(v,t) \to M_n(v)$ independent of the initial $f(v,0)$
exponentially fast in time. We can expect the same to be true for
sufficiently small fields, with the current $u$ linear in $E$. For our
particular model this will also occur for very large $|E|$, since for large
velocities $D(v) \to c$ and for a constant $D(v) = c$ the stationary
solution is $M(v;u,T)$ with $u = E/c$ and $T = T_n = 1$ in our units.
With more realistic electron-neutral collisions the large $|E|$ behavior
will presumably be that of a Druyvestan distribution [6].
\bigskip
\bigskip
\noindent{\bf Acknowledgements}
This research was supported in part by NSF Grant DMS--920--7703, CNR--GUFM
and HURST, and AFOSR Grant AF--92--J--0015.
\bigskip
\noindent {\bf References}
\bigskip
\item{[1]} \ M.Cross and P.C. Hohenberg, Rev. Mod. Phys., {\bf 65}, 881
(1993); P. Berg{\'e}, Y. Pomeau, C. Vidal, {\it Order within Chaos}, J.
Wiley (1984); A.V. Gaponov-Grekhov, M.I. Rabinovich, {\it Nonlinearities in
Action}, Sprinegr (1988).
\item{[2]} \ J. Koplik, J.R. Banavar and J.F. Willemson, Phys. Rev. Lett.
{\bf 60}. 1282 (1988); Phys. Fluids A, {\bf 1}, 781 (1989); D. Rappaport, Phys. Rev.
Lett. {\bf 60}, 2480 (1988).
\item{[3]} \ B. Schmittmann, Intl. J. Mod. Physics, {\bf B4}, 2269 (1990);
B. Schmittmann and R.K.P. Zia, preprint; P. Bak, C. Tang and K.
Wiesenfeld, Phys. Rev. Lett. {\bf 59}, 381 (1987); T.M. Liggett, {\it
Interacting Particle Systems}, Springer Verlag (1985).
\item{[4]} \ A. Rokhlenko, Phys. Rev.A, {\bf 43}, 4438 (1991); A.
Rokhlenko and J.L. Lebowitz, Phys. Fluids B: Plasma Physics, {\bf 5}, 1766
(1993).
\item{[5]} \ c.f.\ R. Balescu, {\it Transport Processes in Plasmas},
North-Holland (1988), Chapter 19.
\item{[6}] \ R.N. Franklin, {\it Plasma Phenomena in Gas Discharges}, Clarendon
Press, Oxford, 1976.
\item{[7]} \ c.f.\ C. Cercignani, {\it The Boltzmann Equation and Its
Applications}, Springer Verlag, 1988.
\vfill \eject
$$b = {a \over c} = 0.1$$
\bigskip
\bigskip
\bigskip
\bigskip
$u$ is the drift velocity
\bigskip
$e = {u^2 \over 2} + {3 \over 2}T$ is the mean energy
\end
The proof of these theorems involves a number of steps, even for the simple BGK model. It is possible to develop analogs of the lemmas proved below for other
more realistic Boltzmann collision kernels, including at least the
case of hard spheres. This, however, requires considerably more labour.
The proofs are organized as follows. In Section 2 we prove moment bounds.
Section 3 contains the proof of two ``interpolation inequalities''. The
first of these will be used to obtain {\it a--priori} smoothness bounds
in Section 4. The second will be used to transform the smoothness
bounds of Section 4 into a lower bound on the variance of our density.
Having assembled these moment, smoothness and interpolation bounds, we
prove a key entropy production inequality in Section 5. This is used to
get quantitative bounds on the tendency of the BGK operator to keep the
density nearly Maxwellian. What we obtain directly is $L^1$ control on the
difference between $f$ and $M_f$, but the smoothness bounds together with the
interpolation bounds allows us to obtain control in stronger norms.
Section 6 contains the proofs of the theorems, which, given the lemmas,
are quite short.
\end