ABOUT THE LINK BETWEEN THE DETAILED DESCRIPTION OF TRANSITIONS IN AN ION AND THE AVERAGE-ION MODELS GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

A BSTRACT. We study the link which exists between microscopic (detailed) models for the evolution of the electronic configurations in a population of ions and the macroscopic (average ions) models. A rigorous asymptotics is presented in situations where it exists (large temperature; almost empty or almost full shells), and numerical simulations are presented.

1. I NTRODUCTION In the last years, the extension of average-ion models to the modeling of plasmas in off-equilibrium conditions has been considered (cf. [DR], [DFDM]). The validity of this class of models, which are meant to give a simplified macroscopic statistical description of the state of a large set of ions as an alternative to the more complex detailed description based on evolution equations associated to microscopic processes in the plasma, is in general justified on the basis of heuristic arguments, since a priori the average-ion models can strictly be used only when the local thermodynamic equilibrium approximation is valid. In this paper we study the link connecting the microscopic detailed description of a set of ions and its average-ion description in off-equilibrium conditions through the analysis of a toy-model involving only simple processes in the plasma. The analysis can be considered somehow complementary to the analysis performed in papers dealing with the validation problem for linear Boltzmann type equations (like [S], [BGW] or [G], just to give a very short list), where the goal is to find a correct simplified description for a set of particles which in our case would be the sea of free particles (electrons or photons) surrounding the ions. We shall give for our toy-model some rigorous (asymptotic) equivalence results and we shall present numerical simulations. In what follows, we shall consider a set of ions which belong to the same species of atoms in a bath of particles (electrons) at Maxwellian equilibrium at a given temperature T . We denote by Z the charge of the nucleus of the considered atomic species. 1

2

GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

We consider the set of bound electrons in each ion and we collect the electrons in subsets which we shall call levels. Levels are defined by grouping electrons with about the same energy, and usually the grouping is built in such a way that the number of levels N for bound electrons is finite: in our simulations, the levels will be indexed according to the principal quantum number n (up to the number N which is a priori fixed), so that they will correspond to the atomic shells, and we shall use indifferently both words (shells or levels) to denote the same object. A configuration ~k = (k1 , . . . , kN ) of an ion is specified by the occupation number (i.e. the number of electrons) ki ∈ N of each level i in the ionic configuration. Each bound electron shell can accommodate a finite number of electrons; we shall denote by Di the maximal number of electrons which can be accommodated in the shell i (Di = 2 i2 in the numerical example that we present). We shall denote as C the set of all ionic configurations ~k. Electrons can switch their energy to a value corresponding to a different level (bound-bound transitions), and be expelled or absorbed by the ion (continuum-bound transitions). We shall include this last kind of transition by modeling the set of free-electrons as the N + 1-th shell; we write then DN +1 = ∞ for coherence. At (what we shall call) the microscopic level, the set of ions is described by the probability to find an ion in the P configuration ~k at time t, which we denote by g~k (t). We have of course ~k g~k (t) = 1, and the evolution equation for g~k (t) is: X X (1) ∂t g~k (t) = B~k′ →~k gk~′ (t) − B~k→~k′ g~k (t), ~k ′ ∈C

~k ′ ∈C

′ where B~k′ →~k is the rate of the transition ~k → ~k. We notice that, because we included among the allowed transitions the processes of ionization and ′ recombination, the transition ~k → ~k does not necessarily preserve the total number of electrons in the configuration ~k. The description of the system can be simplified thanks to the use of a macroscopic model in which the set of ions in different electronic configurations is replaced by a set of ions all in the same electronic configuration (average ions). The electronic configuration of each (and all) ion in this last system is such that the occupation number of each shell of the average ion is the average of the occupation numbers of the corresponding shell of the ions in the original system. At this macroscopic level, the set of ions is described by the collection of populations of levels for the average ion, which we shall denote by {Ph }h≥1 or P~ = (P1 , . . . , PN ), where Ph ∈ [0, Dh ] denotes the (non necessarily

LINK BETWEEN DETAILED DESCRIPTION AND AVERAGE-ION MODELS

3

integer) population of the h-th level of the average ion. In this modeling, given the level populations P~ at any time t, the probability G~k to find an ion in the configuration ~k is then computed as  N  Y Dh P h k h Ph Dh −kh ( ) (1 − ) , (2) G~k = k h Dh Dh h=1 as if ions would be in what we shall call a local equilibrium. In the average ion description, P~ satisfies the following evolution equation (for n = 1, . . . , N): d Pn (3) Pn = An ({Pm }m≥1,6=n )(1 − ) − Bn ({Pm }m≥1,6=n )Pn , dt Dn where we denote by An the total transition rate to the level n from other levels (including the continuum) and by Bn the total transition rate to other levels (including the continuum) from level n. In general, the rates An and Bn are functions of the population of the levels in the form of a sum of coefficients (themselves depending on P~ ) multiplied by Pm or (1−Pm /Dm ) for m 6= n plus a term coming from the ionization-recombination processes. We would like to get an equation of the form (3) as a consequence (in a certain asymptotics) of an evolution equation for the probability g~k of the form (1), in analogy (even though not strictly in the same sense) to the reduction of hierarchies of equations which describe many particles systems to a single equation for a one particle density (Cf. for instance [DMP]) . To this purpose, we shall compare the evolution of the populations of the shells ~ = (f1 (t), . . . , fN (t)), defined, for h = 1, . . . , N, by f(t) X (4) fh (t) = kh g~k (t), ~k∈C

where g~k satisfies eq. (1), with the evolution of P~ , solution of eq. (3). In order to keep things as simple as possible, we consider the evolution of the level populations in ions where the only involved transition processes between levels are excitation and de-excitation (concerning only one bound electron) which are due to collisions with particles in the bath (including the continuum-bound transitions which may change the total number of electrons in an ionic configuration). As a consequence, we do not take into account the radiative transitions (Cf. [MM] for the physics of such transitions, and for example [BGPS] for a mathematical study of the radiative transfer equations) and the two-electrons collisional transitions. In section 2, we describe in detail the microscopic model that we shall study, and we write in section 3 a non closed equation for the populations f~

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

of the shells. This equation can be closed under a factorization assumption which is related to the equilibrium. We study rigorously in section 4 the asymptotics which enable to pass from the macroscopic model towards the average ion model. Numerical illustrations are then provided in section 5. 2. D EFINITION

OF THE MICROSCOPIC MODEL

When needed, we shall use the following notation for sums of vectors:  1≤i m, and to the ionization process when m = N + 1 or to its inverse process (recombination) when n = N + 1. For the continuum-bound, boundcontinuum transitions, we define kN +1 = Z ∗ , DN +1 = ∞, EN +1 = 0. As a consequence of the detailed balance principle, the microscopic transition probabilities satisfy the following conditions: for n < m, n, m = 1, . . . , N,

(10)

c Rmn =

Dn En −Em c e T Rnm , Dm

and for n = 1, . . . , N (11)

c RN +1n = Dn CT e

En T

c RnN +1 ,

where CT is a positive constant (depending only on T ). In our second model, we assume the microscopic transition probabilities ′ c to be of the form (9), where Rnm depends on (m, n,) En (~k) − Em (~k ) and ′ c T . We write Rnm (~k, ~k ) for the sake of simplicity in the sequel. As a consequence of the detailed balance principle, the microscopic transition probabilities satisfy the following conditions: for n < m, n, m = 1, . . . , N,

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

(12) c Rmn (~k, ~k + (1, −1)nm ) =

Dn En (~k+(1,−1)nm )−Em (~k) c ~ T e Rnm (k + (1, −1)nm , ~k), Dm

and for n = 1, . . . , N, (13)

c ~ RN +1n (k) = Dn CT e

En (~ k+(1,−1)nN+1 ) T

c ~ RnN +1 (k + (1, −1)nN +1 ),

where CT is the positive constant appearing also in the first model. 2.3. Evolution microscopic equations. According to these transition probabilities, we can write down the evolution equation (for our different models) of the probability g~k (t). We begin with our first model: (14)  N N X X km − 1 ) 1{kj 6=Dj ,km 6=0} g~k+(1,−1)jm ∂t g~k = (kj + 1)(1 − D m j=1 m=j+1  kj Dj Ej −Em c T ) e −km (1 − g~k × Rjm Dj Dm  N N X X km − 1 Dm − Ej −Em T ) e 1{kj 6=Dj ,km 6=0} g~k+(−1,1)mj + (kj + 1)(1 − D D m j m=1 j=m+1  kj c )g~ × Rmj −km (1 − Dj k N  X + (kj + 1)1kj 6=Dj g~k+(1,−1)jN+1 j=1

 Ej kj c ~ T −Z (f )CT (1 − )Dj e g~k RjN +1 Dj  N  X Ej kj − 1 ∗ ~ c + Z (f)CT (1 − )Dj e T 1kj 6=0 g~k+(−1,1)jN+1 − kj g~k RjN +1 . D j j=1 ∗

c We recall that in this formula, the rates Rjm , etc., depend on the energy ~ levels Ej . Those levels depend on f through eq. (5) and (7).

Then, we write the corresponding formula for our second model:

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(15)  km − 1 ∂t g~k = ) 1{kj 6=Dj ,km 6=0} g~k+(1,−1)jm (kj + 1)(1 − Dm j=1 m=j+1  kj Dj Ej (~k+(1,−1)jm )−Em (~k) c T ) e −km (1 − g~k × Rjm (~k + (1, −1)jm , ~k) Dj Dm  N N X X km − 1 Dm − Ej (~k+(−1,1)mj )−Em (~k) T 1{kj 6=Dj ,km 6=0} g~k+(−1,1)mj + (kj + 1)(1 − ) e Dm Dj m=1 j=m+1  kj c −km (1 − )g~k × Rmj (~k, ~k + (−1, 1)mj ) Dj N  X (kj + 1)1kj 6=Dj g~k+(1,−1)jN+1 + N X

N X

j=1

 Ej (~ k+(1,−1)jN+1 ) k j c ~ T )Dj e g~k RjN −Z (f~) CT (1 − +1 (k + (1, −1)jN +1 ) Dj  N  X Ej (~ k) kj − 1 c ∗ ~ ~ )Dj e T 1kj 6=0 g~k+(−1,1)jN+1 − kj g~k RjN + Z (f) CT (1 − +1 (k). D j j=1 ∗

3. R EDUCTION

OF THE MICROSCOPIC EQUATIONS TO MACROSCOPIC EQUATIONS AND THEIR CLOSURE

3.1. Non closed equations. When we consider our first model, it is possible to write a simplified (non closed) equation for the quantities fh , starting from eq. (14) and making a suitable change of indices, by summing over all possible configurations. It reads (16)  h−1 X  X d kj Dj Ej (f~)−Eh (f~) kh c ~ T ) − kh (1 − ) e fh = kj (1 − g~k Rjh (f ) dt D D D h j h j=1 ~k

+

N X X

j=h+1

 kj kh Dh − Ej (f~)−Eh (f~) c ~ T ) e ) g~k Rhj (f ) − kh (1 − kj (1 − Dh Dj Dj ~k  X Eh (f~) kh ∗ ~ )Dh e T − kh g~k Rhc N +1 (f~). + Z (f ) CT (1 − Dh ~k

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

The corresponding equation for our second model is (17) h−1 X  X d kh c ~ ~ fh = )R (k, k + (−1, 1)jh ) kj (1 − dt Dh jh j=1 ~k  kj Dj Ej (~k+(1,−1)jh )−Eh (~k) c ~ T Rjh (k + (1, −1)jh, ~k) g~k −kh (1 − ) e Dj Dh N X X kh Dh Eh (~k+(1,−1)hj )−Ej (~k) c ~ T + kj (1 − ) e Rhj (k + (1, −1)hj , ~k) D D h j j=h+1 ~k  kj c ~ ~ )R (k, k + (−1, 1)hj ) g~k −kh (1 − Dj hj X Eh (~ k+(1,−1)hN+1 ) kh c ~ T + Z ∗ (f~) CT (1 − RhN )Dh e +1 (k + (1, −1)hN +1 ) Dh ~k  c ~ −kh Rh N +1 (k) g~k .

3.2. Factorized solutions. In the case of our first model, if there exists a factorized solution of (15) on a certain interval of time, i.e. g~k (t) = P h QN ˆh (kh , t) = 1), equation (16) becomes ˆh (kh , t) (and of course D kh =1 g h=1 g closed:  h−1  X d Pj Dj Ej (P~ )−Eh (P~ ) c ~ Ph T Rjh (P ) ) − Ph (1 − ) e Ph = Pj (1 − dt D D D h j h j=1   N X Ph Dh − Ej (P~ )−Eh (P~ ) Pj c ~ (18) T Pj (1 − + − Ph (1 − ) e ) Rhj (P ) D D D h j j j=h+1   ~) E h (P Ph ∗ ~ T )Dh e − Ph Rhc N +1 (P~ ), + Z (P ) CT (1 − Dh PD where Pj (t) := fj (t) = kjj=0 kj gˆj (kj , t). Equation (18) coincides with equation (3) for the average population on the h-th level in the corresponding average-ion model, by defining A and B in a suitable way. Notice that eq. (18) cannot be obtained from a microscopic model (even when a factorized solution is assumed) when a microscopic screening is imposed (that is, for example, in the case of our second model). It also could not be obtained from a microscopic model in which the effective charge would be microscopic (that is, depending on ~k rather than f~).

LINK BETWEEN DETAILED DESCRIPTION AND AVERAGE-ION MODELS

9

3.3. Equilibrium. We now analyze the existence of equilibrium solutions for equations (14), (15) and (18), and the connection between them. Since this is the most relevant case in physical applications, we shall only look for equilibrium solutions which do not depend on the choice of the c transition rates Rnm : this corresponds to looking for probability densities g~k (t) or occupation numbers Ph (t) such that each coefficient in the linear combination of transitions rates on the right-hand side of (14), (15) or (18) is identically equal to 0. In our second model, represented by eq. (15), in order to satisfy all the constraints, compatibility conditions will be required. 3.3.1. Microscopic equilibrium. We shall first look for an equilibrium solution of equations (14) and (15). Setting each coefficient of the linear combination of transitions rates on the right-hand side of (15) equal to 0, we get a system of equations having as compatibility condition the following condition on the energy: (19)

Ej (~k) − Ej (~k + (−1, 1)sN +1) = Es (~k) − Es (~k + (−1, 1)jN +1 )

for s 6= j. Whenever this condition is fulfilled, we can obtain a (non necessarily factorized) microscopic equilibrium solution. Assuming the electron energies of the form presented in (45), condition (19) is unfortunately not satisfied for our second model. For the first model (corresponding to eq. (14)), the equilibrium (factorized) solution is: Eh (f~eq )  N  Y (Z ∗ (f~eq ) CT e T )kh Dh eq (20) g~k = . kh (1 + Z ∗ (f~eq ) C e Eh (Tf~eq ) )Dh h=1 T Note that this solution is given in implicit form. We shall see that (20) can be connected in a simple way to the equilibrium solution of the macroscopic model. 3.3.2. Macroscopic equilibrium. Equation (18) has as equilibrium solution the Fermi-Dirac distribution, given (implicitly) by: (21)

Pheq = Dh

Z ∗ (P~ eq ) CT e

~ eq ) E h (P T

1 + Z ∗ (P~ eq ) CT e

~ eq ) E h (P T

.

When we consider the function  N  Y Dh Pheq kh P eq eq ( (22) g~k = ) (1 − h )Dh −kh , kh Dh Dh h=1

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

where Pheq is given by formula (21), we find that g~keq is given by (20). This means that for our first model, the macroscopic equilibrium can be obtained as an average of the microscopic equilibrium. We expect therefore that for large times, the result of the microscopic and macroscopic simulations coincide. This property is however not shared by our second model, as we shall see in the fourth simulation of section 5. One can verify that for our first model, the equilibrium solution is the only solution of the microscopic equations which remains factorized on an interval of time (that is, all other factorized initial data lose immediately the property of being factorized in the evolution of eq. (14)), so that the closure presented in subsection 3.2 cannot be considered as consistent. As a consequence, we can in general deduce the macroscopic equations from the microscopic ones only in some asymptotics that will be detailed in next section. 4. A SYMPTOTIC

ANALYSIS

We begin with a proposition mainly based on Gronwall’s lemma which enables to make explicit the evolution of the difference between the microscopic and macroscopic descriptions P (for our first model) in terms of its initial value, the correlation matrix ~k kh kj g~k (s) − fh (s) fj (s), and the temperature T of the bath. Proposition 1. We consider N ≥ 2, an integer number, T > 0, and a family (Di )i=1,..,N of numbers of N. We denote D = max(Di ). We also consider a function Z ∗ of f~ and a sequence (En )n∈N of functions of f~ which all lie in W 1,∞ (that is, the space of bounded and Lipschitz-continuous functions). c Then, we take a family of transition rates (Rij )i=1,.,,N ;j=i+1,.,,N +1 which are ~ functions of T and f and lie (for all T ) in the space W 1,∞ with respect to the variable f~. Finally, we take a constant CT > 0. Then for all Tinf > 0 and t1 > 0, one can find a constant K depending only on N, D, ||Z ∗||W 1,∞ , supn=1,..,N ||En ||W 1,∞ , Tinf such that (for any T ≥ Tinf > 0 and t ∈ [0, t1 ]) if P~ := P~ (t) is solution of eq. (18) and (g~k )~k∈C := (g~k )~k∈C (t) is solution of eq. (14), then (23) X

h=1,...,N

|fh (t) − Ph (t)| ≤

 X

h=1,...,N

|fh (0) − Ph (0)|+ K T

−1

XZ j,h

0

t1

 X ds eK ϕ(T ) t , k k g (s) − f (s) f (s) ~ h j h j k ~k

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11

where c c (24) ϕ(T ) := sup ||Rhj (T, ·)||W 1,∞ + (1 + CT ) sup ||RhN +1 (T, ·)||W 1,∞ , h

h,j

and fh is given (knowing g~k ) by formula (4). Proof of Proposition 1: since we are looking to our first model, we can write (25) X  N N X Eh (f~) d c ∗ ~ ~ ~ ~ fh (t) = Ahj (f )fj (t) − Ajh (f) + RhN +1 (f)(Z (f)CT e T + 1) fh (t) dt j=1 j=1 −

N X

∗ ~ ~ hj + Rc Bhj (f)χ hN +1 Z (f ) CT Dh e

Eh (f~) T

j=1

and (26) X  N N X ~) E h (P d c ∗ ~ ~ ~ ~ T Ahj (P )Pj (t) − Ajh (P ) + RhN +1 (P )(Z (P )CT e Ph (t) = + 1) Ph (t) dt j=1 j=1 −

N X

c ~ ∗ ~ Bhj (P~ )Ph Pj + RhN +1 (P )Z (P )CT Dh e

j=1

where χhj (t) =

X

kh kj g~k (t),

~k

(27)

Ahj

 c   Rjh E −E c Dh − j T h = Rhj e Dj   0

j h, j=h

Ej −Eh

Bhj

(1 − e T = Dh

)

Ahj ,

and the evolution of the quantity fh − Ph is given by:

~) E h (P T

,

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

 N  X d ~ ~ (fh (t) − Ph (t)) = Ahj (f)fj (t) − Ahj (P )Pj (t) dt j=1  X  N
 Eh (f~) c ∗ ~ ~ T − Ajh (f~) + RhN + 1 fh (t) +1 (f ) Z (f )CT e j=1

N ~) X
 E (P c ~ ) Z ∗ (P~ )CT e hT + 1 Ph (t) − Ajh (P~ ) + RhN ( P +1

(28)

j=1



N X   − Bhj (f~)χhj (t) − Bhj (P~ )Ph Pj j=1

 c ~ +CT Dh RhN +1 (f )e

Eh (f~) T

c ~ Z ∗ (f~) − RhN +1 (P )e

~) E h (P T

 Z ∗ (P~ ) .

We can rewrite (28) as (29) N X d (fh (t) − Ph (t)) = Ahj (f~)(fj (t) − Pj (t)) dt j=1 N X E (f~) c ~)(Z ∗ (f~)Ce hT + 1)](fh (t) − Ph (t)) Ajh (f~) + RhN ( f −[ +1 j=1

−

N X

+

N X

j=1

j=1

+

~ hj (t) − Ph Pj ) Bhj (f)(χ

[Ahj (f~) − Ahj (P~ )]Pj (t)

N nX  j=1

Ajh (P~ ) − Ajh (f~)



~)  c o E (P E (f~) ~)(Z ∗ (f~)CT e hT + 1) Ph (t) ~ )(Z ∗ (P~ )CT e hT + 1) − Rc + RhN ( P ( f +1 hN +1

N X ~)     E (P E (f~) ~ − Rc ~ P h P j + C T Dh R c ~ hT Z ∗ (f) ~ )e hT Z ∗ (P~ ) . ( P + Bhj (P~ ) − Bhj (f) f)e ( hN +1 hN +1 j=1

LINK BETWEEN DETAILED DESCRIPTION AND AVERAGE-ION MODELS

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Therefore, (30) X h

|fh (t) − Ph (t)| ≤

X h

|fh (0) − Ph (0)| + 2N sup ||Ahj (T, ·)||∞ h,j

Eh T

c ∗ + sup ||RhN +1 (T, ·)(Z CT e h

+ sup ||Bhj (T, ·)||∞ h,j

+ 1)||∞

Z tX 0

j

Z tX 0

j

|fj (s) − Pj (s)|ds

|fj (s) − Pj (s)|ds

N Z t X X [kh kj − fh (s)fj (s)]g~k (s) ds 0

j,h=1

+2N D sup ||Ahj (T, ·)||Lip h,j

~k

Z tX 0

c ∗ +D sup ||RhN +1 (T, ·)(Z CT e h

+N D 2 sup ||Bhj (T, ·)||Lip h,j

j

Eh T

+ 1)||Lip

c ∗ +D sup ||RhN +1 (T, ·)Z CT e h

Eh T

Z tX 0

Z tX 0

|fj (s) − Pj (s)|ds

j

j

|fj (s) − Pj (s)|ds

|fj (s) − Pj (s)|ds

||Lip

Z tX 0

|fj (s) − Pj (s)|ds,

j

so that  X X |fh (t) − Ph (t)| ≤ |fh (0) − Ph (0)| + 2N D sup ||Ahj (T, ·)||W 1,∞ h

h,j

h

Eh

c ∗ c T || +2D sup ||RhN W 1,∞ + D sup ||RhN +1 (T, ·)||W 1,∞ +1 (T, ·)Z CT e h h  Z tX +N D 2 sup ||Bhj (T, ·)||Lip |fj (s) − Pj (s)|ds h,j

+ sup ||Bhj (T, ·)||∞ h,j

Then, we observe that sup ||Ahj (T, ·)||W 1,∞ ≤ D h,j

0

N X

j,h=1

0

~k

c sup ||Rhj (T, ·)||W 1,∞ h,j

suph

e

||Eh ||∞ Tinf

  suph ||Eh ||Lip 1+2 , Tinf

h ||∞ suph ||Eh ||∞ 2 suph ||ETinf , e T h ||∞ suph ||Eh ||W 1,∞ suph ||ETinf e ≤ 2 ||Ahj (T, ·)||W 1,∞ T

||Bhj (T, ·)||∞ ≤ 2 ||Bhj (T, ·)||Lip

j

Z t X ds. [k k − f (s)f (s)]g (s) ~ h j h j k

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GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

≤ 2D

c sup ||Rhj (T, ·)||W 1,∞ h,j

h ||∞ suph ||Eh ||W 1,∞ 2 suph ||ETinf e T

  suph ||Eh ||Lip 1+2 , Tinf

Eh

c ∗ c ∗ T || ||RhN W 1,∞ ≤ CT ||Z ||W 1,∞ ||RhN +1 (T, ·)||W 1,∞ +1 (T, ·)Z CT e   h ||∞ suph ||Eh ||Lip suph ||ETinf . e × 1+ Tinf As a consequence, for some constant K which depends only on N, D, Tinf , ||Z ∗||W 1,∞ and suph ||Eh ||W 1,∞ , X X |fh (t) − Ph (t)| ≤ |fh (0) − Ph (0)| h

+K ϕ(T )

Z tX 0

j

h

|fj (s)−Pj (s)|ds+K T

−1

N Z t X X [kh kj −fh (s)fj (s)]g~k (s) ds,

j,h=1

where

0

~k

c c ϕ(T ) = sup ||Rhj (T, ·)||W 1,∞ + (1 + CT ) ||RhN +1 (T, ·)||W 1,∞ . h,j

Then, thanks to Gronwall’s lemma (and for all t ∈ [0, t1 ]): X |fh (t) − Ph (t)| ≤ h

X h

|fh (0)−Ph (0)|+K T

−1

N Z X

j,h=1

and Proposition 1 is proven.

0

t1

 X ds eK ϕ(T ) t , [k k −f (s)f (s)]g (s) ~ h j h j k ~k

We can deduce from this proposition the equivalence of the microscopic and macroscopic descriptions when the temperature is large, under conditions which are satisfied by the rates (which are those of [DR]) taken in the simulations of Section 5. Corollary 1. (H IGH T EMPERATURE LIMIT ) Under the same assumption as in Prop. 1, and for initial data such that Ph (0) = fh (0) (that is, “well prepared” initial data), h = 1, . . . , N, if c lim sup kRnm (T, ·)kW 1,∞ < +∞

(31) (32)

T →∞ n,m

c lim CT sup kRnN +1 (T, ·)kW 1,∞ < +∞,

T →∞

n

then, for large T and for any t1 > 0 (33)

sup

N X

t∈[0,t1 ] j=1

|fj (t) − Pj (t)| = O(T −1).

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Proof of Corollary 1: since Prop. 1 is valid and fh (0) = Ph (0) for all h, N X

(34)

j=1

|fj (t) − Pj (t)| ≤ N 2 D 2 K T −1 t1 eKϕ(T )t ,

where ϕ is given by (24), so that, from (31) and (32), we get (33). This ends the proof of Corollary 1. Remark: ThoughPwe also have (thanks to the analysis of equilibria in Section 3) limt→∞ j |fj (t) − Pj (t)| = 0, it doesn’t seem possible to take t1 = +∞ in estimate (33). We now turn to another type of asymptotics, namely the situation in which all shells are almost full or almost empty during the evolution of the plasma. Corollary 2. Under the assumptions of Proposition 1, if, for all t ∈ [0, t1 ], fhi (t) fhi (t) < ε, for i = 1, . . . , N1 and 1 − D < ε for i = N1 + 1, . . . , N (for Dhi hi P some N1 ∈ {1, .., N}), and if h=1,...,N |fh (0) − Ph (0)| ≤ C1 ε, then for all t ∈ [0, t1 ], N X

(35)

h=1

where

|fh (t) − Ph (t)| ≤ C2 ε,

  2 −1 3 C2 = C1 + N K T D t1 eK ϕ(T ) t ,

and K, ϕ are the constants appearing in Proposition 1 (and formula (24)). Proof of Corollary 2: We first observe that under our assumption, for i = 1, .., N1 (and any δ ∈]0, Dhi [), X Dh i ε, g~k (t) ≤ δ ~k:kh ≥δ i

and for i = N1 + 1, .., N, X

~k:kh ≤Dh −δ i i

g~k (t) ≤

Dh i ε. δ

Then, for t ∈ [0, t1 ], h, j = 1, .., N, X X (36) (kh kj − fh (t)fj (t))g~k (t) = 1h∈{1,..,N1} (kj − fj ) kjh g~k (t) ~k

kjh ≥1

16

GUIDO CAVALLARO, LAURENT DESVILLETTES, AND VALERIA RICCI

+1h∈{N1 +1,..,N }

X

(kj − fj ) (kjh − Djh ) g~k (t),

kjh ≤Djh −1

so that thanks to the estimate above, X ≤ D 3 ε. (37) (k k − f (t)f (t))g (t) ~ h j h j k ~k

From (23) and (37) we get then (38)  X  X 2 −1 3 |fh (t)−Ph (t)| ≤ |fh (0)−Ph (0)|+N K T D ε t1 eK ϕ(T ) t . h=1,...,N

h=1,...,N

This ends the proof of Corollary 2. It is of course difficult to guarantee that the shells will remain almost full or almost empty on a long interval of time, it is however at least possible to show that for a small interval of time, it remains so if it is true initially. This is the point of the following proposition: Proposition 2. Under the same hypothesis as in Proposition 1, and for initial data such that Ph (0) = fh (0), h = 1, . . . , N (that is, “well prepared” initial data), if (39) (40)

Phi (0) < ε for i = 1, . . . , N1 Dh i Ph (0) < ε for i = N1 + 1, . . . , N 1− i Dh i

for some N1 ∈ {1, .., N}, then (41)

N X h=1

|fh (t) − Ph (t)| ≤ C (εt + t2 ) eK ϕ(T ) t ,

where K and ϕ are the constants appearing in Proposition 1 (and forc mula (24)), and the constant C depends on T , supn,m kRnm (T, ·)kW 1,∞ , ||Z||W 1,∞ , N, suph kEh kW 1,∞ and D. Remark: We can in particular consider as initial datum g~k (0) the microscopic equilibrium solution (22) for a temperature T ∗ s. t., for each Pheq , ∗ P eq (T ∗ ) either hDh < ε or 1 − PhD(Th ) < ε, with ε small. This is what is done in the two first simulations of Section 5. Proof of Proposition 2:

LINK BETWEEN DETAILED DESCRIPTION AND AVERAGE-ION MODELS

17

We have for the correlation matrix (for any h, j = 1, .., N): (42) X X | (kh kj − fh (t)fj (t))g~k (t) − (kh kj − fh (0)fj (0))g~k (0)| ≤ ~

~

k X k
˙ ˙ t sup (kh kj − fh (s)fj (s))g˙~k (s) − fh (s)fj (s) + fj (s)fh (s) g~k (s) ≤ s∈[0,t]

~k

h suph kEh k∞ T 4D 2 N (1 + ND) e i suph kEh k∞ T +(4ND 3 + 2D 2 ) (1 + ||Z ∗||∞ CT e ) t c sup kRnm (T )k∞ n,m

:= c1 t,

c and c1 depends on supn,m kRnm (T, ·)k∞ , T , ||Z ∗ ||∞ , D, N,suph kEh k∞ . According to the proof of Corollary 2, for h = 1, . . . , N, X ≤ D 3 ε. (k k − f (0)f (0)) g (0) ~ h j h j k ~k

Therefore, we get the bound: Z t X c1 2 t. (43) (kh kj − fh (t)fj (s)) g~k (s) ds ≤ D 3 ε t + 2 0 ~k

We use then Proposition 1 to get N X h=1

|fh (t) − Ph (t)| ≤ K T

−1

  c1 2 K ϕ(T ) t 3 N D εt+ . t e 2 2

Finally, Proposition 2 is proven. Remark: Of course, both Corollary 2 and Proposition 2 are valid (with P i fi ,