V~~lumc 42.4. number 3 PHYSICS LETTERS 4 December 1972

ONTHEEXISTENCEOFATWO-DIMENSIONALCLASSICALPLASMA

J.J. GONZALEZ and P.C. HEMMER Inslirurr for Iwrc~tisk f:lsikt. :VTfi. I9lircrsiry of’ Trcmfht~itn. Srwwaj

Received 17 Octohcr 1972

‘I’hc csiutenl.r rlf the partitilm t’unrtitrn of I classical plasma for tcmperaturcs above a critical value Tc is proved.

Consider a two-dimensional classical plasma consist- ing of N positively and N negatively charged particles with charge of absolute value q(2m,I )‘I’ interacting with each other through the Coulomb potential q(ri.5)’ -qiqj InIfi qlwhere 4i()“E,))“’ is the charge of the particle i.

Above a critical temperature TC E y2/2k the follow- ing equation of state was derived by several authors [I -41 under the assumption that the partition func- tion exists.

p=pk(T- f c,

where p is the number of particles per unit volume.

Below T, the configuration integral diverges and one is forced. in the classical treatment. to introduce a hard core of finite radius d. But the equation of state exists in the limit d -, 0 and is given [4] by

p=f pkT (T<T.;.).

If no hard core is present the condition T > q is necessary for the existence of the partition function, but the question of its sufficiency has been raised several times [3.5] and is answered affirmatively in the present note. We give a rigorous proof that the partition function

exists whenever o < 2. Here, V is a two-dimensional volume, Eij = qiqj fq2 and o = flq2 = 2T,/T.

The convergence of the configuration integral is proved by means of the inequality

(2)

where now we have characterized positively charged particles by indices I. . . . .A’ and negatively charged ones by I’. . . ,A”. The sum in the right-hand side is carried out over all N! permutations ii. . . . . <& of

I’. . . . .N’. The left hand side of eq. ( 2) is simply an- other form of the integrand in eq. ( 1). The inequality (7) provides a majorant of the configuration integral for 0 < 2:

Q Q z”tN-1 )W N! r’l-W I N (3)

Physically speaking our inequality shows that coales- cing in many-particle clusters is not more singular than formation of a neutral pair: the attraction be- tween opposite charged particles is successfully bal- anced by the repulsion between like charged ones.

To prove the inequality consider a possible config. uration and select out of all possible combinations of pairs of particles with opposite charges those two par- ticles whose distance is least (if there are several pairs satisfying this requisite any choice will do). Index these particles by 1 and 1’. Excluding this pair. select out of the neutral pairs built out of the 2N - 2 re- maining particles two particles whose mutual distance is minimal and index them by 2 and 2’. Repeat this procedure N times. Using the triangle inequality one hasfori<k

(4)

and therefore

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