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On “critical points” in the two-dimensional classical jellium
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
ON “CRJTICAL POINTS” IN THE TWO-DIMENSIONAL CLASSICAL JELLIUM S. JOHANNESEN Matematisk Institutt, Universitetet i Oslo, Blindern, Oslo 3, Norway
and D. MERLINI Mathematisches Institut 1, Ruhr-Universitdt Bochum, 4630 Bochum, Germany and Centre de Physique Théorique, Université d’Aix Marseille II, 13288 Luminy Cedex 9, France Received 10 August 1982 Revised manuscript received 20 October 1982
Using the results of a careful extrapolation method applied to systems with small particle-number N, we give an estimate for the location of a possible critical point in the two-dimensional classical jellium. Our computation yields the criti2/kT 137, in excellent agreement with the results of computer experiments. cal value ‘y~= e
In recent years there has been a considerable interest in the study of two-dimensional plasmas with long range logarithmic interaction between the particles, in particular the two-dimensional classical jellium (classical point electrons in a neutralizing charge background) usually called the OCP model [1—4].A vanety of theoretical techniques have been used to investigate the behavior of the system in the plasma—fluid equilibrium state in the weak [5] and in the strong coupling regime [6—8];moreover the harmonic approximation for a “triangular crystalline state” has also been computed [9]. Very recently, some computer experiments with Monte Carlo [10] as well as with Molecular Dynamics techniques [11], reveals that the model under consideration undergoes a first order fluid—solid phase transition at finite temperature (like the three-dimensional case and the two-dimensional surface plasma), but a theoretical proof is still lacking. The results of the computer experiments give a possible freezing transition located in the range ‘v~ 140 [10] and “~ 135 [11], where = e2/kTis the twodimensional plasma parameter. In this note we present a very simple method of computation of the fluid ~‘
free energy which is independent of the computer experiment techniques and is based on a careful extrapolation of the exact computations of the free energy of the system with few (N = 1, 2, 3,4) electrons. The method is discussed in detail in ref. [12] *1 To localize a possible critical point in the system we then use the standard argument which consists in comparing the free energy of the fluid with that of the “solid” obtained by other methods. We found that the fluid freezing point is located in the region y~= e2/kTc 137 in excellent agreement with the results of refined computer experiments. To begin, let us consider the model defined on a disk of radiusR in the plane. The main point is to obtain an accurate Ansatz for the free energy density of the model in the fluid phase. The excess free energy fCX (with respect to the H-stability bound —B and the perfect gas limit In p 1) is defined by: 4’ex I F B~ 1 1/ ~ L7f(1’ j —Qn~ )j~’y, (1 where exp(—~3fN)lnQ andB =(~1n~rp +~)[3]. -
—
—
~
*1
—
This work has been presented by one of us (D.M.) at the workshop on Statistical mechanics of ionic matter, held
1
Postal address.
0 031-91 63/82/0000—0000/$02.75 © 1982 North-Holland
at Les Houches (March—April 1982).
21
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
Table 1 ~
yex(7)/~
——
jex(7)/7 —~-—_____________
2 8 10 14
0.415529700 0.229999474 0.177959464 0.146737916
________________________
0.415529700 0.231207197 0.178649591 0.147221005
f8x(7)17
~
_______________________
0.415529700 0.23060333 0.17830452 0.14697945
________
350 II
Q is the partition function, 3 l/kT the inverse ternperature, —e the electron charge,N the particlenumber and p N/Vthe density; —B is the H-stability bound (i.e. H ~ —Be2N) and ~ = e2/kT the plasma parameter. In table 1 we present the results of our extrapolation to large N (thermodynamic limit) for ~ at the discrete values y = 2, 8, 14, 20. In the table fe~((
7)
and yex(.1) are respectively lower and upper estimate while f~X(y) is their mean value. The results of table 1 have been obtained with the help of the exact computation of the free energy for systems just consisting of N = 2,3,4 particles and the detailed treatment has been given in ref. [12]. For clarity we briefly explain here the procedure adopted to obtain these results. We consider a sequence of systems with small particle-number N = 2, 3, 4, neutral and embedded in a circular domain of volume VN = N/penergyf~(’y) at fixed density p and computeThen exactly their free = —(ln QN)/(IW). an estimation of the limit, linIN fN(7) f~(y),(called the thermodynarnic limit) can be obtained by the fN f +a(y)N~lnN+N~b(’y)+
Ansatz [12]:
where f,,,a, b can be determined from the exact computations of fN which was carried out forN = 2, 3,4, and the results are the numbers given in table 1. The Ansatz for the free energy of the fluid we consider, is very similar to the exact solution of the model in the so-called STLS truncation scheme of the equilibrium hierarchy [3] (other Ansätze of course possi. ble), and given by:
+
= A1 I-v + A2 + (~a ‘y — b) ln [/(y [2(a + b)/] ln(7 + 2) .
+
2)] (2)
Notice that in the STLS scheme a = 0 [11]. We can now compute f~(y)i.e. A1, A2, a, b using our nonperturbative results at ‘y = 2, 8, 14, 20 given in table 1 22
~
320 0~ ~t0
820
830
833
130
13?
1+0
145
(
Fig. 1. —
and plot feX(y)/y, fex(7)/7 andf~(7)/7as a function of ‘y in fig. 1. These are our estimates for the excess free energy of the model in the fluid phase. Looking for a critical point we then use the standard argument cited above which consists in comparing the free energy branch of the “fluid phase” given by eq. (2) and the harmonic approximation for the freeenergy branch of the triangular crystalline “solid” [9] (which is believed to be very accurate for large y) given by: tln 0 262 3 fs(k’YJ~/7 = 0 00062 + ~ 7 7. In eq. (3)E 0 = 0.00062 is the excess internal 2(~ln energyirp of the perfect triangular lattice [9], i.e. E + e +~)E 0.f5(7)/yis also plotted in fig. 1. The intersection of the curves in fig. 1 gives then a lower respectively upper estimate y~= 133 and = 140, while the estimate using the average f~X/7 gives the critical value ‘‘~= 137. Our results are in excellent agreement with the results obtained by Molecular Dynamics (‘va = 135) and Monte Carlo (‘vt = 140) methods. (Notice the scale in fig. 1). To conclude, we note here the following: clearly the above argument is not complete in fact the triangular lattice is the most stable one since it has lower energy then the hexagonal and square lattice at zero temperature [21; nevertheless if a freezing transition for the fluid into a “solid” takes place at y~,then other possible transitions between the three different “solids” should also be investigated, since the nature —
.
Volume 93A, number 1
PHYSICS LETTERS
of the fluid—solid transition which emerge from cornputer experiment studies is still unknown. In relation with the remark above, we shall mention that the free energy of the jellium in the thermodynamic limit is independent of the free or the Dobrushin type boundary conditions (obtained by putting outside the vessel a crystalline configuration of fixed charges and background [13]), which could favorize one of the “solids”. We thus think that a careful computation of the unique thermodynamic limit for the free energy along the method introduced in ref. [12], should give some more insight on the occurrence (or absence) of singularities, as a manifestation of possible phase transitions taking place in the system, and work is in progress [14]. The authors have the pleasure to thank all the members of the Centre de Physique Théorique de Luminy, Universit6 d’Aix Marseille II and especially Professor Mohamed Mebkhout for the kind hospitality. One of us (D.M.) thanks also the “Deutsche Forschungsgemeinschaft” for financial support.
20 December 1982
References [1] D. Merlini, in:
Strongly coupled plasmas, ed. G. Kalman (Plenum, New York, 1978). [21 R.R. Sari, D. Merlini and R. Calinon, J. Phys. A9 (1976) 1539. [3] R.R. Sari and D. Merlini, J. Stat. Phys. 14(1976) 91. [4) R. Calinon, Ph. Choquard and J. Navet, in: Ordering in two-dimension, ed. S.K. Sinha (North-Holland, Amsterdam, 1980). [5] C. Deutsch, H.E. De Witt and Y. Furutani, Phys. Rev. A20 (1979) 2631. [61 R. Calinon et al., Phys. Rev. A20 (1979) 327. [7] P. Bakshi eta!., Phys. Rev. A20 (1979) 339. [8] P. Bakshi et al., Phys. Rev. A23 (1981) 1925. [9] A. Alastuey and B. Jancovici, J. Phys. (Paris) 4 (1981) 21. [10] J.M. Caillol et al., J. Stat. Phys. (1982), to be published. [11] S.W. De Leew et al., Physica A, (1982), to be published. [121 S. Johannesen and D. Merlini, submitted to J. Phys. A (1982). [13] S. Albeverio, D. Durr and D. Merlini, preprint (1982), submitted to J. Stat. Phys. [14] S. Johannesen and D. Merlini, in preparation.
23
PHYSICS LETTERS
20 December 1982
ON “CRJTICAL POINTS” IN THE TWO-DIMENSIONAL CLASSICAL JELLIUM S. JOHANNESEN Matematisk Institutt, Universitetet i Oslo, Blindern, Oslo 3, Norway
and D. MERLINI Mathematisches Institut 1, Ruhr-Universitdt Bochum, 4630 Bochum, Germany and Centre de Physique Théorique, Université d’Aix Marseille II, 13288 Luminy Cedex 9, France Received 10 August 1982 Revised manuscript received 20 October 1982
Using the results of a careful extrapolation method applied to systems with small particle-number N, we give an estimate for the location of a possible critical point in the two-dimensional classical jellium. Our computation yields the criti2/kT 137, in excellent agreement with the results of computer experiments. cal value ‘y~= e
In recent years there has been a considerable interest in the study of two-dimensional plasmas with long range logarithmic interaction between the particles, in particular the two-dimensional classical jellium (classical point electrons in a neutralizing charge background) usually called the OCP model [1—4].A vanety of theoretical techniques have been used to investigate the behavior of the system in the plasma—fluid equilibrium state in the weak [5] and in the strong coupling regime [6—8];moreover the harmonic approximation for a “triangular crystalline state” has also been computed [9]. Very recently, some computer experiments with Monte Carlo [10] as well as with Molecular Dynamics techniques [11], reveals that the model under consideration undergoes a first order fluid—solid phase transition at finite temperature (like the three-dimensional case and the two-dimensional surface plasma), but a theoretical proof is still lacking. The results of the computer experiments give a possible freezing transition located in the range ‘v~ 140 [10] and “~ 135 [11], where = e2/kTis the twodimensional plasma parameter. In this note we present a very simple method of computation of the fluid ~‘
free energy which is independent of the computer experiment techniques and is based on a careful extrapolation of the exact computations of the free energy of the system with few (N = 1, 2, 3,4) electrons. The method is discussed in detail in ref. [12] *1 To localize a possible critical point in the system we then use the standard argument which consists in comparing the free energy of the fluid with that of the “solid” obtained by other methods. We found that the fluid freezing point is located in the region y~= e2/kTc 137 in excellent agreement with the results of refined computer experiments. To begin, let us consider the model defined on a disk of radiusR in the plane. The main point is to obtain an accurate Ansatz for the free energy density of the model in the fluid phase. The excess free energy fCX (with respect to the H-stability bound —B and the perfect gas limit In p 1) is defined by: 4’ex I F B~ 1 1/ ~ L7f(1’ j —Qn~ )j~’y, (1 where exp(—~3fN)lnQ andB =(~1n~rp +~)[3]. -
—
—
~
*1
—
This work has been presented by one of us (D.M.) at the workshop on Statistical mechanics of ionic matter, held
1
Postal address.
0 031-91 63/82/0000—0000/$02.75 © 1982 North-Holland
at Les Houches (March—April 1982).
21
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
Table 1 ~
yex(7)/~
——
jex(7)/7 —~-—_____________
2 8 10 14
0.415529700 0.229999474 0.177959464 0.146737916
________________________
0.415529700 0.231207197 0.178649591 0.147221005
f8x(7)17
~
_______________________
0.415529700 0.23060333 0.17830452 0.14697945
________
350 II
Q is the partition function, 3 l/kT the inverse ternperature, —e the electron charge,N the particlenumber and p N/Vthe density; —B is the H-stability bound (i.e. H ~ —Be2N) and ~ = e2/kT the plasma parameter. In table 1 we present the results of our extrapolation to large N (thermodynamic limit) for ~ at the discrete values y = 2, 8, 14, 20. In the table fe~((
7)
and yex(.1) are respectively lower and upper estimate while f~X(y) is their mean value. The results of table 1 have been obtained with the help of the exact computation of the free energy for systems just consisting of N = 2,3,4 particles and the detailed treatment has been given in ref. [12]. For clarity we briefly explain here the procedure adopted to obtain these results. We consider a sequence of systems with small particle-number N = 2, 3, 4, neutral and embedded in a circular domain of volume VN = N/penergyf~(’y) at fixed density p and computeThen exactly their free = —(ln QN)/(IW). an estimation of the limit, linIN fN(7) f~(y),(called the thermodynarnic limit) can be obtained by the fN f +a(y)N~lnN+N~b(’y)+
Ansatz [12]:
where f,,,a, b can be determined from the exact computations of fN which was carried out forN = 2, 3,4, and the results are the numbers given in table 1. The Ansatz for the free energy of the fluid we consider, is very similar to the exact solution of the model in the so-called STLS truncation scheme of the equilibrium hierarchy [3] (other Ansätze of course possi. ble), and given by:
+
= A1 I-v + A2 + (~a ‘y — b) ln [/(y [2(a + b)/] ln(7 + 2) .
+
2)] (2)
Notice that in the STLS scheme a = 0 [11]. We can now compute f~(y)i.e. A1, A2, a, b using our nonperturbative results at ‘y = 2, 8, 14, 20 given in table 1 22
~
320 0~ ~t0
820
830
833
130
13?
1+0
145
(
Fig. 1. —
and plot feX(y)/y, fex(7)/7 andf~(7)/7as a function of ‘y in fig. 1. These are our estimates for the excess free energy of the model in the fluid phase. Looking for a critical point we then use the standard argument cited above which consists in comparing the free energy branch of the “fluid phase” given by eq. (2) and the harmonic approximation for the freeenergy branch of the triangular crystalline “solid” [9] (which is believed to be very accurate for large y) given by: tln 0 262 3 fs(k’YJ~/7 = 0 00062 + ~ 7 7. In eq. (3)E 0 = 0.00062 is the excess internal 2(~ln energyirp of the perfect triangular lattice [9], i.e. E + e +~)E 0.f5(7)/yis also plotted in fig. 1. The intersection of the curves in fig. 1 gives then a lower respectively upper estimate y~= 133 and = 140, while the estimate using the average f~X/7 gives the critical value ‘‘~= 137. Our results are in excellent agreement with the results obtained by Molecular Dynamics (‘va = 135) and Monte Carlo (‘vt = 140) methods. (Notice the scale in fig. 1). To conclude, we note here the following: clearly the above argument is not complete in fact the triangular lattice is the most stable one since it has lower energy then the hexagonal and square lattice at zero temperature [21; nevertheless if a freezing transition for the fluid into a “solid” takes place at y~,then other possible transitions between the three different “solids” should also be investigated, since the nature —
.
Volume 93A, number 1
PHYSICS LETTERS
of the fluid—solid transition which emerge from cornputer experiment studies is still unknown. In relation with the remark above, we shall mention that the free energy of the jellium in the thermodynamic limit is independent of the free or the Dobrushin type boundary conditions (obtained by putting outside the vessel a crystalline configuration of fixed charges and background [13]), which could favorize one of the “solids”. We thus think that a careful computation of the unique thermodynamic limit for the free energy along the method introduced in ref. [12], should give some more insight on the occurrence (or absence) of singularities, as a manifestation of possible phase transitions taking place in the system, and work is in progress [14]. The authors have the pleasure to thank all the members of the Centre de Physique Théorique de Luminy, Universit6 d’Aix Marseille II and especially Professor Mohamed Mebkhout for the kind hospitality. One of us (D.M.) thanks also the “Deutsche Forschungsgemeinschaft” for financial support.
20 December 1982
References [1] D. Merlini, in:
Strongly coupled plasmas, ed. G. Kalman (Plenum, New York, 1978). [21 R.R. Sari, D. Merlini and R. Calinon, J. Phys. A9 (1976) 1539. [3] R.R. Sari and D. Merlini, J. Stat. Phys. 14(1976) 91. [4) R. Calinon, Ph. Choquard and J. Navet, in: Ordering in two-dimension, ed. S.K. Sinha (North-Holland, Amsterdam, 1980). [5] C. Deutsch, H.E. De Witt and Y. Furutani, Phys. Rev. A20 (1979) 2631. [61 R. Calinon et al., Phys. Rev. A20 (1979) 327. [7] P. Bakshi eta!., Phys. Rev. A20 (1979) 339. [8] P. Bakshi et al., Phys. Rev. A23 (1981) 1925. [9] A. Alastuey and B. Jancovici, J. Phys. (Paris) 4 (1981) 21. [10] J.M. Caillol et al., J. Stat. Phys. (1982), to be published. [11] S.W. De Leew et al., Physica A, (1982), to be published. [121 S. Johannesen and D. Merlini, submitted to J. Phys. A (1982). [13] S. Albeverio, D. Durr and D. Merlini, preprint (1982), submitted to J. Stat. Phys. [14] S. Johannesen and D. Merlini, in preparation.
23