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CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
THERMODYNAMIC PROPERTIES OF NONIDEAL STRONGLY
DEGENERATE HYDROGEN PLASMA
Pavel R. Levashov1, Vladimir S. Filinov1, Vladimir E. Fortov1, and M. Bonitz2
1
Institute for High Energy Densities, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412, Russia
2
Fachbereich Physik, Universitdt Rostock, D-18051 Rostock, Germany
Abstract. The results of numerical simulation of strongly coupled hydrogen plasma are presented.
Direct fermionic path-integral Monte-Carlo method is used for calculations. It allows one to compute pressure, energy, and pair distribution functions of hydrogen plasma over a wide range of densities and temperatures. AtT= 104 K and density range 0.1-1.5 g/cc phase transition which vanishes
at higher temperatures is predicted. This density range covers the region of abrupt increase of an
electrical conductivity under shock compression experiments.
INTRODUCTION
generate proton-electron plasma in a broad range of
densities and temperatures.
Thermodynamic properties of Fermi systems are
of great interest in many fields, including plasmas,
astrophysics, solids and nuclear matter [1, 2]. The
most interesting phenomena, such as metallic hydrogen, plasma phase transition, bound states, etc.,
occur in situations where both quantum and Coulomb effects are relevant. Among the most promising theoretical approaches to such systems are pathintegral quantum Monte Carlo (PIMC) method [3,
4]. A remarkable progress has been recently
achieved in applying these techniques to Fermi
systems [4, 5]. However, these simulations are substantially hampered by the so-called fermion sign
problem. Additional assumptions such as fixed node
and restricted path concepts have been introduced to
overcome this difficulty [4]. It can be shown, however, that such assumptions do not reproduce a correct ideal Fermi gas limit [6].
Recently, we have presented a new path integral
representation which avoids additional approximations (direct path integral Monte-Carlo) [7-9]. In
this work we apply the PIMC method to the computation of thermodynamic properties of dense de-
DIRECT PIMC METHOD
It is well known that the thermodynamic properties of a quantum system are fully determined by
the partition function Z. For a binary mixture of Ne
electrons and Nf protons, Z can be written as
Z(N99Ni9V9p)=Q(Ne9Ni9fl)/Ne\Ni\9
(1)
Here, # = {<li,q 2 >•••,<!#/ } is the coordinates of the
protons and a = {o^ ,. . .,<rNe } and r = {rb. . ,,rNe } are
the electron spins and coordinates, respectively,
/3 = l/kBT , kB - Boltzman constant. The density
matrix pin Eq. (1) is represented in the common
way by a path integral [10]:
(± 1)"
'} (2)
119
where AJ3 = J3/(n + l) and A2A=27rti2A/]/me . Further, r("+l^ = r , cr' = cr ; i. e., the electrons are represented by fermionic loops with the coordinates
(beads) [r] = [r,r(1^,...,r(^,r]. The electron spin
gives rise to the spin part of the density matrix 5,
whereas exchange effects are accounted for the
permutation operator P and the sum over the permutations with parity KP . Following [3], we use a
modified representation (3) of the high- temperature
density matrices of the r. h. s. of Eq. (2) which is
suitable for efficient direct fermionic PIMC simulation of plasmas. With the error of the order
£ ~ (flRy)2 %/(n + 1) vanishing with a growing
number of beads, we obtain the approximation
From the above expressions (l)-(3) it can be
easily found the equation of state:
= d\nQ/dV = [a/3VdlnQ/da]a=l
CALCULATION RESULTS
In our simulations we used Ne = Af/ =50. To test
the MC procedure, we have considered a mixture of
ideal electrons and protons for which the thermodynamic quantities are known analytically. The
agreement, up to the degeneracy parameter % as
large as 10, has been very good and improved with
increasing number of particles [7]. In the case of
interacting electrons and protons we have performed a series of calculations over a wide range of
the classical coupling parameter F and degeneracy
X for temperatures T> 10000 K. The analysis of
pair distribution functions has shown a number of
interesting phenomena, i.e. formation (T= 2-104K,
n = 1022 cm'3) and decay (T= 5-104 K, n = 1022
cm"3) of molecules, pairing of electrons
(r=5-10 4 K, n = 5-1025 cm"3), ordering of protons
(T= 5-104 K, n = MO27 cm"3) and others [8, 9].
In this work we present the calculated thermodynamic properties of hydrogen at constant temperatures r= 104 and 5-104. Fig. 1 demonstrates pressure and energy vs. concentration at r=5-10 4 K.
The results are in a good agreement with restricted
path integral calculations at slightly higher temperature 6.25-104 K [12]. We have observed similar
dependencies at higher temperatures.
However at T= 104 K the properties of hydrogen
plasma are significantly different (Fig. 2). We have
found the region of bad convergence to an equilibrium state in density range 0.1-1.5 g/cc. In this region the pressure values are negative; such behavior
is typical for Monte-Carlo simulations of metastable
systems. Note that there are no such peculiarities on
isotherms at T= 5-104 K and higher. We treat these
facts as the existence of phase transition with critical point. There are several confirmations of this
conclusion. Experimentally it has been registered an
abrupt increase of an electrical conductivity (by 4-5
orders of value) under shock compression in the
1
where # - degeneracy parameter, t/7 , £/f , and (7f
denote the sum of the binary interaction Kelbg potentials 0ab [11] between protons, electrons at
vertex /, and electrons (vertex /) and protons, respectively.
In Eq. (3) ^ ^expf-Tr^l 2 ] arises from the ki-
netic energy density matrix of the electrons with
index /?, and we introduced dimensionless distances
between neighboring vertices on the loop,
The exchange matrix is given by
expj-—Hr D n
k=l
As a result of the spin summation, the matrix
carries a subscript s denoting the number of electrons having the same spin projection.
120
10'
iio
10Z
10U
1
21CT
10"
10"
10'
icr
1022 .3 1024
w, cm
101
1024 /i, cm'3
-1
-1
-2
10"
icr
10'
10"
«, cm""-3
10*
FIGURE 1. Pressure and internal energy for hydrogen plasma at
T= 5-104 K. 1 - direct PIMC simulation of this work, 2 - ideal
plasma, 3 - restricted PIMC computations at T= 6.25-104 K [12].
icr
10
FIGURE 2. Pressure and internal energy for hydrogen plasma at
T= 104K. 1 - direct PIMC simulation of this work, 2 - ideal
plasma, 3 - restricted PIMC computations [12], 4 -density functional theory [15].
density range 0.3-0.5 g/cc [13, 14]. Calculations of
thermodynamic properties of hydrogen in the metallic phase on the basis of density functional formulation of the free energy of the ion-electron
plasma showed that the long-wavelength ion density fluctuations were enhanced as the density
dropped [15]. Therefore at the lowest concentration
explored n = 4.77-1023 cm"3 the ion-ion pair structure and effective potential exhibited an unusual
behavior, which the authors of Ref. [15] interpreted
as a precursor to an incipient metal-to-insulator
transition. It can be seen from Fig. 2 that this concentration value lies near the boundary of the region
of negative pressures of this work. In the same region we have also observed the splitting of the system of particles into drops. On Fig. 3 it is shown a
typical snapshot of a Monte-Carlo cell at T= 104K
andw = 2-1023cm~3.
CONCLUSIONS
In summary, we presented results for thermodynamic properties of strongly coupled quantum
plasma using a modified path integral representation for the //-particle density matrix. This representation allows one to avoid additional approximations for the density matrix and to perform efficient direct fermionic simulations for temperatures
above approximately 10000 K. The method can be
121
3. Zamalin, V. M., Norman, G. E., and Filinov, V. S.,
The Monte-Carlo Method in Statistical Thermodynamics, Nauka, Moscow, 1977.
4. Ceperley, D., in The Monte Carlo and Molecular
Dynamics of Condensed Matter Systems, edited by
K Binder and G. Ciccotti, SIF, Bologna, 1996,
pp. 447-482.
applied to molecular and atomic systems as well as
to partially and fully ionized plasmas. It is possible
to simulate the effects of dissociation, pressure
ionization, proton ordering and others. In this work
we have studied the region of probable phase transition. Further investigations will focus on more
precise investigations of Mott transition region to
explain an anomalous behavior of shock compressed deuterium [16, 17].
T=10QQOK
5. Classical and Quantum Dynamics of Condensed
Phase Simulation, edited by B. J. Berne,
G. Ciccotti, and D. F. Coker, World Scientific,
Singapore, 1998.
6. Filinov, V. S., J. Phys. A, 34, 1665-1677 (2001).
7. Filinov, V. S., Levashov, P. R., Fortov, V. E., and
Bonitz, M., «Thermodynamic Properties of Correlated Strongly Degenerate Plasmas», in Progress in
Nonequilibrium Green's Functions, edited by M. Bonitz, World Scientific, Singapore, 2000.
8. Filinov, V. S., Fortov, V. E., Bonitz, M., and
Kremp, D., Physics Letters A 274, 228-235 (2000).
9. Filinov, V. S., Bonitz, M., and Fortov, V. E., JETP
Letters 72, 245-248 (2000).
10. Feinman, R. P., and Hibbs, A. R., Quantum Mechanics and Path Integrals, McGraw-Hill, New York,
1965.
11. Kelbg, G., Ann. Phys. (Leipzig) 12, 219 (1963); 13,
354 (1963); 14, 394 (1964).
12. Militzer, B., and Ceperley, D. M., Phys. Rev. E 63,
066404(2001).
13. Weir, S. T., Mitchell, A. C., and Nellis, W. J., Phys.
Rev. Lett. 76, 1860-1863 (1996).
14. Teraovoi, V. Ya., Filimonov, A. S., Fortov, V. E.,
Kvitov, S. V., Nikolaev, D. N., and Pyalling, A. A.,
PhysicaB 265, 6-11 (1999).
15. Xu, H., and Hansen, J. P., Phys. Rev. E 57, 211-223
(1998).
16. Da Silva, L. B. et al., Phys. Rev. Lett. 78, 483-486
(1997).
17. Coffins, G. W. et al., Science 281, 1178 (1998).
n
/;f
FIGURE 3. Snapshot of a Monte-Carlo cell at n=2-\023 cm"3 and
r=104K. Black circles are protons, dark and light broken lines
are representations of electrons as fermionic loops with different
spin projections.
ACKNOWLEDGEMENTS
We acknowledge W. Ebeling, D. Kremp, and
W. D. Kraeft for stimulating discussions. We also
wish to thank D. Ceperley and B. Militzer for useful
critical remarks.
REFERENCES
1. Kraeft, W. D., Kremp, D., Ebeling, W., and Ropke,
G., Quantum "Statistics of Charged Particle Systems,
Akademie-Verlag, Berlin, 1986.
2. Strongly Coupled Coulomb Systems, edited by
G. Kalman, Pergamon Press, 1998.
122
ADDENDUM
Editors' comment: Questions raised during the review of the preceding paper are here presented together
with the authors' reply for the benefit of the readership,
Comments on uThermodynamic Properties of
Nonideal Strongly Degenerate Hydrogen Plasma"
In agreeing to review the conference paper of
Pavel Levashov et. al. I did not anticipate becoming involved in an apparently well developed
debate about path integral methodologies. It may
be useful however, to publicly state some reservations about this contribution. Although the paper
is well written and reports on a difficult computation there are severe doubts about their method
and conclusions.
Physical implausibility of reported phase transition
In addition to several puzzling effects mentioned
in passing: an electron pairing phenomena at densities and temperatures where molecules are dissociating; a reported proton ordering (Wigner lattice
at F w 50 ?) at the same temperature but higher
density; Levashov et. al. report negative energies
per electron of over 1.5 Hartree at number-densities around 1024cm~3. These unphysical energies
(nearly 3 times atomic or molecular binding energies) are said to signal a metastability in the dense
hydrogen plasma.
It seems more symptomatic of an inadequate
treatment of Fermi statistics. It is well documented
both theoretically [1] and in path integral simulations [2] that failure to fully account for Fermi
statistics leads to instability and large negative
binding energies.
Authors assertions about RPIMC
Restricted path integral Monte Carlo (RPIMC)
is based on replacing the usual Bloch equation for
the many body density matrix
(l)
with antisymmetric initial condition
(2)
123
by the same equation, dp/d/3 = —Hp and initial
condition supplemented by the boundary condition
(3)
where the domain Q(Ro, /?) is defined as all points
R such that /?(R? RQ; /?) > 0. This domain is guaranteed to exist since the diagonal density matrix
/>(RO,RQ;/?) is positive.
The solution to the original Bloch equation
clearly satisfies the augmented system, p — p, by
construction and from the uniqueness property of
the heat equation with a source-sink term [3] it is
the only solution. This augmented system now allows probabilistic methods to be used to sample
the positive p. When the boundary condition is
replaced by the nodes of a trial density matrix the
algorithm becomes approximate. While this may
emphasizs the importance of work on constructing
accurate trial density matrices for physically interesting systems it does not apply to the free Fermi
gas where the exact density matrix and thus exact
nodes are known.
The comment at the end of paragraph one of the
paper is thus incorrect. Supporting numerical results can be found in reference [4].
Treatment of the Fermion sign problem
Fermi statistics must be imposed on hydrogen
plasma at any temperature to prevent the system
from collapsing [1], [2]. The impracticality of the
most straightforward way of doing this, sampling
from the Bosonic density matrix where all permutations are positive and reweighting odd permutations with a minus sign to obtain the Fermion density matrix, is obvious even from the free particle
case. The figure below showing the ratio of Bose to
Fermi partition functions at various temperatures
and system sizes indicates the impossible accuracy
required. For example for ten particles at a degeneracy of five, T/Tpermi = 1/5, the cancellation
between odd and even permutations is complete to
one part in 1012. This is the essence of the Fermion
sign problem in quantum Monte Carlo simulations.
[2] J. Theilhaber and B. J. Alder, PRA, 43, 4143
(1991).
[3] O. A. Ladyzhenskaya, V. A. Solonnikov, and N,
N. UraPceva, Linear and Quasi-linear Equations
of Parabolic Type. Izdat. Nauka, Moscow (1967). I
thank Gerald Hedstrom for his comments on this
question.
[4] Burkhard Militzer, Path Integral Simulations of Hot
Dense Hydrogen, PhD thesis, Univ. of Illinois, 2000
(section 2.6.6)
[5] D. M. Ceperley, Path integral Monte Carlo methods
for fermions'm Monte Carlo and Molecular Dynamics of Condensed Matter Systems, Ed. K. Binder
and G. Ciccotti, Editrice Compositori, Bologna,
Italy, 1996. (or www.ncsa.uiuc.edu/Apps/CMP)
free particle
10
20
30
40
50
60
70
80
90
100
FIG. 1. Ratio of Fermi to Bose partition functions
for noninteracting particles at indicated temperatures
and number of particles in the periodic cell. The
straight line behavior is due to the extensivity of the
free energies even for these size systems.
Levashov et. al. do not do this but instead antisymmetrize one step in the path integral. This
correctly antisymmetrizes the density matrix but
does not remove the sign problem and should also
limit phase space sampling since exchanging particles must be within a thermal deBroglie wavelength
corresponding not to the temperature but to the
temperature multiplied by the number of steps in
the path integral. This approach has been tried
several times previously and further discussion of
it's difficulties can be found in reference [5] section
XB.
In summary the results are intriguing but the reliability of the method is questionable.
Roy Pollock
LLNL
[1] Elliot H. Lieb, RMP, 48, 553 (1976).
124
Authors' Reply
results have been obtained by one of us for
mesoscopic electron clusters [3].
3) Phase transition in hydrogen plasma. In
earlier work (e.g. [6] and references in [2, 5]), a first
order phase transition in dense plasmas (PPT) was
frequently predicted by simple chemical models
only. Further, some possible indications for a PPT
have been found in recent density functional studies
at 0.8 g/cc [7]. Our results give the first evidence
for the PPT obtained within a physical picture
(starting with free electrons and protons) simulation
of first-principle character. The referee apparently
misunderstood our discussion; evidence for the PPT
is not derived from the low energy values [see 4)
below] but from the convergence behavior of the
DPIMC scheme that is typical for canonical MC
simulations in a two-phase region. For example, we
observed energy fluctuations between two values,
negative pressures and formation of droplets. Our
analysis of these droplets revealed that they contain
de-localized electrons (see Fig. 3 of our paper) like
in metals [2]. The droplet size increases with density.
While the general issue of the PPT is still open
and requires extensive further simulations, it is instructive to point to another similar system - dense
electron-hole plasmas in semiconductors. There, at
comparable coupling and degeneracy, a similar first
order phase transition is well established experimentally [8, 9], and droplet formation is directly
observed. We have performed simulations for those
systems as well and obtained droplet formation in a
good agreement with experimental data [10].
4) The obtained low energy values in the instability region are indeed surprising at first sight. For
an explanation, we stress that the observed droplets
are very small, having a characteristic size of 2-4
Bohr radii and containing only from 10 to 50 electron-proton pairs. This means that each proton
strongly interacts simultaneously with many quantum electrons with a characteristic energy of the
order of 1 Ry. Due to their strong repulsion, most of
the protons are located at the droplet surface. These
strong surface effects are the main reason for the
low energy values. We expect that in a macroscopic
The Authors are grateful to Roy Pollock for his
careful reading of our paper and his insightful and
constructive criticism. We appreciate the opportunity to address a few important issues. As we will
show most of them are caused by misunderstandings or missing details of the calculations. In this
comment we'll provide for our point of view on
some problem distinguished by the referee.
1) Sign problem in DPIMC. We fully agree
with the referee's general statements about the sign
problem. Indeed, the drastic loss in simulation efficiency with increasing degeneracy parameter %
(electron density times DeBroglie wavelength
cubed) is the main obstacle for direct PIMC calculations. For simulations of dense plasmas we, therefore, developed a new approach, which is essentially more efficient than previous DPIMC attempts.
We do not sample all N\ individual permutations
(this would indeed be prohibitive). Instead we have
introduced the exchange determinant which allows
for a drastically more efficient DPIMC procedure,
for details see [1]. The correctness of the treatment
of the Fermi statistics in our DPIMC simulations
has been carefully checked for an ideal Fermi gas
[1, 2], Further, the approach has been successfully
applied to interacting fermions in plasmas and
quantum dots [1, 3]. Finally, comparisons with restricted PIMC simulations of Ceperley et al. show
remarkable quantitative agreement for degeneracy
parameter as high as % = 10 [1]. Further, we note
that the referee's conclusion about the "impracticality" of DPIMC simulations that he derived from the
small ratio of the two partition functions is not justified. Similar problems (vanishing ratio of two partition functions) appear in a variety of classical systems and have been successfully overcome by
means of standard umbrella MC techniques for
more than 25 years, see e.g. [4].
2) Probable Wigner crystallization. The referee incorrectly interpreted our results; we reported
about proton ordering, not crystallization at F= 54,
where the pair distribution functions [2, 5] indicate
a strongly ordered liquid-like structure. Crystallization is observed at five times lower temperature,
where F~ 260. Further, accurate crystallization
125
system this negative energy contribution will decrease drastically. This effect should be observable
in simulations (which are presently underway) with
large particle numbers. Interestingly, if we do not
allow for spatially inhomogeneous configurations in
the simulations, our energy values increase to approximately 1 Ry per proton that is very close to the
RPIMC data.
5) The RPIMC method and the "fixed node
approximation" (FNA) [11] describe rather well
the thermodynamic properties of strongly coupled
plasmas at "weak and moderate" degeneracy. Yet
there are many open questions. According to the
definition of the partition function, the integration
has to be performed over the whole configuration
space. In contrast, in the FNA the region of integration Q(RQifi) is restricted to a domain, which is
given by the condition that p(R,RQ,/3)>Q (p is the
"trial density matrix" in the notation of the referee).
We agree with the referee that inside <Q(/?0,/?) the
exact and FNA density matrices coincide- both
satisfy the Bloch equation. However, the restriction
on the region of integration in the partition function
leads to a number of contradictions. In [12] one of
us has given an analytical proof that the exact grand
partition function of ideal fermions cannot be reproduced in the FNA, and the Trotter formula for
the matrix elements (ME) of the density matrix is
not satisfied. The latter can be easily shown by consideration of two fermions in the ID case. In coordinate representation of the operator identity
expH?£] = exp[-/?£/2]2 on the left and right
hand sides (l.h.s and r.h.s.) all ME are Gaussian
exponents. To obtain the final Gaussian exponent
on the l.h.s. one has to integrate over intermediate
variables within infinite limits on the r.h.s. On the
other hand, any restriction on the region of integration on the r.h.s. gives rise to an error function depending on the limits of integration.
For two fermions to take into account the Fermi
statistics we should consider the sum of two
permutations (identical and nonidentical), while in
the "fixed node approximation" only the identical
permutation should be taken into account. This
means that the coordinate of the first fermion along
the path should always be larger (or smaller) than
the coordinate of the second one. In the FNA, the
path integral representation of the ME on the l.h.s.
integral representation of the ME on the l.h.s. may
be identically transformed into an integral of the
product of the path integral representations of two
ME on the r.h.s. but with the above-mentioned restriction on the region of integration. So if the Trotter formula is valid, and if the FNA of ME on the
r.h.s. equals the anti-symmetrized sum of the Gaussian exponents, the error function (appearing after
integration on the r.h.s) can be presented as an algebraic combination of the Gaussian exponents, which
contradicts the transcendental character of the error
function.
In conclusion, we regard the systematic comparison results of alternative independent methods as
crucial for future progress in dense plasma theory.
We are grateful to H. DeWitt, W. Ebeling and B.
Militzer for many fruitful discussions.
The Authors
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126