Critical exponents for structural phase transitions in a complex plasma

J. D. E. Stokes∗, S. V. Vladimirov∗ and A. A. Samarian∗
∗School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
Abstract. The critical instability of two dust particles levitating in the complex plasma sheath of a radio-frequency discharge
is considered. It is shown that the two-particle systemhas a critical point where the alignment symmetry is continuously broken
as the system parameter is varied. The associated critical exponents are derived and found to belong to the Ising universality
class. Another universality class is suggested for symmetry breaking of the radial and vertical confinement potentials.
Keywords: Hamiltonian, Plasma Sheath, Dust Particles, Phase Transitions, Symmetry Breaking
PACS: 52.27.Lw,52.35.Py,52.80.Pi
The theory of critical phenomena has mostly been explored from the perspective of the statistical thermodynamics. In
the so-called extensive system regime, where the number of interacting particles is of the order of Avogadro’s number,
the assumption of an infinite uniform system is justified. In non-extensive systems where the number of particles is
fewer than ∼ 103, the thermodynamic limit can no longer be assumed, since the extent of the interparticle interaction
is comparable the size of the system. Complex plasma provides an ideal medium for studying phase transitions in
non-extensive systems, with as little as two particles displaying extremely rich physics such as spontaneous symmetry
breaking and universality.
The Order Parameter Exponent b
Below a critical radial confinement wr,c. The Hamiltonian has a stable equilibrium with the dust particles horizontally
aligned (Dz = 0). Taylor expanding the Hamiltonian about the equilibrium position in the vertical interparticle
separation order parameter Dz we obtain.

The critical exponents for the horizontal alignment instability are the same as those of the mean field theory for
thermodynamic systems such as the Ising model and Van derWaals theory. The dust system exhibits both a continuous
(second order) phase transition at the critical resonant frequency, as well as discontinuous phase transitions induced by
asymmetric wake fields. The exponents are independent of the plasma parameters such as the Debye length. Although
we made use of the a the point-charge approximation to model the ion wake distribution, the results should apply
equally well for any wake distribution since they depend only on the local approximation to wake potential up to third
The second order phase transition is easily observed for two identical particles in a uniform discharge. In order to
observe the discontinuous first order phase transition between the oblique equilibria, consideration must be given to
the experimental realisation of the wake charge asymmetry DQw. The wake asymmetry may come about by virtue
of the non-uniformity of the discharge, however, dynamical wake charging in the plasma is necessary to observe the
In terms of spatial symmetry breaking of the confinement fields, the critical frequency ratio Y for two identical
dust particles provides an order parameter to describe the extent of the symmetry breaking by the dust-induced wake
fields. The remarkable simplicity of the order parameter scaling Y ∼ (−q )1/2 near the critical wake charge tempts us
to define other critical exponents for the susceptibility and the response. The fact that the exponents b , g and d satisfy
theWidom equality gives strong support to the notion of universality. It is not clear, however, if the predicted exponent
of a = 1/2 can be derived from the Hamiltonian.
The role of fluctuations has not been considered. Thermal fluctuations in thermodynamic systems are responsible
for changing the critical exponents from their mean field values. Fluctuations of grain charge in the dust system may
allow us to define a correlation function, giving deeper insight into the universality class of this system.
The fact that the horizontal alignment instability belongs to the Ising universality class for thermodynamic systems
suggests a common symmetry underlying these seemingly disparate systems. The new universality class for breaking
of the external confinement symmetry needs to be explored in greater detail so that it can be compared to other systems
which share the same symmetry.
1. V. Steinberg, R. Sütterlin, A. V. Ivlev, and G. Morfill, Phys. Rev. Lett., 86, 4540 (2001).
2. L. D. Landau and E. M. Lifschitz, Lehrbuch der theoretischen Physik (Band V, Akademie, Berlin, 1966).

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