**Turbulence and Spatiotemporal Chaos**

Turbulence is a very common phenomenon and observed everywhere, mostly in fluid systems. Turbulence occurs when a strong external field is applied to the systems. For the phenomenon, statistical physics and hydrodynamics are powerful tools to describe its mechanism and property as well as nonlinear dynamics. However it is not well understood yet because of strong nonlinearity and tremendous degrees of freedom. Recently great development related to concept and classification of turbulence has been obtained, e.g. a new mathematical concept for chaos and detailed classification of turbulence such as fully-developed, weak, defect turbulence, spatiotemporal chaos (STC) and so on. The STC which is a spatially and temporally disordered structure is frequently observed in a typical nonequilibrium open systems such as electroconvection of nematic system [Cross 1993]. Very recently, the soft-mode turbulence (SMT) which is a quite novel STC has been found in electroconvection in nematic liquid crystals. Here, the description of the SMT will be given after explaining the electroconvection in nematic liquid crystals.

**Electroconvection of Nematic Liquid Crystals**

The term of liquid crystals indicates an intermediate state between crystalline solid and isotropic liquid. The main difference between liquid crystals and isotropic liquids is that the liquid crystals have anisotropy. The form of liquid crystals must be geometrically anisotropic in shape, like a rod or a disk. The most type of transition from liquid crystals to isotropic liquid is realized by thermal. The main typical liquid crystals namely nematic liquid crystals have a high degree of long-range orientational order of the molecule.

Due to the anisotropy in nematics, two directions can be defined, namely parallel and perpendicular to the director. Therefore, all macroscopic parameters of nematics can be defined using the anisotropic behavior, such as dielectric anisotropy , electric conductivity anisotropy and diamagnetic anisotropy. For example, a material of MBBA (p-methoxy-benziliden-p’-n-butyl-annyline) has the following anisotropies: dielectric anisotropy < 0 and electric conductivity anisotropy > 0 [de Gennes 1993].

For the director of nematic liquid crystals ± n, there are three types of distortion, namely splay, twist and bend .

**Planar and Homeotropic Alignments**

Before explaining electroconvection of nematic liquid crystals, it should be mentioned here that there are two typical alignments of nematic liquid crystals in the media, namely planar and homeotropic alignment. In the planar alignment, the director n for the planar has an initial direction n0 = (1, 0, 0). From the view in the x–y plane, the continuous rotational symmetry in the planar system is broken. On the other hand, in the homeotropic alignment, the initial direction for the director is n0 = (0, 0, 1). From the view in the x–y plane, the initial director is uniform. Therefore the homeotropic system has a continuous rotational symmetry.

In the planar system, when the ac voltage V is applied below a critical voltage for convection, the director n0 is tilted in x-z plane so that the director becomes n = (cos F, 0, sin F) where F is the tilted angle of the director with respect to the x-axis. C-director is defined as the projection of the director in the x–y plane. Here, the system is in quiescent (non-convective) state. When the ac voltage is increased above the critical voltage Vc, electroconvection occurs.

A stripe convective pattern called the Williams domain appears in the planar system. The stripe pattern can be represented by a uniform wavevector q which is parallel to C-director.

The so-called Carr-Helfrich effect can explain the mechanism of electroconvection in the planar system beyond threshold (see Fig. 1.5) [Carr 1969; Helfrich 1969]. In addition, the mechanism in the electroconvection in the planar system can also be described by the time-dependent Ginzburg-Landau equation [Hijikuro 1975].

Many types of pattern which depend on e and frequency of the applied voltage have been found and researched in the planar system, such as defect turbulence and dynamic scattering mode (DSM) [Kai 1989]. In the planar system, by the increase of the control parameter e, a type of STC called defect turbulence appears from the Williams domain at a positive e. Here, the route of the STC is quiescent state – ordered pattern – spatiotemporal chaos [Kai 1989].

In the homeotropic system, on the other hand, below the critical threshold for convection , there exists the so-called Fréedericksz transition. Below the Fréedericksz transition point , the director n is not tilted so that the continuous rotational symmetry is not broken. Beyond VF but below Vc, the director n is tilted. Since the continuous rotational symmetry is spontaneously broken, the C-director behaves as a Goldstone mode [Hertrich 1992; Tribelsky 1996].

When the voltage V is increased beyond Vc, electroconvection occurs in the homeotropic system. Here, a type of STC called soft-mode turbulence (SMT) occurs. The SMT is induced by nonlinear interaction between the Goldstone mode and the convective mode [Kai 1996].

It is important to be mentioned here that in the homeotropic system the Goldstone mode exists. The Goldstone mode comes from the spontaneous breaking of the continuous rotational symmetry. The Goldstone mode leads to the unusual spatiotemporal chaos directly from the quiescent state. Therefore, when the Goldstone mode is suppressed, such unusual spatiotemporal chaos cannot occur.

An important property of the SMT is that the critical point for spatiotemporal chaos is the same as that for convection. Therefore the SMT is an unusual STC which directly appears from a quiescent state via single supercritical bifurcation. Also the SMT shows softening of its macroscopic fluctuations near the convective threshold. For taking a similarity with equilibrium systems, the SMT has several advantages as follows [Kai 1996; Tamura 2002].

1. When e decrease into zero, some statistical variable such as correlation time and correlation length approach to infinity. The properties are similar to that in equilibrium systems.

2. With respect to the dimension of systems and the degree of freedom, an analogy between the SMT and a suitable model in equilibrium systems called the conventional two-dimensional (2D) XY model can be done.

By considering the above advantages, SMT can be used as a bridge between non equilibrium open systems and equilibrium systems.

**2D XY Model and Kosterlitz-Thouless Transition**

In equilibrium statistical physics, there are several spin models to describe properties of magnetic systems. One of them is called the conventional 2D XY model. The “2D” and “XY” correspond to dimension of system and degree of freedom, respectively.

The so-called Mermin-Wagner theorem guarantees that the model cannot show a long-range correlation [Mermin 1966]. The Kosterlitz–Thouless (KT) transition is a special transition in the 2D XY model which separates low and high temperature regime [Kosterlitz 1973]. In the low temperature regime, the correlation function is power-law. On the other hand, in the high temperature regime the correlation function is exponential:

There is no long range correlation for any finite temperature in this model, in accordance with the Mermin-Wagner theorem. Therefore, in the conventional 2D XY model, there is no order-disorder transition for any finite temperature, including the Kosterlitz-Thouless transition point. In addition, defects appear only in high temperature regime in form of vortex. The correlation length and the defect in high temperature regime will be useful for the next discussion

As described above, the SMT is a quite novel spatiotemporal chaos because it directly appears from a quiescent state via single supercritical bifurcation. The SMT is induced by nonlinear interaction between the Goldstone mode and the convective mode. Also it shows softening of its macroscopic fluctuations near the convective threshold. However statistical details of the SMT are not yet well understood. Therefore, the understanding of the SMT may be obtained using small extension of conventional statistical physics. Here, there are important keywords which should be considered namely phase transition and symmetry. In the conventional 2D XY model induced by thermal fluctuation, there is no order-disorder transition. Therefore, it is important to investigate the existence of order-disorder transition in the SMT due to nonlinear interaction and nonthermal fluctuations. Moreover, since symmetries play an important role in a system, then it should be researched what the role of a kind of symmetry in the SMT. Under these backgrounds, the present study has focused to investigate statistical properties of the SMT including phase transition and symmetry.

From the above main problems related the phase transition and the symmetry, the following questions can be derived. Due to the difference of symmetry interaction between the OR and the NR regimes what the differences between the properties of SMT pattern in OR and NR regime are. How the Goldstone modes in SMT behave when they are suppressed or unsuppressed? What is the kind of statistical properties in a homeotropic nematics when the system is influenced an external field?