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Fröhlich-Edwards Theory

This theory, developed by Fröhlich and Edwards, explains the low-temperature dynamics of liquid crystals by treating the molecules as coupled harmonic oscillators, describing their collective behavior.
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The statement of the theorem

Let qj(t)\mathbf{q}_{j}(t) be the displacement of the jj-th molecule from its equilibrium position, and ωj\omega_{j} be its natural frequency. The collective dynamics are modeled by the coupled equations of motion for the generalized coordinates q=(q1,…,qN)\mathbf{q} = (\mathbf{q}_1, \dots, \mathbf{q}_N): \nd2qdt2+Γdqdt+Kq=Fext(t)\frac{d^2 \mathbf{q}}{dt^2} + \Gamma \frac{d \mathbf{q}}{dt} + \mathbf{K} \mathbf{q} = \mathbf{F}_{ext}(t) \nwhere Γ\Gamma is the damping matrix, and K\mathbf{K} is the coupling matrix derived from the harmonic potential energy V(q)=12qTKqV(\mathbf{q}) = \frac{1}{2} \mathbf{q}^T \mathbf{K} \mathbf{q}. The low-temperature dynamics are analyzed by solving this system in the frequency domain.