Active nematic elastomers are a class of active materials that possess the elasticity of a rubber, and the orientational symmetry of a liquid crystal. Their constituent elements are typically elongated, cross-linked and active. The cross-linking of the elements leads to an elasticity that prevents the material to ow like a liquid. These elements are active in a sense that they continuously consume and dissipate energy, creating a state that is far-from-equilibrium. Active nematic elastomers may be a good physical model for biological systems such as the metaphase spindle, a complex biological machine that is made of an integrated assembly of microtubules and molecular motors. These motors not only cross-link the microtubules, but also actively slide them against each other, creating a highly dynamic, non-equilibrium state. The metaphase spindle, like other non-equilibrium structures in biology, has important functions to perform. During mitosis, the spindle is responsible for (1) capturing the sister chromatids, (2) bringing all the sister chromatids to the equator of the mother cell, and (3) segregating the daughter chromosome to the opposite poles of the cell. Thus, a fundamental challenge to biological physics is to understand the complex dynamics of the spindle, and similar systems, using the tools of non-equilibrium statistical mechanics. In this Thesis, we develop and explore a phenomenological model for an active nematic elastomer. We formulate the dynamics of this phenomenological model by incorporating the contribution of the active elements to the standard formulation of the hydrodynamic equations of a passive system. In a coarse-grained picture, the activity is taken into account as an extra active stress, proportional to the alignment tensor, added to the momentum equation of an otherwise passive nematic elastomer. Having obtained the equations of motion of an active nematic elastomer, we then investigate the response of the system to an external field by means of examining the structure and the stability of the modes. An active nematic elastomer has eight modes, in which six modes are propagating and two modes are massive. Out of the six propagating modes, two modes are in the longitudinal direction, linked to the density waves, and the other four modes are in the transverse direction, linked to the shear waves. The nature of these propagating modes transitions from dissipative and oscillatory, and vice versa, depending on the length scales. In particular, their stability is largely determined in the hydrodynamic limit, by a competition between the stabilizing effect of the elasticity and the destabilizing effect of the activity. In fact, the activity renormalizes the elastic coefficients down to even a negative value in some cases and thus, rendering the system linearly unstable. This is in contrast to the well-known instability of an active nematic liquid crystal, which is always linearly unstable. We then map out and discuss the stability phase diagram of the active nematic elastomer. Next, we compute and study various equal-time correlation functions of an active nematic elastomer, assuming that the noise spectra are thermal in origin. We find that they can be conveniently arranged into two terms. The first term has the exact mathematical structure of the equal-time correlation functions of a passive nematic elastomer, albeit with certain coefficients renormalized by activity. The second term, which is proportional to the activity, represents the non-equilibrium nature of an active nematic elastomer, and manifestly breaks the Fluctuation-Dissipation Theorem. We also find that (1) the displacement-displacement correlation function decays inversely with the square of the wave number for both the compressible and incompressible nematic elastomer, similar to that of a passive nematic elastomer, with elastic coefficients renormalized by the activity. (2) The density-density correlation function approaches a constant at the long wave-length limit, since the conservation of mass links the density to the rate of changes of the displacement in the longitudinal direction. (3) The director-displacement correlation function is purely imaginary, and thus the director is locked to the displacement with a (π/2) phase-shift. (3) The director-director correlation function approaches a constant value in the long-wavelength limit, instead of decaying inversely with the square of the wave number, like it would for a liquid crystal. This is because of the massive mode stems from the coupling energy, and it indicates that director in the large length scale is locked to a specific angle. These theoretical results are in qualitative agreement with the experimental measurements of the spindle. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection