### Abstract:

In this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. n this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. More specifically, we investigate the entanglement entropy of a massive scalar field in 1+1 dimensions at nonzero temperature. In the small mass (m) and temperature (T) limit, we put upper and lower bounds on the two largest eigenvalues of the covariance matrix used to compute the entanglement entropy. We argue that the entanglement entropy has exp(-m/T) scaling in the limit T<<m. Additionally, we calculate thermal corrections to Renyi entropies for free massless fermions on RxS^(d-1). By expanding the density matrix in a Boltzmann sum, the problem of finding the Renyi entropies can be mapped to the problem of calculating a two point function on an n-sheeted cover of the sphere. We map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the Renyi entropies. At large length scales hydrodynamics is a useful way to study quantum field theories. We review recent interest in the Riemann problem as a method for generating a non-equilibrium steady state. The initial conditions consist of a planar interface between two halves of a system held at different temperatures in a hydrodynamic regime. The resulting fluid flow contains a fixed temperature region with a nonzero flux. We briefly discuss the effects of a conserved charge. Next we discuss deforming the relativistic equations with a nonlinear term and how that deformation affects the temperature and velocity in the region connecting the asymptotic fluids. Finally, we study properties of a non-equilibrium steady state generated when two heat baths are initially in contact with one another. The dynamics of the system in question are governed by holographic duality to a blackhole. We discuss the "phase diagram" associated with the steady state of the dual, dynamical black hole and its relation to the fluid/gravity correspondence.