Note on the thermodynamics and the speed of sound of a scalar field

We investigate the correspondence between a perfect fluid and a scalar field and show a possible way of expressing thermodynamic quantities such as entropy, particle number density, temperature and chemical potential in terms of the scalar field φ and its kinetic term X. We prove a theorem which relates isentropy with purely kinetic Lagrangian. As an application, we study the evolution of the gravitational potential in cosmological perturbation theory.