  In this article, we present an approach for the thermodynamics of phase
oscillators induced by an internal multiplicative noise. We analytically derive
the free energy, entropy, internal energy, and specific heat. In this
framework, the formulation of the first law of thermodynamics requires the
definition of a synchronization field acting on the phase oscillators. By
introducing the synchronization field, we have consistently obtained the
susceptibility and analyzed its behavior. This allows us to characterize
distinct phases in the system, which we have denoted as synchronized and
parasynchronized phases, in analogy with magnetism. The system also shows a
rich complex behavior, exhibiting ideal gas characteristics for low
temperatures and susceptibility anomalies that are similar to those present in
complex fluids such as water.
