Consistency between dynamical and thermodynamical stabilities for perfect fluid in $f(R)$ theories

We investigate the stability criterions for perfect fluid in $f(R)$ theories which is an important generalization of general relativity. Firstly, using Wald's general variation principle, we recast Seifert's work and obtain the dynamical stability criterion. Then using our generalized thermodynamical criterion, we obtain the concrete expressions of the criterion. We show that the dynamical stability criterion is exactly the same as the thermodynamical stability criterion. This result suggests that there is an inherent connection between the thermodynamics and gravity in $f(R)$ theories. It should be pointed out that using the thermodynamical method to determine the stability for perfect fluid is simpler and more directly than the dynamical method.