I was reviewing my lessons when I read this definition on Thermal Energy:
The sum of potential energy and kinetic energy is equal to Thermal Energy
But isn't this the same as the Hamiltonian? So, instead of saying "taking the Hamiltonian of ..." can I say "taking the Thermal Energy of ..."? Sorry for my ignorance.
If you integrate the Gibbs equation $dU=TdS+ \sum_k Y_kdX_k$ with the assumption of $X_k$ and $U$ are extensive and $Y_k$ are intensive, then the function $U(S,X_k)$ are first order homogeneous and there follows that $U=TS+\sum_kY_kX_k$. This can be interpreted as the total internal energy of a body containing two terms $TS$ and $\sum_kY_kX_k$. The former $TS$ is the internal thermal energy, the latter is the internal everything else reflecting the various interactions the system may participate in, such as mechanical $X_1=V, Y_1=-p$, electrical $X_2=q_e, Y_2=\phi_e$, chemical, magnetic, etc. This has nothing to do with the Hamiltonian for $U$ is a static (timeless) energy while the Hamiltonian is a dynamic quantity so much so that even if $H$ is explicitly time independent it describes the time evolution of other dynamic quantities.
