Andrius Kulikauskas: I need to understand how the fundamental thermodynamical relation is related to the free energy principle.
Fundamental Thermodynamical Relation
{$ \mathrm{d}U=\delta Q-\delta W $}
{$ \mathrm{d}U=T\,\mathrm{d}S-P\,\mathrm{d}V $}
Apparently, energy is that which is externally concentrated (unambiguous) yet internally dispersed (ambiguous). Thus energy measures agency.
In terms of Helmholtz free energy
{$F=U-TS$}
{$\textrm{d}F= \textrm{d}U - T\,\textrm{d}S - S\,\textrm{d}T = T\,\mathrm{d}S-P\,\mathrm{d}V - T\,\textrm{d}S - S\,\textrm{d}T = -P\,\mathrm{d}V - S\,\textrm{d}T $}
{$\textrm{d} F=-S\,\textrm{d} T-P\,\textrm{d} V$}
Readings