What is energy?

No one knows and anyone who says they do is lying.

nope

in thermal equilibrium there is energy but no capability to do work

Idk man, God maybe? Magic. that's what it essentially is to me. Understood magic, but fucking magic yo

Energy is quantitative causality.

gradients

>appear to have
Not appears, HAS zero energy. It might still have momentum, tho, if the particles have mass.

Energy is a scalar quantity associated with temporal symmetry in the laws of physics. The functional form which describes the energy of some configuration of matter is the Hamiltonian, and the classical behavior of matter under the influence of forces is described by Hamilton's equations. In quantum theory, the Hamiltonian generates the flow of time - in the sense that the information of a system at any time can be exactly specified (in principle) by the Hamiltonian and the information of the system at any previous time.

Any quantum system with a continuous temporal symmetry will maintain the same value of Energy which describes it at all times, i.e. it is a conserved quantity.

That's misleading, since in thermodynamics work is associated with pressure or volume changes. At any finite temperature you'll have local fluctuations of pressure/volume of a characteristic size which undoubtedly change the energy density. A system in equilibrium is not a static system at all.

>The capacity to do work.

this.

dafuq is this picture