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Let's say I have two opposite sides of a box with temperature $T_{1}$ and $T_{2}$ respectively. And I have calculated the temperature in the region between the two sides. So, just like the equation $$\boldsymbol{E=-\nabla\phi}$$ where E represent the flow of potential, do we also have a similar kind of equation in Thermodynamics that represent the flow of temperature.

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Firstly, energy is not a flow of potential. Potential does not flow.

Secondly, you have the equation of heat, where the flow of heat is granted because of the gradient in temperature. Flow of temperature has no meaning

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