1. I was thinking about analogies between the fields of electric force, pressure force in liquids and heat flow and am wondering whether the heat flow is grad T or grad Q.

Ive come to the conclusion that it must be grad T for the following reason:

Consider two objects of the same mass and form, at the same temperature, but with one object having a higher thermal capacity than the other, so both have the same temperature, but one has more thermal energy than the other.

If I put the two objects into contact there will be no change in temperature, so no flow of heat.

If I put the two objects together when they have different temperatures there will be heat flow.

By this reasoning I come to the conclusion that heat flow = grad T.

Im interested in any comments because I fuck up physics alot.

2.

3. Temperature is a measure of the average energy of the parts of the systems (be those parts molecules, atoms, nuclei, or electrons). The statistical definition of temperature is T = 1/(dS/dU) This definition leads to all the properties of temperature we know and like - although that may not be readily apparent.

Heat flows from high to low temperature. This being because the enthropy of a system must increase or be maintained under all processes (2LoT). Suppose I have two temperature reservoirs with T1 > T2 and an amount of energy dQ moves through heat exchange (with the T1 -> T2 direction being the positive one).

The enthropy change for the entire system is dS = dQ/T2 - dQ/T1

Since dS >= 0 the heat must flow from T2 to T1, T2 needs not have higher total thermal energy than T1 (which it could have if it had greater mass or a higher heat capacity, for instance).

Temperature is an intesive quantity: If you place two heat reservoirs of the same temperature in contact, the new reservoir remains the same temperature as the two individual reservoirs. OTOH, two identical heat reservoirs put together will have twice the thermal energy of one or the other - since energy is an extensive quantity.

Heat is energy that gets redistributed through thermal contact (the technical definition is that heat changes the energy distribution - as opposed to work, which moves energy by shifting the entire distribution up or down the energy axis). Note that 'heat' denotes only the energy that's actually transferred.