# How should a heat current be defined for two component system with unequal numbers

I'm trying to calculate a heat current in a linear response regime of a very weakly-interacting two component low temperature Fermi gas with imbalanced population. I think there are two standard ways to define a heat current.

(1) Energy current in a frame where there is no bulk mass current (assume the system is Galilean invariant) Then, let's express the energy current to be $J_{E}$ and the mass current of species $i$ to be $J_{i}$. Then, we have $$J_{heat} = J_{E} \quad \text{(in J_{1} + J_{2} = 0 frame)}$$ In a lab frame, you can show that $$J_{heat} = J_{E} - \frac{(e + P)}{n_{1} + n_{2}}(J_{1} + J_{2})$$ where $e$ is the energy density and $P$ is the pressure. In low temperature, note that $(e + P) = n_{1}\mu_{1} + n_{2}\mu_{2}$. The origin of this definition comes from the fact that an energy current consists of two component; a convective flow from bulk motion of fluid and a thermal current. We usually measure "thermal" current maintaining no mass current flows. Then we call it a "heat" current.

(2) From "reversible" thermodynamic relation, $$dQ = TdS = dE - \sum_{i}\mu_{i}dN_{i}$$ Then, the heat current "seems" to follow the usual expression, $J_{heat} = J_{E} - \mu J$. $$J_{heat} = J_{E} - \mu_{1}J_{1} - \mu_{2}J_{2}$$ However, when $\mu_{1} \neq \mu_{2}$, in the frame there is no bulk motion of fluid, $J_{1} + J_{2} = 0$, the heat current is $$J_{heat} = J_{E} - (\mu_{1} - \mu_{2})J_{1} \neq J_{E}$$ Note that if we only work in a linear response regime, entropy transport need not necessarily irreversible. We can transport "local" entropy from one side to the other side maintaining the total entropy of the system unchanged. In usual systems, this issue does not come up since $\mu_{\uparrow} = \mu_{\downarrow}$ in most metallic system. But when we polarize a Fermi system, we encounter this issue of defining heat current.

How should I define the heat current here? Is is just a matter of choice or is there a standard way to define a heat current in population imbalanced multi-component fliud?

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In a multi-component system the energy current in the frame where the mass current vanishes (the rest frame) not only has a $\nabla T$ term related to heat transport, but also a $\nabla(\delta\mu)$ term related to diffusion. That's discussed in most standard text books (see, for example, the discussion of diffusion in Lanau's hydro book).