# Dissipation when the temperature is not constant

Consider a process where some chemical species diffuses from one part of a system (which I'll call $A$) to another ($B$) at a rate $r$ $\text{mol}\cdot \mathrm s^{-1}$. If the system's temperature is constant and homogeneous, we say that energy is dissipated at a rate $$D = r(\mu_B - \mu_A),$$ where $\mu_A$ and $\mu_B$ are the chemical potentials of the particles in parts $A$ and $B$ of the system respectively. The dissipation is always positive, because $D=T d_i\!S/dt$, where $d_i\!S/dt$ is the rate at which entropy is produced within the system due to the transport process.

However, if the temperatures of the two parts of the system are different then the second law says $$\frac{d_i\!S}{dt} = r\left( \frac{\mu_B}{T_B} - \frac{\mu_A}{T_A} \right) \ge 0,$$ which means that the above expression for $D$ can be negative if $T_A>T_B$.

So my questions are

• Is the term "dissipation" generally thought to be meaningful in systems that don't have a constant, homogeneous temperature?
• If yes, what is the correct expression for it in the above scenario?
• Most importantly, does anyone know of a reference where the concept of dissipation in systems with a non-constant or non-homogeneous temperature is discussed?
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I wonder if crossposting to a private beta is allowed.. Chem.SE would like this 'un, but you don't have an account there. I can do it, but I don't know if it's allowed.. – Manishearth May 3 '12 at 17:20
I have an account there now :) (I'd committed to the proposal but oddly didn't get an email to say it'd gone live, so I didn't know until you mentioned it.) What's the best way to cross-post it - should I just post a new question there with the same text? – Nathaniel May 3 '12 at 17:38
#account check your spam folder, if its not there then file a bugreport on MSO #crosspost: only if ypu can make the two questions distinct meta.stackexchange.com/questions/71938/… Otherwise, we can migrate.. On Phy.SE we have more activity, but less users who know this stuff. On chem, less overall activity but more users who can answer thia. So the better site(for getting an answer) is debatable. I say the post is a better fit on chem, but maybe not its ability to get an answer quickly. – Manishearth May 3 '12 at 17:59
I'd say go ahead and migrate it. There seems to be an impressive knowledge of chemical thermodynamics over there, and it would be good to see more questions like this on that site. – Nathaniel May 3 '12 at 20:45
Flag it (for ♦ moderator attention) if you want a migration. Anyway, I already did that :) – Manishearth May 4 '12 at 0:45