Local and global detailed balance I'm taking a course on nonequilibrium statistical mechanics and I encountered the terms local and global detailed balance. I'm a bit confused about what is their exact definition and what is the difference.
The only definition of detailed balance I know is for Markov processes and goes as
"There exists a distribution $\rho$ for which $k(x,y)\rho(x) = k(y,x)\rho(y)$ for all states x and y." This distribution is then called the equilibrium distribution (and it is of course stationary).
 A: In non-eq stat mech the comment above isn't what they mean. 
We have to establish 3 separate things:


*

*Balance


This is the condition that all transitions into and out of a state 'balance' ie
$$ \sum_x p(x)k(x,y)=p(y) \quad \forall y$$
This ensures a stationary state, but nothing else


*Detailed balance


This is the condition that all transitions between any pair of states balance, as it's name suggests, 'in detail'. This means (for just normal even variables w.r.t. time reversal eg position, node, state etc. [not momenta, magnetic fields]) that there is a stationary state and there is no stationary current in that distribution (and no entropy production) ie
$$p(x)k(x,y)=p(y)k(y,x) \quad \forall x,y$$


*Local detailed balance


This is a somewhat different concept which, although related, adds some physical properties over and above just the properties of stochastic processes.
If you have detailed balance and the system is physical you have the distribution being the equilibrium distribution so that
$$ \frac{k(x,y)}{k(y,x)}=\frac{p^{eq}(y)}{p^{eq}(x)}=\exp\left(-\frac{\Delta U}{k_BT}\right) $$
where $U$ is the internal energy of the system. The environment is deemed to be providing the stochastic behaviour as a heat bath or similar with a defined temperature (i.e. at equilibrium itself) which isn't doing work on the system (it's not changing it's Hamiltonian) so the change in internal energy is heat to/from the environment (depending on your sign convention). As such you have
$$ \frac{k(x,y)}{k(y,x)}=\exp(\Delta S) $$
where $\Delta S$ is the entropy in/associated with the environment from the transition $x\to y$.
Local detailed balance is the condition that this relationship between the ratio of transition rates to the exponential of entropy production in the environment holds without caring whether the system obeys detailed balance (i.e. has a stationary state which is the equilibrium state). 
Historically the reason why it is considered 'local' detailed balance is because you can break detailed balance by connecting a particle (or whatever) to two equilibrium baths with different temperatures (or chemical potentials in the grand canonical ensemble) and yet reliably identify entropy production to one of the baths 'locally' as the heat exported over the temperature which is still well defined. More generally what you are saying, given some general perturbing field that is causing non-equilibrium behaviour (i.e. you can't tie it down to a Hamiltonian; examples include non-conservative (path dependent) forces ) locally can be treated as though arising from some potential which you can write down, associate with an energy change, which you identify as heat and as such connect to entropy production in the bath. First instance I can find of people talking about it is here: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.28.1655. Perhaps more intuitively you might consider it as: the system does not care and cannot distinguish whether a force arises from a Hamiltonian or not as long as you consider small enough times (And therefore one would assume locales: ie local detailed balance) and so the heat looks the same; simply string a load of these small changes in this pseudo hamiltonian together and you get the result for a whole path.
If you have detailed balance you have balance and local detailed balance. In non-eq stat mech you generally always have local detailed balance.
