TITLE: Physical interpretation of Robin boundary conditions
QUESTION [19 upvotes]: In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann 
$D_{\nu} u|_{\partial\Omega}\equiv 0$
or Robin (for $\alpha \in \mathbb{R}$) 
$(D_{\nu} u + \alpha u)|_{\partial \Omega} \equiv 0$.
I know that, for example for the heat equation, Dirichlet eigenvalues correspond physically to the boundary being in contact with a (large) heat bath at $T=0$. Or, in the Laplace equation, if we're interested in the modes supported by $\Omega$ (as a drum), Dirichlet boundary conditions can be thought of keeping the boundary from moving.
Neumann boundary conditions, for the heat flow, correspond to a perfectly insulated boundary. For the Laplace equation and drum modes, I think this corresponds to allowing the boundary to flap up and down, but not move otherwise.

My question is: what sort of physical interpretations are there for the Robin boundary conditions? Wikipedia says that they are related to electromagnetic problems, but gives no details. I'd be happy with answers that are not necessarily physics-related, for example, if there was somewhere that Robin boundary conditions naturally arise in a mathematical context, I'd be interested to know about that as well.

REPLY [2 votes]: Robbins conditions correspond to immersing one end of the rod in a constant temperature bath. The temperature bath does not have to match the initial temperature at the end of the rod.  The immersed end of rod may come into thermal equilibrium with the bath, or a constant temperature difference may come about (to ensure a constant heat flux). 
In thermodynamics, reference is always made to work reservoirs (ideal mechanical energy sources that can do work reversibly without any entropy increase) and reference is made to heat reservoirs that can transfer heat without any change in temperature (of the reservoir).  Such a heat reservoir can be attained by making it a water/ice mixture - so that the temperature of the bath is maintained due to the phase change occurring.  
The Robbins condition itself can be interpreted as a statement of Newton's law of heating (cooling): the rate of heat transfer is directly proportional to the difference in temperature (between the hot and cold object).
dQ/dt ̇  =  h(To - T)   Newton’s law of heat transfer
Here dQ/dt represents the rate of heat transfer between the end of rod and the temperature bath.
The constant h is a thermal boundary layer coefficient whose reciprocal represents IMPEDANCE to the transfer of heat.
To is the temperature of the bath.
T is the temperature of the end of the rod.
We can get a Robbins condition of this by invoking Fourier's first law of heat conduction:  
q = -k dT/dx
Here q is the heat flux and equals (1/A)(dQ/dt) where dQ/dt is the rate of heat transfer and A is the cross sectional area of the solid rod.
Combining Newton's law of heat transfer with Fourier's first law of conduction yields
-kA dT/dx  =  h(To - T)  right end immersed
kA dT/dx = h(To - T)  left end immersed 
An equivalent form is 
dT/dx - aT = b
or
dT/dx + aT = b
where a and b are constant dependent on k, A, h and To.
The concept of impedance is hidden in a Robbins condition  because Newton's law of heating (cooling) is hidden.  The Robbins condition comes about from applying Newton's law.  
Newton's law of cooling is an impedance equation and the reciprocal of h (the thermal boundary layer coefficient) acts like a resistance.  A good analogy is to compare the rate of heat transfer to current, the temperature difference to voltage, and the thermal boundary layer to a resistor.