Negative amount of particles in statistical mechanics

[Catria]

Suppose that you have $N = \left(\frac{\partial U}{\partial \mu}\right)_{S,V} < 0$, supposedly the number of particles, even though the actual number of particles is greater than zero. This means that you can have, in a system subjected to a grand canonical ensemble, less than 0 particle for statistical physics purposes (or less catastrophically a non-integer number of particles), yet the actual number of particles is an integer greater than 0. Or would it otherwise mean that negative numbers of particles are physically possible (albeit as dark matter since standard model particles have all been detected in positive numbers)?

I fail to understand how can the stat-mech number of particles, which can be non-integer, or negative even, represent something different from an actual physical quantity. I knew $\mu$ represented the internal energy per particle, however.

 

[PeterDonis]

Suppose that you have $N = \left(\frac{\partial U}{\partial \mu}\right)_{S,V} < 0$, supposedly the number of particles, even though the actual number of particles is greater than zero.

Do you have an actual case in mind where this happens? Or a reference describing one?

 

[DrDu]

Don't you habe to consider $\Omega=U-\mu N$ instead of U? Remember that you are considering an ensemble average, so fractional numbers aren't that peculiar. Whether negative values for N Marke sense vor not, depends on the system, e.g. considering positrons AS negative amount oft electrons.