Something that has bothered me for a while regards the interpretation of chemical potential for different statistics. While I understand its meaning in metals (and its relation with the Fermi surface), I cannot quite relate this definition with the thermodynamic chemical potential, defined as the change in energy of the system when one particle is added to it (or according to Wolfram Demonstrations, " It can be interpreted, for example, as the ability of the system to perform phase transitions or chemical reactions, or its tendency to diffuse").
1) Are those concepts (thermodynamic vs Fermi-Dirac chemical potential) related or they should be thought as different things?
2) Am I missing something trivial or this cited demonstration is misleading? It is mentioned there that the x coordinate corresponds to (E-u). Shouldn't the function blow up for x = 0 independent of temperature, then? I do not understand the shift for negative values as temperature increases.