Grand Canonical Partition function

[WWCY]

Hi everyone,

I understand that the grand-canonical partition function is given by
$$Z = \sum_i e^{-\beta(E_i - \mu N_i)}$$
Is there any interpretation to the quantity $E_i - \mu N_i$ here? In the canonical ensemble this would simply be energy of the $i$th state, so I suppose this would be "energy" of some kind in the GCE?

Thanks in advance.

 

[Christopher Grayce]

The chemical potential (mu) is the marginal energy cost of adding one more mole of particles to the system. Another way to think of it is that it's the energy (per mole) of the highest occupied many-body orbital, or the Fermi energy. So mu*Ni is some kind of "typical" energy of the state with Ni moles: the number of moles times the marginal energy cost per mole of the state.

That means Ei - mu*Ni is the deviation from this "typical" cost. If that quantity is positive, the energy of state i is higher than "typical" for Ni moles, and the Boltzmann exponential weights it low, and if that quantity is negative, the energy of state i is lower than "typical" for Ni moles, and the Boltzmann factor weights it higher.

 

[stevendaryl]

Hi everyone,

I understand that the grand-canonical partition function is given by
$$Z = \sum_i e^{-\beta(E_i - \mu N_i)}$$
Is there any interpretation to the quantity $E_i - \mu N_i$ here? In the canonical ensemble this would simply be energy of the $i$th state, so I suppose this would be "energy" of some kind in the GCE?

Thanks in advance.

Here's an intuition for the grand canonical partition function: Imagine that you have a small system that is allowed to interact with a much larger system (which might consist of many copies of the same small system). The boundary between the small system and the large system is permeable, so energy and particles can be exchanged with the large system.

Let's suppose that the small system initially has a temperature that is very different from the large system, and initially has a particle density that is much different. Then after the interact for a while, the hotter of the two systems will get cooler and the colder of the two systems will get warmer until they reach equilibrium. By definition, they are at equilibrium when they have the same temperature, which means that on the average, no energy flows from one system to the other. Similarly, particles will tend to move from the system with the highest concentration to the system with the lowest concentration. By definition, they are at equilibrium when they have the same chemical potential, which means that on the average, no particles flow from one system to the other. Particles go in both directions, and so does energy, but on the average, as much energy enters the small system as leaves it and as many particles enter as leave.

So even though the chemical potential has the dimensions of "energy", I would say that it's more of an accounting of entropy than of energy. However, an entropy change can be converted into an equivalent energy change using the temperature. From thermodynamics:

$\Delta S = \frac{\Delta E}{T} - \mu \frac{\Delta N}{T}$

So adding a single particle to the small system changes the entropy by the same amount as removing an amount of energy $\mu$.

It might be confusing at first why adding particles would lower the entropy (the minus sign shows that it does, as long as $\mu \gt 0$). But think about it this way: If you increase the number of particles while keeping the energy constant, that means that the amount of energy per particle goes down, which tends to lower the entropy. (I believe it's possible for $\mu$ to be negative in certain circumstances, so this rule of thumb isn't always valid).

 

[Christopher Grayce]

Chemical potential is negative under almost all circumstances, as we normally define energy zero to be when particles are infinitely far from each other, not interacting.