Gibbs Distribution

Recall that the Boltzmann factor allowed us to determine the ratio of the probability that a system is in a state with energy e₁ to the probability that the system is in a state with energy e₂ if the system is in thermal contact with a reservoir at temperature t.The ratio was

formula

We now want to generalize this to a system that is in thermal and diffusive contact with a reservoir.Consider the following system

drawing

Let N be the number of particles in S, which has an energy e_S.Let the total number of particles be N₀, and the total energy by U₀.Then the number of particles in the reservoir is U₀ - e_S.As before, we can define the probability that the system S is in a state associated with energy e_S and has N particles to be

i.e., the probability is proportional to the number of states accessible to the reservoir times the number of states accessible to the system.Bt if we specify that the system is in a certain state associated with energy e_S, this just becomes

and so the ratio of probabilities becomes

(12.1)

We still need to determine g(U-e_S,N₀-N).Recall that

so the probability becomes

where Ds = s(U₀-e₁,N₀-N₁) - s(U₀-e₂,N₀-N₂).Since the reservoir is large compared to the system, we can calculate the entropy of the reservoir to be

formula

and thus, to first order

(12.2)

We can get the final form by using the definitions and .The Ds becomes

(12.3)

and so the ratio of the probabilities becomes

(12.4)

We call a term of the form exp[(Nm-e)/t] a Gibbs factor.

Gibbs Sum

We can determine the absolute probability by normalizing the probability.Proceeding as before, we get

(12.5)

where Z is called the grand sum, or Gibbs sum, and is defined to be

(12.6)

We can use (12.5) to find the expectation value of various physical measurements, just as before.If X(e_s,N) is some physical measurement which depends on the energy of the state and the number of particles, then the expectation value is given by

(12.7)

where ASN stands for all N and S.

Number of States

Calculate <N>.

(12.8)

It is useful to define the notation

(12.9)

l is called the absolute activity.In terms of it, the grand sum can be written as

(12.10)

and the expectation value of N, <N>, becomes

(12.11)

Energy

Now determine U = <e_S>.First look at <Nm - e>.

formula

So,

(12.12)

or, if we let b = 1/t,

(12.13)

Example:

Consider the absorption of O₂ by Myoglobin.Myoglobin has the property that it either has a single O₂ molecule attached, or else there are no O₂ molecules attached.If we let e be the energy of an absorbed O₂ molecule relative to an O₂ molecule at rest at an infinite distance, then the grand sum is

If the energy must be added to remove the molecule from the Myoglobin, then e will be negative.The expectation value of N is then

(12.14)

This is the fraction of Myoglobin that is occupied by O₂.If we assume that m(O₂) = m(ideal gas of O₂), then since the O₂ molecules which have attached to the Myoglobin will be in diffusive equilibrium with the O₂ molecules in the liquid, we have

formula

Divide this by t and exponentiating, we get

(12.15)

Substituting this into (12.14), N becomes

(12.16)

where .This is called the Langmuir absorption isotherm when it is used to describe the absorption of gases on the surface of solids.