Results from statistical physics

Consider a closed system that is kept at a certain temperature, T. The system has several possible

energy states, E. The probability, f, to find the system in the state is then given by

=1

Where k is the Boltzmann constant and has the value = 1.38 1023 J/K = 8.617 105 eV/K.

=

We obtain the average energy of the system by multiplying the energy of a state with the probability

of finding the system in that state and then sum over all states, i.e.

=

=1

Example: We have a system with four states. The energies of the states are given by

E1 = 0.01 eV

E2 = 0.05 eV

E3 = 0.05 eV

E4 = 0.1 eV

1. What is the probability that the system is in state 3 at 300 K?

Solution:

Begin by calculating Z.

Z = 0.679 + 0.145 + 0.145 + 0.021 = 0.990

3 =0.145

0.990= 0.146.

Answer: The probability is 14.6%.

2. What is the probability that the system has the energy 0.05 eV at 300 K?

Solution:

There are two states with energy 0.05 eV. The probability of finding the system with an energy 0.05

eV is then equal to the sum of the probabilities of finding the system in one of those states, i.e.

0.146 + 0.146 = 0.292.

A more general way to find that result is to write

=0.05 =

, where g gives the number of states with the energy E and is called the

degeneracy. In our example g = 2.

Answer: The probability is 29.2%.

The average energy of the system can also be written

=1

=

We also find

=

2

Inserting this result in the expression for the average energy we obtain

=2

= 2

In large systems it is sometimes possible to get an analytical expression for Z as a function of T.

In these cases the last expression for the average energy is simple to use.

Systems described by one-particle states

Consider a system of identical particles kept at a temperature T. The number of particles in the

system can vary. The particles are found in different one-particle states, j, with energy Ej.

The statistics of this systems depends on whether the particles are bosons or fermions.

Bosons

Bosons are particles with integer spin (s = 0, 1, 2, etc.).

The average number of particles in a one-particle state, j, is given by

=1

1

where is the chemical potential. This relation is called a Bose-Einstein distribution. We see that

there can e any number of bosons in a one-particle state. The average number of particles is given by

= = 1

1

Fermions

The other type of particles is called fermions. Fermions are particles with odd half-integral spin (like

1/2, 3/2, and so forth). One important example is the electron.

The average number of particles in a one-particle state, j, is given by

=1

+ 1

where is the chemical potential. This relation is called a Fermi-Dirac distribution. We see that fj can

only have values between 0 and 1. This is consistent with the rule that two electrons (and in general

two fermions) can never be in the same one-particle state. The average number of particles is given

by

= = 1

+ 1

We are usually not interested in systems where the number of particles can vary. It is possible to

show that in large systems the relative fluctuation of the number of particles is small so we will

assume that = . However, if we would like to study the effect of changing the temperature but

keeping the number of particles constant, we see that the number of particles would increase when

the temperature is raised according to the relations above. In order to compensate for this effect we

must require that the chemical potential, , is dependent on temperature.

The figure 1 below shows the Fermi-Dirac distribution at different temperatures. Note that as the

temperature goes towards zero, the Fermi-Dirac distribution function approaches a step function.

This means that there are particles in all states up to an energy equal to the chemical potential.

There are no particles in states with energy higher than the chemical potential at T = 0 K.

Figure 1. Fermi-Dirac distribution function (picture taken from Wikipedia).