A.4 Open Systems at Constant Temperature: Grand-Canonical Description

When a system is in contact with a reservoir of heat and particles much larger than the system, the reservoir imposes on the system its temperature T and its chemical potential ц. The system is in the "grand-canonical situation." The energy and the number of particles of the system fluctuate by exchange of heat and particles with the reservoir. The system reaches equilibrium when its temperature is equal to T and its chemical potential equal to ц of the reservoir.

The grand-canonical probability of the microscopic state / is given by

where

Z is called "grand-partition function.” The ensemble of microscopic states obeying the probability (A. 19) is called "grand-canonical ensemble.”

Z plays an important role in the calculation of principal properties of the system. As In Z appears often in the calculation, we define a new function J, called "grand potential,” by

We can express the following quantities as functions of J : ^[1]

• Average energy:

We deduce

• Grand-canonical pressure p: p is defined by

• Grand-canonical entropy: Using Р/ of (A.19), we have

We can show that in the spontaneous evolution when an external constraint is removed, the system tends to equilibrium in the sense of decreasing J. The new equilibrium is attained when J is minimum (see demonstration in Ref. [88]).

A.4.1 Applications

We consider again a classical ideal gas studied above by the microcanonical and canonical descriptions. We show below that the grand-canonical description gives the same results. We calculate the grand-partition function, the average number of particles of the system, the energy and the pressure as follows:

• The grand-partition function:

Z = '^2,_Ne^NP'ⁱZ[T, N, V) where Z is the partition function. We have Z = z^N/N =; therefore,

Z = Y^n=о e^"z^w/W! = Ew=oU^z)^W/^W! = exp(dz) where A = e^" (fugacity).

• The average number of particles:

We have

We obtain thus the same equation of state for the gas as before.

A.5 Fermi–Dirac and Bose–Einstein Statistics

We can use the grand-canonical description to demonstrate the Fermi-Dirac and Bose-Einstein distributions.

We consider a system of identical, independent and indiscernible particles. In such a hypothesis, each particle has the same "list” of individual states. We can define each individual state by the number of particles present in that state. Let к be an individual state of energy 6k and n_k its number of particles. The total energy and the total number of particles in the microscopic state / of the system are

The grand-partition function reads

where, for a given distribution of particles {n*} on individual states, we make the sum in the argument of the exponential, then we repeat for another distribution (njj until all distributions have been considered. This procedure is equivalent to the sum on the states /. In doing so, we can express Z as

where

It is noted that the sum in z_k is performed on all possible values of n_k for the level 6_k. We distinguish two cases:

• Bosons (particles of spin 0 or integer):

In this case, n_k = 0, 1, 2, • • • (no limit). We have

• Fermions (particles of spin half-integer):

In this case, n_k = 0, 1. We have

To calculate the average number of particles n_k of the state of energy e_k, we use (A.22), (A.28) and (A.30):

We obtain

Using (A.32) and (A.33), we have, for the boson case, and for the fermion case,

The distributions /(e)^B£ and /(e)^£D are called "Bose-Einstein and Fermi-Dirac distributions," respectively.

A.6 Phase Space: Density of States

A.6.1 Definition

The phase space is defined by the number of degrees of freedom which characterize the microscopic states of the system. In a quantum case, each state is defined by some quantum numbers. For example, each of the microscopic states of a free particle in a box is defined by a wave vector к which is quantified by the boundary conditions, the state of an electron in an atomic orbital is given by four quantum numbers (n, /, m/, m_s). In the case of a system of classical particles each of the microscopic states of the system is defined by the momentum and the position of each particle, p, and r,. These variables constitute the phase space. The sum on the microscopic states is taken over all of these variables.

We consider a system of N classical particles in three dimensions. The number of degrees of freedom is 6N because each particle is defined by 6 variables: three of p, and three of r,. We attribute a “volume" of 6N dimensions for each microscopic state in the phase space. The elementary volume occupied by a particle is chosen as small as allowed by the uncertainty principle of quantum mechanics. The smallest elementary volume occupied by a particle is equal to [Vh]⁶ = h³ where h is the Planck constant. This choice is made to discretize the classical "continuous” phase space in order to count the number of states: it suffices to divide a chosen volume in the phase space by the elementary volume to find the number of states contained in that volume. To find results for classical particles, we let h -*■ 0 at the end of the calculation.

The sum on the microscopic states is written as an integral in the classical phase space as follows:

We consider now a quantum system. When the size of the system is large (thermodynamic limit) we can consider the energy as a continuous variable. We can replace the sum on discrete microscopic states к by an integral on the energy e but we have to take into account the degeneracy of each energy. For a continuous energy, the degeneracy is the density of states. We write

where e₀ is the lowest energy and p(e) the density of states. According to the studied case, we replace in this integral n_k by n®^£ or n^Fk^D.

A.6.2 Density of States of a Free Particle in Three Dimensions

For a free electron in a box of linear dimension Lin three dimensions, the Schrodinger equation with the periodic boundary conditions gives the following solution for the energy:

with

where n, = 0, ±1, ±2,____

LetA/”(e) be the number of microscopic states of energy < e. The number of states between e and e + Ae is thus A^r(e + Де) - Af(e). The density of states p(e) is defined by

This is the number of states of energy e, namely the degeneracy in the continuous energy case. We calculate Af(e): in the phase space defined by (k_x, k_y, k_z) where each state is defined by a point (k_x, k_y, k_z), the volume of each state is (2л/L)³ [see Eq. (A.41)]. The number of microscopic states of energy < e is the volume of the sphere of radius к = J^r divided by the volume of a state:

We deduce

where Q. = 1? the system volume, and m the particle mass. If the particle has a spin s then the density of states is

(2s + 1) is the spin degeneracy. For electrons, s = 1/2.

A.7 Properties of a Free Fermi Gas at T = 0

At T = 0, / = 1 for E < до = Ef, and / = 0 for E > д₀. One replaces, therefore, the upper limit in the integrals (A.54) and (A.56) by Ef and one replaces f(E) = l/[exp(/l(F - д)) + 1] by 1.

A.7.1 Fermi Energy

Equation (A.54) becomes

Using p(E) = AE^l/1 [see (A.45)], one obtains

where

from which,

using В given by (A.48).

This result shows that the Fermi energy depends on the density of fermions n = у of the gas. Since the Fermi energy is the chemical potential at Г = 0, the chemical potential in general is closely related to the particle density.

A.7.2 Total Average Kinetic Energy

The total average kinetic energy at Г = 0 is

from which, one has Using (A.47) one gets

One sees that the energy of a free Fermi gas is not zero T = 0, in contrast to the case of a classical ideal gas.

A.8 Properties of a Free Fermi Gas at Low Temperatures

The exact formula for the total number of particles for a large system is:

or

The average energy is or

A.8.1 Sommerfeld’s Expansion

One considers the following integral:

where /?(£■) is a function with finite derivative with respect to E at any order. At low temperatures, one can show that this integral can be expanded in powers of T as follows (see Exercise 4 of Chapter 1):

where /j("i(£")|я=м >^s the n-th derivative of h(E) with respect to E, taken at E = ц.

A.8.2 Chemical Potential, Average Energy and Calorific Capacity

Using (A.58) for the integral in (A.54) one obtains to the order of T²:

where В is given by (A.48). Since N does not vary with T, by equalizing (A.47) and (A.59) one obtains

This equation shows that the chemical potential д is a function of temperature.

In the same manner, using (A.58) for (A.56) with h(E) = Ep[E), one gets

where д has been replaced by (A.60). The average energy thus increases as Г² at low temperatures.

The calorific capacity of a free Fermi gas at low T is, therefore,

One sees that C_v is linear in T. Note that for a classical ideal gas C_v = 3Nke/2, independent of T.

A.9 Free Fermi Gas at the High-Temperature Limit

When T is very high, one shows that the free Fermi gas becomes a classical ideal gas: the quantum nature disappears at high temperatures.

The energy (A.56) becomes at high T

where one has neglected 1 in the denominator because e^^£ -*) » i at high T. Replacing p(E) by (A.45) and putting и = f)E, one has

where one has used the definition of the Г function. Similarly, Eq. (A.54) becomes at high T

From these two equations, one finds

This is the equation obtained for a classical ideal gas. The free Fermi gas loses, thus, the quantum nature at high temperatures.

Appendix В

• [1] Average number of particles: