Statistical Mechanics

We now discuss the connection between the microscopic states of a system and the thermodynamic quantities. We start by assuming that we have a quantum system with discrete energy levels. The passage to classical mechanics will be discussed later. We take as our defining equation the `information theoretic' definition of the entropy:

$$S = -k_B \sum p_i \ln p_i,$$ (1.29)

where $p_i$ refers to the probability for the system to exist in the quantum state $i$. The constant $k_B$ is Boltzmann's constant which has units of energy divided by temperature. Our guiding principle will be to maximize the entropy under various constraints.

First consider an isolated system. If we maximize Eq. 1.29 with respect to the probabilities, we find that $p_i$ is a constant. Let us imagine that there are exactly $\Omega = \Omega(E)$ states at energy $E$ (or this many states in a small range of energies centered about $E$). The sum of all probabilities must be one (this is a constraint), so $p_i = 1/\Omega$ and

$$S = k_B \ln \Omega(E)$$ (1.30)

This formula connects the microscopic degeneracy to the thermodynamic variable $S$. From Eq. 1.15, we find that

$$\left(\frac{\partial \ln\Omega}{\partial E}\right)_{V,{\vec N}} = \frac{1}{k_B T}$$ (1.31)

This is a fundamental equation of statistical mechanics which relates the temperature to the change in the logarithm of the degeneracy with energy. Since the degeneracy must increase with energy, the temperature is a positive quantity. Note that the combination $k_BT \mbox{(change in)}\ln\Omega$ is thermodynamically related to an energy change. The discussion so far relates to the microcanonical partition function: the case where the system is completely isolated from the environment.

Next allow the system to exchange energy across a bounding wall which separates it from the surroundings. We maximize the entropy under the constraint that the average energy of the system is a constant. To perform this optimization with constraints, we employ the method of Lagrange multipliers. We consider the entropy expression modified by two constraints (one for the average energy, one for the normalization):

$$S' = -k_B \sum p_i \ln p_i - \lambda \sum p_i E_i - \gamma \sum p_i$$ (1.32)

Notice that $\lambda$ has units of one over temperature. When the extremum is obtained, the probability is then given by

$$p_i = \mbox{constant} \times e^{-\lambda E_i/k_B}$$ (1.33)

The normalization constraint makes the probability

$$p_i = \frac{e^{-\lambda E_i/k_B}}{\sum e^{-\lambda E_i/k_B}}$$ (1.34)

The sum in the denominator is over all states of the system, and we refer to the sum as the canonical partition function $Q = Q(T,V,{\vec N})$. If we sum over energy levels instead of over all states, then the partition function becomes

$$Q(T,V,{\vec N}) = \sum \Omega_ie^{-\lambda E_i/k_B}$$ (1.35)

Now, what is the connection between the partition function $Q$ and thermodynamics? To make this connection, we utilize the maximum term theorem, which states that the logarithm of the sum is essentially equal to the logarithm of the largest term in the sum. This result occurs physically because the degeneracy $\Omega(E,V,{\vec N})$ is a rapidly growing function of $E$, while the exponential term decreases rapidly with $E$ (so as to make the terms for high energy go to zero in the sum). Somewhere in the middle of the energy range sits the (sharp) maximum. Then

$$-k_BT \ln Q \approx -k_BT \ln \Omega_{max} + T\lambda E_{max}.$$ (1.36)

Above we have made the association between $k_B\ln\Omega$ and the entropy. If we differentiate Eq. 1.36 with respect to $E_{max}$, we find

$$0 = -1 + T\lambda ,$$ (1.37)

so $\lambda = 1/T$. It is then plausible to take the step

$$-k_BT \ln Q = -TS + E,$$ (1.38)

Therefore,

$$-k_B T \ln Q(T,V,{\vec N}) = A$$ (1.39)

the Helmholtz free energy. Once we have this connection, other thermodynamic quantities can be derived. For example, the system pressure is

$$-\left(\frac{\partial A}{\partial V}\right)_{T,{\vec N}} = p... ..._BT\left(\frac{\partial \ln Q}{\partial V}\right)_{T,{\vec N}}$$ (1.40)

Now that we have the maximum term tool for connecting a partition function to a thermodynamic quantity, we can construct other partition functions which pertain to differing physical circumstances. In addition to modeling a different set of experimental conditions, alternative partition functions may be more suitable for computations of system properties. Here we examine a system which allows heat and particle exchange with the environment, leading to the grand canonical partition function:

$$\Xi(T,V,\vec{\mu}) = \sum_i \sum_{\vec{N}} \Omega_i e^{-E_i/... ...{N}/k_BT} = \sum_{\vec{N}} \vec{z}^{\vec{N}} Q(T,V,{\vec{N}}),$$ (1.41)

where $\vec{z}$ is the set of all absolute activities $\exp(\mu_i/k_BT)$. Here $\vec{\mu}$ is the set of chemical potentials for all of the system components. The `dot product' leads to a sum of $\mu_i N_i$ terms, one for each component. If we employ the maximum term method as above, we find the following thermodynamic connection:

$$pV = k_BT \ln \Xi (T,V,\vec{\mu})$$ (1.42)

This partition function is thus ideally suited for computations of the pressure equation of state.

So far we have expressed the partition functions as discrete sums over the quantum energy levels of the system. For most systems considered in this book, however, classical mechanics is an appropriate level of theory. In the transition from quantum statistical mechanics to the classical picture, the discrete sums are replaced by integrations over particle momenta and coordinates. The transition can be handled rigorously via Wigner distributions, for example, and is discussed in several texts. We simply give the results here.

For a classical system with the set $\vec{N}$ particles, the canonical partition becomes

$$Q(T,V,\vec{N}) = \frac{1}{\prod N_i!}\prod\left( \frac{2\pi m_i k_B T}{h^2} \right)^{3N_i/2} Z_{\vec N}$$ (1.43)

where

$$Z_{\vec N} = \int dx_1 dx_2 \cdots dx_{\vec N} e^{-U(x_1,x_2,\cdots,x_{\vec N})/k_BT},$$ (1.44)

$h$ is Planck's constant, and $m_i$ is the particle mass. The expression containing Planck's constant involves the thermal de Broglie wavelength:

$$\left(\frac{h^2}{2\pi m_i k_B T} \right)^{1/2} = \Lambda_i$$ (1.45)

Since there is a product of $3N$ de Broglie wavelength terms, those units cancel the units from the integration variables in $Z_{\vec N}$ to give a unitless $Q(T,V,{\vec N})$ in Eq. 1.43.

The energy $U(x_1,x_2,\cdots,x_{\vec N})$ is the classical potential energy of interaction of all the particles in the system for a given configuration. If we absorb the terms involving Planck's constant into the activity, we arrive at

$$\Xi(T,V,\vec{\mu}) = \sum_{\vec{N}} \frac{\vec{z}^{\vec{N}}} {\prod N_i!} Z_{\vec{N}}(T,V),$$ (1.46)

This classical form for the grand canonical partition function naturally leads to series expansions for virial equations of state due to its close resemblance to typical power series.