Subsections

• Energy and Quadratic Forms
• Partition Function and Thermodynamic Functions
  • Partition function
  • Change of Variables
  • Thermodynamics
• Equipartition Theorem and Its Applications
• What's Wrong With Equipartition Theorem?

Equipartition Theorem

Energy and Quadratic Forms

Suppose we have l degrees of freedom (generalized coordinates) q₁, q₂,..., q_l and corresponding momenta p₁, p₂,..., p_l, and all q's are close to their equilibrium values (take them to be zero, and q_i will be deviations from equilibrium positions): $q_i\approx0$. We want to calculate thermodynamics of such a system.

Example:

A simple model of a crystal--masses m on springs with coefficients k. All masses sit near their equilibrium positions! We have 3N degrees of freedom $\mathbf{r}_i$, i=1,..., N.

$\mbox{ \InputIfFileExists{crystal.pstex_t}{}{} }$

Kinetic energy:

$$K(\tilde p) = \sum_{i=1}^{l} \frac{p_i^2}{2m_i}$$

Potential energy: let's expand

$$U(\tilde q) = U(0) + \sum_i a_i q_i + \frac12\sum_{i,j} k_{ij} q_i q_j + \ldots$$

The first term does not matter--we'll drop it! The second term is zero (we are at equilibrium!)

Harmonic approximation:

cubic and higher terms are small, and  

$$U(\tilde q) = \frac12\sum_{i,j} k_{ij} q_i q_j$$ (1)

This is quadratic form.

In harmonic approximation both potential and kinetic energy are quadratic forms of coordinates and momenta.

Example:

For our crystal

$$K(\tilde \mathbf{p}) = \sum_i \frac{\mathbf{p}_i^2}{2m},\qua... ...m_{\text{neighbors}} \frac{k(\mathbf{r}_i-\mathbf{r}_j)^2}{2}$$

Expanding U, we recover quadratic form (1)

Partition Function and Thermodynamic Functions

Partition function

 

$$Q =\frac{1}{(2\pi\hbar)^{l}}\int \exp\bigl(-\beta K(\tilde p) - \beta U(\tilde q)\bigr)\, d\tilde p\,d\tilde q$$ (2)

with $\beta=1/k_B T$

Note:

In this lecture we use partition function instead of configuration integral and k_B instead of k for Boltzmann constant (why?)

Change of Variables

$$q_i = \sqrt{k_BT}q'_i,\qquad p_i=\sqrt{k_BT}p'_i$$

New energy:

$$E = k_BT\left(\frac{{p'_i}^2}{2m} + \sum_{i,j} \frac{k_{i,j}q'_i q'_j}{2}\right) = k_BTE'$$

New differentials:

$$d\tilde q=(k_BT)^{l/2}d\tilde q',\quad d\tilde p=(k_BT)^{l/2}d\tilde p'$$

Classical partition function:

$$Q =(k_BT)^{l}\frac{1}{(2\pi\hbar)^{l}}\int \exp\bigl(-E(\tilde p',\tilde q'\bigr)\, d\tilde p'\,d\tilde q'$$

The integral does not depend on temperature!

Thermodynamics

Free energy:

$$A = -k_BT\ln Q = -k_BT\ln X - lk_BT\ln T$$

with X not dependent on T. Entropy, energy and heat capacity:

$$\begin{gathered} S = -\frac{\partial A}{\partial T} = k_B\l... ...BT\ C = T\frac{\partial S}{\partial T} = lk_B \end{gathered}$$

We have l degrees of freedom, and energy is lk_BT: one k_BT for each degree of freedom!

What happens, if for some q_i potential energy is zero (rotational and translational degrees of freedom)? Then it contributes k_BT/2.

Equipartition Theorem and Its Applications

In harmonic approximation for classical systems each degree of freedom contributes k_BT to energy and k_B to heat capacity unless potential energy is identically zero. In the latter case it contributes k_BT/2 and k_B/2 correspondingly.

Examples:

1.

Monoatomic ideal gas. 3N degrees of freedom. Energy 3k_BTN/2, heat capacity 3k_BN/2

2.

Diatomic ideal gas, no vibrations. 5N degrees of freedom, energy 5k_BNT/2, heat capacity 5k_BN/2

3.

Diatomic ideal gas, vibration is allowed. 6N degrees of freedom, one of them vibrational. Energy 7k_BNT/2, heat capacity 7k_BN/2

4.

Solid body. 3N vibrational degrees of freedom. Energy 3k_BNT, heat capacity 3k_BN. Molar heat capacity 3R (Dulong & Petit law).

What's Wrong With Equipartition Theorem?

Equipartition theorem is fine--except sometimes it does not work!

1.

It predicts different results for gases whether we allow vibration or not--but should not it be always allowed?

2.

It predicts heat capacity 3k_BN for solids. This is right--at high temperatures. For diamond ``high'' means $T\gt 2400\,\mathrm{K}$, sometimes it is 100-$300\,\mathrm{K}$.

Reason:

Vibrations are not classical. They are quantum. We need to include quantum mechanics to describe them properly!