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- Energy and Quadratic Forms
- Partition Function and Thermodynamic Functions
- Equipartition Theorem and Its Applications
- What's Wrong With Equipartition Theorem?

Suppose we have *l* *degrees of freedom* (generalized
coordinates) *q _{1}*,

**Example:**- A simple model of a crystal--masses
*m*on springs with coefficients*k*. All masses sit near their equilibrium positions! We have 3*N*degrees of freedom ,*i*=1,...,*N*.

Kinetic energy:

Potential energy: let's expand The first term does not matter--we'll drop it! The second term is zero (we are at equilibrium!) This isIn harmonic approximation both potential and kinetic energy are quadratic forms of coordinates and momenta.

**Example:**- For our crystal
Expanding
*U*, we recover quadratic form (1)

(2) |

**Note:**- In this lecture we use partition function instead of
configuration integral and
*k*_{B}instead of*k*for Boltzmann constant*(why?)*

Free energy:

with
What happens, if for some *q*_{i} potential energy is zero (rotational
and translational degrees of freedom)? Then it contributes *k*_{B}*T*/2.

In harmonic approximation for classical systems each degree of freedom
contributes *k*_{B}*T* to energy and *k*_{B} to heat capacity unless
potential energy is identically zero. In the latter case it
contributes *k*_{B}*T*/2 and *k*_{B}/2 correspondingly.

Examples:

- 1.
- Monoatomic ideal gas. 3
*N*degrees of freedom. Energy 3*k*_{B}*TN*/2, heat capacity 3*k*_{B}*N*/2 - 2.
- Diatomic ideal gas, no vibrations. 5
*N*degrees of freedom, energy 5*k*_{B}*NT*/2, heat capacity 5*k*_{B}*N*/2 - 3.
- Diatomic ideal gas, vibration is allowed. 6
*N*degrees of freedom, one of them vibrational. Energy 7*k*_{B}*NT*/2, heat capacity 7*k*_{B}*N*/2 - 4.
- Solid body. 3
*N*vibrational degrees of freedom. Energy 3*k*_{B}*NT*, heat capacity 3*k*_{B}*N*. Molar heat capacity 3*R**(Dulong & Petit law)*.

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 3
*k*_{B}*N*for solids. This is right--at high temperatures. For diamond ``high'' means , sometimes it is 100-.

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

© 1997

Wed Oct 1 00:45:35 EDT 1997