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- General Problem
- Energy for a Classical System
- Same or Different Particles?
- Factorization
- When This Does Not Work?
- Kinetic part. Maxwell Distribution
- Internal Part
- Epitome

Suppose we have *N* *identical* particles 1, 2,.... We
want to describe their thermodynamics--i. e. calculate
*partition function.*

**Mixtures:***N*particles of kind 1,_{1}*N*particles of kind 2, ...._{2}

**Quantum Answer:**- I have a number of quantum states for each
particle. The state is the set of
*occupation numbers**n*,_{1}*n*, ..._{2} **Consequence:**- For quantum mechanics the grand canonical ensemble
and potential are
*more*natural than canonical ensemble and free energy! **Classical Answer:**- I need coordinate
*q*_{i}and momentum*p*_{i}for each of my particles*plus*information about the internal degrees of freedom for each particle.

- Kinetic energy. In the non-relativistic case it depends
*only*on the momenta: - Potential energy. In the classical case it depends
*only*on the coordinates: - Internal energy. Depends on the internal degrees of freedom (often they are quantum--even if everything else is classical!):

**Boltzmann principle:**- If the particles are identical,
exchanging them will
*not*change the state. This is a consequence of Quantum Mechanics.

(2) |

Since

we can write:(3) |

**Kinetic part****Internal part****Configuration integral**

- 1.
- Large quantum effects: we must include them honestly instead of
as a kludge!
*But*the kludge works for large temperatures (if the distance between quantum levels is ) - 2.
- Relativistic effects: at velocities about light speed or high gravitation fields we cannot separate kinetic and potential energy!
- 3.
- Polarisability: sometimes we cannot separate
*U*and . - 4.
- In non-Cartesian coordinates
*K*might depend both on and --we cannot separate it! In the systems with rigid bonds we must use non-Cartesian coordinates.

Since

the kinetic part is fully factorizable: with*Vector* momentum :

**Quantum interpretation:**- particles have wavelengths depending on the momentum
*p*. Thermal wavelength corresponds to . **Validity of Boltzmann Distribution:**- Classical formulae work if

Since partition function is factorized, the probability to have momentum

For velocity or, since This is calledAverage velocity Average kinetic energy per molecule: Another definition of temperature: in classical systems

The exact value depends on the *molecular structure*.

It is always factorizable:

with and depends only onThe kinetic part is calculated exactly--for all classical systems.

Calculation of the internal part is mostly a Quantum Mechanics problem.

Classical Statistical Thermodynamic is about calculation of the
*configuration integral*

© 1997

Wed Sep 17 22:58:45 EDT 1997