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Subsections

# Many-Particle Systems

## General Problem

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

Mixtures:
N1 particles of kind 1, N2 particles of kind 2, ....

General formula:
Gibbs law
 (1)
What is a state in this formula?

I have a number of quantum states for each particle. The state is the set of occupation numbers n1, n2, ...
Consequence:
For quantum mechanics the grand canonical ensemble and potential are more natural than canonical ensemble and free energy!
I need coordinate qi and momentum pi for each of my particles plus information about the internal degrees of freedom for each particle.
We will discuss only classical case.

## Energy for a Classical System

• 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!):

## Same or Different Particles?

Boltzmann principle:
If the particles are identical, exchanging them will not change the state. This is a consequence of Quantum Mechanics.
For distinguishable particles classical partition function is:

However, they are identical. States are overcounted--divide by the number of permutations:
 (2)
This is called Boltzmann Statistics

## Factorization

Since

we can write:
 (3)
Kinetic part

Internal part

Configuration integral

## When This Does Not Work?

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.

## Kinetic part. Maxwell Distribution

Since

the kinetic part is fully factorizable:

with

Vector momentum :

and

has the dimension of length. It is called thermal de Brogile wavelength.

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 called Maxwell distribution

Average velocity

Average kinetic energy per molecule:

Another definition of temperature: in classical systems measures the average kinetic energy of thermal motion.

In classical systems kinetic part can be separated from other parts and calculated exactly!

## Internal Part

The exact value depends on the molecular structure.

It is always factorizable:

with

and depends only on T!

## Epitome

The 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

Next: Ideal Gas Up: Systems with Many Particles. Previous: Systems with Many Particles.

© 1997 Boris Veytsman and Michael Kotelyanskii
Wed Sep 17 22:58:45 EDT 1997