A brief introduction to how we can determine macroscopic properties from microscopic models using Statistical Thermodynamics

The central quantity of statistical thermodynamics is the Partition Function Q:

(eqn 1)

where E_j is the energy of microstate j. In the so-called "canonical ensemble" E_j is the internal energy (potential plus kinetic).

Macroscopic state variables are expressed as averages of the respective microscopic quantities, so for a generic property x, its macroscopic average (where the <>'s indicate the "ensemble average") is:

(eqn 2)

So we need the probabilities p_j. These are defined as:

(eqn 3)

Note that: (eqn 4)

Consider a few examples:

  1. The macroscopic internal energy:


    (eqn 5)

    observe that (eqn 6) (denominator)

    and (eqn 7)

    so I can write (eqn 8)

  2. Pressure, P:

    (eqn 9)

    where the micrscopic pressure is

    (eqn 10)

    (eqn 11)

    We note that:

    (eqn 12)

    (eqn 13)

  3. Entropy, S:

    Here a good place to start is the Gibbs Entropy Formula:

    (eqn 14)

    which upon substitution for p_j becomes

    (eqn 15)

    With some manipulation:

    (eqn 16)

    Finally, we get: (eqn 17)

For more details, an excellent source is D.A. McQuarrie, "Statistical Mechanics", Harper and Row: New York (1976), Chapter 2.
MIT Home Page | UP one level | Ask the Professor
Last modified 3/18/99 - GCR