THE UNIVERSITY OF BRITISH COLUMBIA

PHYSICS 455

Lecture # 4 :

Mon. 13 Jan. 1997

Temperature & Thermodynamics

READING: You should be done with Ch. 1 and into Ch. 2 by now.

I. Recapitulate Equilibrium:

1. Switchover from Discreet to Continuum Description: The model quantum mechanical systems used to illustrate the essential content of statistical mechanics actually have discrete energy levels with different quantum degeneracy factors. When we go to define the temperature, however, we speak in terms of a continuous derivative of the multiplicity g as a function of energy U. This description is actually invalid for small systems where, to be rigourous, we would have to define the temperature in terms of discrete jumps between discrete energy levels U_i with discrete values of g_i. You may have to deal with such systems someday if you work in the forefront of nanotechnology, but for the time being we shall relax into the traditional limit of large systems where we can apply the continuous approximations safely.

   The way we do this is as follows: we assume U is only known (or specified) within some uncertainty dU and that dU (while very small) is much larger than the separation between any discrete energy levels in its range. We must then think of g(U) = D(U) dU in terms of the density of states D(U) per unit energy.

2. Definition of temperature of a system.

3. is a meaningful concept only for predicting the most probable configuration (how the energy gets shared) of two systems in thermal contact. However, this can be very powerful!

4. Example: two systems of spin-½ particles

5. Possibility of negative temperature (hotter than infinite temperature!) in any system that can only hold a limited amount of energy - usually g = 1 (or some small number, in the case of degenerate quantum states) when U = 0 and again when U reaches its upper limit, so there must be a peak in (U) somewhere in between. At that value of U the inverse temperature goes through zero.

II. Laws of Thermodynamics:

• Zeroth Law:

  If two systems are in thermal equilibrium with (have the same temperature as) a third system, they are in thermal equilibrium with (have the same temperature as) each other.

  (Well, duh!)

• First Law:

  Energy is conserved; heat is a form of energy.

  (``You can't get something for nothing.'')

  More details on this later.

• Second Law:

  If a closed system is in a configuration that is not the equilibrium (most probable) configuration, the most probable consequence is that the entropy of the system will increase monotonically with time.

  (``You can't break even.'')

  This is actually somewhat circular, as it is our only way of defining the ``arrow of time.'' I did indulge in a short philosophical digression on this subject. Again, ``more later.''

• Third Law: (Nernst's Theorem)

  The entropy of a system approaches a constant as its temperature approaches zero.

  (We are always talking about absolute temperature here, of course. See the textbook's Appendix B on thermometry.)

  This Law is a little arcane. We may get back to it later or we may not. In any case the notion of temperature smoothly approaching zero is invalid for small systems (see above).

III. Graphical Device for Illustrating Definition of & 2^nd Law:

The condition of maximum net entropy (most probable configuration) is satisfied at the value of U₁ (and U₂ = U - U₁) where the slopes are the same.

If the combined system starts in some other configuration, . . . (see the Second Law).

IV. Model System: N Spin-½ Particles in a Magnetic Field

1. Spin Excess: 2s N - N

2. Magnetization: M = 2s m , where m is the magnetic moment of one particle.

3. Energy: U = -M B , where B is the magnetic field on the particles.

4. High Temperature Limit: Drawing on the approximation g(s,N) g(0,N) exp(-2s²/N), valid for s N, we can express the entropy in terms of the energy as

   - U² / 2 m² B N .

5. How Sharp is g(s,N)? (see ``Digression'' in text, pp. 37-39)

   When two spin systems are in thermal contact, the sharpness of the combined g = g₁(s₁,N₁) ^. g₂(s₂,N₂) can be defined in terms of the fractional deviation 2s₁ / N₁ at which g drops from its peak value g_peak (defining the most probable configuration) to [let us say] 1/e² 5% of g_peak.

   This deviation is shown to be given by 2/N₁ if N₁ N₂ (the smaller system determines the sharpness).

   For small systems this is not all that sharp (20% for a system of 100 spins) but when N₁ = 10²⁰ (not atypical for a small sample of magnetic material) the sharpness is on the order of 10^-10, which (to paraphrase Boltzmann) is ``the practical equivalent of fixed.''

   We shall shortly move on to a general assumption that the thermal equilibrium value of a statistical quantity is indistinguishable (for all intents and purposes) from the actual, instantaneous value. This is not mathematically true, of course, so it is important to bear in mind what conditions must be satisfied in order to make it ``good enough.''