STATISTICAL THERMODYNAMICS

1. STATISTICAL THERMODYNAMICS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email wlawton@math.nus.edu.sg Tel (65) 874-2749 Fax (65) 779-5452

2. SECOND LAW REVISITED Lord Kelvin : A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible Rudolph Clausius : A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible (this principle implies the previous one) Martian Skeptic : What temperature ?

3. SECOND LAW REVISITED Definition : Body A has higher temperature than body B ( ) if, when we bring them into thermal contact, heat flows from A to B. Body A has the same temperature as B ( ) if, when we bring them into thermal contact, no heat flows from A to B and no heat flows from B to A. ( [A] := {U | U A} ) Enrico Fermi : (Clausius Reformulated) If heat flows by conduction from a body A to another body B, then a transformation whose only final result is to transfer heat from B to A is impossible.

4. SECOND LAW REVISITED

5. SECOND LAW REVISITED Definition : Absolute Thermodynamic Temperature Choose a body D If then then then In a reversible process

6. SECOND LAW REVISITED System : Cylinder that has a movable piston and contains a fixed amount of homogeneous fluid States (Macroscopic) : Region in positive quadrant of the (V = volume, T = temperature) plane. Functions (on region) : V, T, p = pressure Paths (in region) : Oriented curves Differential Forms : can be integrated over paths WORK HEAT

7. SECOND LAW REVISITED Definition : Entropy Function S (by thermal equilibrium and by thermal isolation)

8. FIRST LAW REVISITED such that Therefore

9. FIRST & SECOND LAWS COMBINED Therefore, the basic (but powerful) calculus identity Yields (after some tedious but straightforward algebra)

10. IDEAL GAS LAW (Chemists) Boyl, Gay-Lussac, Avogardo amount of gas in moles ideal gas constant ideal gas temperature in Kelvin (water freezes at 373.16 degrees)

11. JOULE’S GAS EXPANSION EXPERIMENT We substitute the expression for p (given by the ideal gas law) to obtain and observe that the outcome of Joule’s gas expansion experiment

12. IDEAL GAS LAW (Physicists) number of molecules of gas Boltzmann’s constant

13. GAS THERMODYNAMICS Experimental Result : (dilute gases) Therefore

14. GAS KINETICS Monatomic dilute gas, m = molecular mass average kinetic energy / molecule

15. GAS KINETICS Photon gases Maxwell Equipartition of Energy

16. EQUIPARTITION Number of ways of partitioning N objects into m bins with relative frequencies (probabilities) is Stirling’s formula yields where denotes Shannon’s information-theoretic entropy

17. EQUIPARTITION If the bins correspond to energies, then and therefore (nearly) C, is maximized, subject to an energy constraint by the Gibbs distribution Therefore and free energy Maxwell dist.

18. THIRD LAW Nernst : The entropy of every system at absolute zero can always be taken equal to zero inherently quantum mechanical discrete microstates, a quart bottle of air has about & Maxwell’s demon : may he rest in peace Time’s arrow : probably forward ???

19. REFERENCES V. Ambegaokar, Reasoning about Luck H. Baeyer, Warmth Disperses and Time Passes F. Faurote, The How and Why of the Automobile E. Fermi, Thermodynamics R. Feynman, Lectures on Physics, Volume 1

20. REFERENCES H. S. Green and T. Triffet, Sources of Consciousness, The Biophysical and Computational Basis of Thought K. Huang, Statistical Mechanics N. Hurt and R. Hermann, Quantum Statistical Mechanics and Lie Group Harmonic Analysis C. Shannon and W. Weaver, The Mathematical Theory of Communication