STATISTICAL THERMODYNAMICS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65) Fax (65)

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 ?

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.

SECOND LAW REVISITED

Definition : Absolute Thermodynamic Temperature Choose a body D Ifthen In a reversible process

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 WORKHEAT

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

FIRST LAW REVISITED such that Therefore

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

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

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

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

GAS THERMODYNAMICS Experimental Result : (dilute gases) Therefore

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

GAS KINETICS Photon gases Maxwell Equipartition of Energy

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

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

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 ???

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

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