Energy

Simple System Statistical mechanics applies to large sets of particles. Assume a system with a countable set of states.  Probability p n  Energy level E n The entropy and mean energy follow from the distribution.

Canonical System A canonical system assumes no material exchange. The entropy is expressed in terms of the energy of states.  Link to mean energy The linkage can be expressed as a differential.

Boundaries Statistical systems are bounded.  Macroscopic measures  Population N  Volume V Heat energy may be exchanged through the boundary.  Permissive – diathermal  Preventive – adiabatic diathermal Q adiabatic N, V

Energy Shift The differential for energy can be expanded in terms of the system properties.  Number, volume  Other total measures  Not temperature The summed change with volume is the pressure P. The summed change with number is the chemical potential .

Entropy Shift The differential change in entropy can now be expressed in terms of the other changes.  Use U for mean total energy  Replace T = 1/k  The differential implies that entropy is a function of the macroscopic variables.

Extensive Variables Extensive variables measure the size of a system.  Eg. energy, volume, number An extensive function scales linearly with the extensive variables. Entropy is ideally an extensive function.

Conjugate Forces The partial derivatives associated with the extensive variables are conjugate forces.  Pressure with volume  Chemical potential with particle number Other extensive variables beyond the simple system have conjugate forces.  Multiple chemical potentials  Magnetic field with magnetization

Equation of State When the entropy is known for a system, the partial derivatives give the fundamental relationships. Written in terms of observables, this becomes an equation of state.  For example ideal gases  Three equations of state Sackur-Tetrode

Third Law of Thermodynamics Scaling implies if an extensive variable is zero, then entropy is zero.  No information uncertainty  Fails for ground state energy Nernst’s theorem states that the temperature approaches zero, the entropy becomes constant.  Zero for quantum  Arbitrary but assigned zero for classical distribution ground state E 0  E 1 >> 1

Equilibrium Axioms The conditions for thermostatic equilibrium are: 1.Macroscopic equilibrium states of simple systems are determined by the extensive variables. 2.All the information is contained in the entropy. 3.The entropy is a continuous, differentiable function. 4.The entropy is monotonically increasing in energy and invertible for energy. 5.The entropy approaches zero when temperature approaches zero.