Teaching

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1. Equilibrium Thermodynamics I: Introduction [tsc1]

• Thermodynamic system and thermodynamic state
• State variables
• Equations of state
• Thermodynamic equation of state for a classical gas
• Encoding thermodynamic information
• Thermodynamic contacts
• Zeroth law of thermodynamics
• First law of thermodynamics
• Second law of thermodynamics
• Third law of themodynamics
• Thermodynamic processes
• Differentials

Exercises:

• Exact and inexact differentials I [tex5]
• Fast heat [tex143]
• Expansion and compression of nitrogen gas [tex144]
• Bathtub icebreaker [tex145]
• Exact and inexact differentials II [tex146]
• Exact and inexact differentials III [tex168]

Additional materials:

• Thermodynamics overview [tln2]
• Equations of state for ideal gas and real fluid [tsl12]
• Physical constants [tsl47]
• Relevant textbooks [tln90]  

2. Equilibrium Thermodynamics II: Engines [tsc2]

• Carnot engine
• Efficiency of Carnot engine
• Maximum efficiency of heat engine
• Absolute temperature
• Entropy
• Internal energy
• Reversible processes in fluid systems
• Gasoline engine (Otto cycle)
• Diesel engine
• Escher-Wyss gas turbine
• Stirling engine

Exercises:

• Entropy change caused by expanding ideal gas [tex1]
• Heating the air in a room [tex2]
• Carnot engine of a classical ideal gas [tex3]
• Carnot engine for an ideal paramagnet [tex4]
• Adiabates of the classical ideal gas [tex7]
• Idealized Otto cycle [tex8]
• Work extracted from finite heat reservoir in infinite environment [tex9]
• Work extracted from finite heat reservoir in finite environment [tex10]
• Mayer’s relation for heat capacities of the classical ideal gas [tex12]
• Room heater: electric radiator versus heat pump [tex13]
• Idealized Diesel cycle [tex16]
• Roads from 1 to 2: isothermal, isentropic, isochoric, isobaric [tex25]
• Positive and negative heat capacities [tex26]
• Ideal-gas engine with two-step cycle I [tex106]
• Ideal-gas engine with two-step cycle II [tex107]
• Joule cycle [tex108]
• Idealized Stirling cycle [tex131]
• Absolute temperature from measurements [tex134] 
• Circular heat engine I [tex147]
• Circular heat engine II [tex148]
• Square heat engine [tex149]
• Work performance and heat transfer [tex155]

3. Equilibrium Thermodynamics III: Free Energies [tsc3] 

• Fundamental equation of thermodynamics
• Analogy with mechanical equilibrium
• Free energy in a mechanical system
• Free energy in a thermodynamic system
• Thermodynamic potentials for a fluid system
• Differentials of thermodynamic potentials
• Facts about thermodynamic potentials
• Thermodynamic functions for fluid system
• Substitutions for magnetic system
• Maxwell’s relations
• Free energy stored and retrieved
• Response functions
• Thermal response functions
• Mechanical response functions
• Magnetic response functions
• Isothermal and adiabatic processes
• Conditions for thermal equilibrium
• Stability of thermal equilibrium

Exercises:

• Retrievable and irretrievable energy put in heat reservoir [tex6] 
• …

Additional materials:

• Legendre transform [tln77] 
• Alternative set of thermodynamic potentials [tln9]
• Useful relations between partial derivatives [tln6]
• Jacobi transformations [tln21]

4. Equilibrium Thermodynamics IV: Applications [tsc4]

• Classical ideal gas
• Van der Waals gas
• Cooling a gas by free expansion (Joule effect)
• Cooling a gas by throttling (Joule-Thomson effect)
• Entropy of mixing in classical ideal gas
• Ideal paramagnet
• Adiabatic demagnetization
• Ideal paramagnetic gas
• Photon gas
• Rubber band elasticity
• Inhomogeneous systems

Exercises:

• How not to modify the ideal gas equation of state [tex11]
• Entropy and internal energy of the classical ideal gas [tex14]
• Thermodynamic potentials of the classical ideal gas [tex15]
• Chemical potential of the classical ideal gas [tex17]
• Sound velocity in the classical ideal gas I [tex18]
• Thermodynamics of an ideal paramagnet I [tex19]
• Thermodynamics of an ideal paramagnet II [tex20]
• Thermodynamics of an ideal paramagnet III [tex21]
• Thermodynamics of a classical ideal paramagnetic gas I [tex22]
• Thermodynamics of black-body radiation [tex23]
• Carnot cycle of thermal radiation [tex24]
• Heat capacities of the van der Waals gas [tex27]
• Determining C_V of condensed matter [tex28]
• Assembling thermodyamic information [tex29]
• Joule coefficient of van der Waals gas [tex31]
• Joule-Thomson coefficient of van der Waals gas [tex32]
• Effects of first virial correction on ideal gas properties [tex33]
• Ideal gas heat capacity by design [tex35]
• Thermodynamics of a real paramagnet [tex36]
• Internal energy and entropy of van der Waals gas [tex38]
• Rubber band heat engine [tex39]
• Equation of state and adiabate of an elastic band [tex40]
• Reconstructing the equation of state of a fluid system [tex42]
• Reconstructing the equation of state of a gas [tex43]
• Sound velocity in the classical ideal gas II [tex99]
• Hydrostatic pressure [tex132]
• Thermodynamics of a classical ideal paramagnetic gas II [tex133]
• Polytropic process of classical ideal gas [tex138]
• Heavy piston I [tex141]
• Isothermal atmosphere [tex150]
• Adiabatic atmosphere [tex151]
• Homogeneous atmosphere [tex152]
• Heavy piston II [tex170] 
• Effect of mixing on chemical potential [tex173]

Additional materials:

• Joule-Thomson inversion curves [tsl1]
• Osmotic pressure [tln26]
• Entropy landscape of paramagnetic salt [tsl2]
• Mechanocaloric and thermomechanical effects [tln34] 

5. Thermodynamics of Phase Transitions I [tsc5]

• Typical solid-liquid-gas phase diagram
• Additional and alternative phases
• Classification of phase transitions
• Discontinuous transition
• Continuous transition
• Order parameter
• Phase coexistence: Gibbs’ phase rule
• Clausius-Clapeyron equation
• Effects of a uniform gravitational field

Exercises:

• Entropy of supercooled liquid [tex30]
• Coexistence line of continuous phase transition [tex37]
• Heat capacity of vapor in equilibrium with liquid phase [tex41]
• Melting or freezing? [tex51]
• Triple-point phase changes I [tex52]
• Abnormal phase behavior [tex54]
• Phase coexistence of ammonia [tex55]
• Discontinuous transition: change in internal energy [tex123]
• Latent heat and response functions [tex124]
• Dry ice [tex125]
• Cooling down? Heating up? [tex153]
• Triple-point phase changes II [tex156]
• Effects of heat input [tex159]

Additional materials:

• Phase diagram of H₂O [tsl4]
• Liquid crystal phases [tsl51]
• Ferrimagnetic phases [tsl49]
• Ordering of surfactant molecules [tsl50]

6. Thermodynamics of Phase Transitions II [tsc6]

• Law of corresponding states (using van der Waals equation of state)
• Maxwell construction (using Gibbs potential or Helmholtz potential)
• Nucleation of droplets or bubbles (coexistence line, spinodal line)

Exercises:

• Dieterici equation of state [tex34]
• Latent heat and heat capacies at superconducting transition [tex44]
• Mean-field ferromagnet I [tex45]
• Mean-field ferromagnet II [tex46]
• Structural transition of iron [tex53]

Additional materials:

• Helium liquids and superfluidity [tln33]
• Superconducting transition [tln35]
• Thermodynamics of a ferromagnet [tsl5]
• Mean-field ferromagnet [tln84]  

7. Kinetic Theory I [tsc7]

• Statistical concept of uncertainty
• Statistical concept of information
• Statistical uncertainty and entropy
• Kinetics of classical ideal gas
• Maxwell velocity distribution
• Boltzmann equation
• H-function
• H-theorem
• H-theorem and irreversibility

Exercises:

• Statistical uncertainty: verification of criteria [tex47]
• Information regarding a census of birds [tex48]
• Information of sequenced messages [tex61]
• Pressure and mean-square velocity in classical ideal gas [tex49]
• Maxwell velocity distribution (Maxwell’s derivation) [tex50]
• Maxwell distribution in D dimensions [tex56]
• Energy distribution for N ideal gas atoms [tex57]
• Maxwell velocity distribution (Boltzmann’s derivation) [tex58]
• Ideal-gas entropy and Boltzmann’s H-function [tex59]
• Maxwell distribution from variational principle [tex60]
• Doppler broadening of atomic spectral lines [tex63]

8. Kinetic Theory II [tsc9]

• Gas container with tiny hole
• Leakage from container with heat conducting walls
• Leakage from container with insulating walls
• Particle flow and energy flow between containers
• Kinematic pressure and interaction pressure
• Kinetic forces and mobility
• Collision rate and mean free path

Exercises

• Ideal gas atoms escaping from a container I [tex62]
• Isotope separation via diffusion [tex65]
• Ideal gas atoms escaping from a container II [tex176]
• Ideal gas atoms escaping from a container III [tex177]
• Toward thermal equilibrium via particle transfer [tex64]
• Interaction pressure produced by Gaussian interparticle potential [tex66]
• Average force of particle beam on heavy hard sphere [tex68]
• Mobility of a hard sphere in a dilute gas [tex69]
• Collision rate in a classical ideal gas [tex70]
• Mean free path of particle in classical ideal gas [tex71]
• Rate of chemical reaction in gas phase [tex67]
• Effect of escaping particles on temperature of 1D ideal gas [tex72]

9. Microcanonical Ensemble [tsc10]

• Classical Hamiltonian systems
• Points and trajectories in phase space
• Probability density in phase space
• Probability flow in phase space
• Classical Liouville operator
• Stationarity condition for phase-space probability density
• Density operator
• Quantum time evolution
• Stationarity condition for density operator
• Gibbs entropy
• Phase-space volume allocated per quantum state
• Microcanonical ensemble
• Aspects of significance
• Simple applictions
• Entropy of mixing revisited
• Negative temperatures

Exercises:

• Classical ideal gas [tex73]
• Array of classical harmonic oscillators [tex74]
• Array of quantum harmonic oscillators I [tex75]
• Array of quantum harmonic oscillators II [tex126]
• Quantum paramagnet [tex127]

10. Canonical Ensemble I [tsc11]

• Extremum principle
• Canonical partition function
• Systems of noninteracting particles
• From phase-space density to Maxwell velocity distribution
• Ensemble averages
• Energy fluctuations and heat capacity
• Classical ideal gas (relativistic)
• Inhomogeneous systems
• Partition function and density of states

Exercises:

• Nonrelativistic ideal gas [tex76]
• Ultrarelativistic ideal gas [tex77]
• Ultrarelativistic ideal gas in two dimensions [tex154]
• Relativistic ideal gas I: canonical partition function [tex91]
• Relativistic ideal gas II: entropy and internal energy [tex92]
• Relativistic ideal gas III: heat capacity [tex93]
• Classical ideal gas in uniform gravitational field [tex79]
• Gas pressure and density inside centrifuge [tex135]
• Irreversible decompression [tex136]
• Irreversible heat exchange [tex137]
• Reversible decompression [tex139]
• Reversible heat exchange [tex140]
• Heavy piston I [tex141]
• Ideal gas partition function and density of states [tex81]
• Relative momentum of two ideal-gas particles [tex80]

11. Canonical Ensemble II [tsc12]

• Vibrational heat capacities of solids
• Theory of Dulong and Petit
• Theory of Einstein
• Atoms interacting via harmonic forces
• Theory of Debye
• Paramagnetism of localized magnetic dipoles
• Langevin paramagnetism
• Two-level system
• Brillouin paramagnetism
• Fluctuations in a magnetic system
• Gases with internal degrees of freedom
• Translational motion (classical)
• Rotational motion (classical)
• Rotational motion (quantum)
• Vibrational motion (quantum)
• Fine structure
• Orthohydrogen and parahydrogen

Exercises:

• Array of classical harmonic oscillators [tex78]
• Array of quantum harmonic oscillators [tex82]
• Vibrational heat capacity of a solid [tex83]
• Anharmonic oscillator and thermodynamic perturbation [tex104]
• Classical paramagnet [tex84]
• Quantum paramagnet (two-level system) [tex85]
• Quantum paramagnet (three-level system) [tex157]
• Quantum paramagnet (Brillouin function) [tex86]
• Ising trimer [tex142]
• Fluctuation in a magnetic system [tex109]
• Classical rotational entropies of HCl and N₂ gases [tex88]
• Classical rotational free energies of NH₃ gas [tex87]
• Quantum rotational heat capacity of a gas at low temperature [tex89]
• Quantum rotational heat capacity of a gas at high temperature [tex90]

Additional materials:

• Vibrational heat capacities of solids [tsl29]
• Paramagnetic salts [tsl30]
• Thermodynamic perturbation expansion [tln80]

12. Grandcanonical Ensemble [tsc13]

• Extremum principle
• Grandcanonical partition function
• Density fluctuations and compressibility
• Gentle introduction to quantum statistics
• Permutation symmetry
• Occupation number representation
• Canonical partition function (for quantum gases)
• Grandcanonical partition function (for quantum gases)
• Grand potential
• Average number of particles and state occupancies
• Entropy and state occupancies
• Internal energy and state occupancies
• Fluctuations of state occupanices
• Density of states
• Occupancy of 1-particle states

Exercises:

• Classical ideal gas [tex94]
• Ultrarelativistic ideal gas [tex169]
• Density fluctuations [tex95]
• Density fluctuations and compressibility [tex96]
• Energy fluctuations and thermal response functions [tex103]
• Occupation number fluctuations [tex110]
• Density of 1-particle states [tex111]
• Maxwell-Boltzmann gas in D dimensions [tex112]
• Some fantasy gas [tex171]
• Ideal lattice gas [tex172]
• Entropy and internal energy from state occupancies [tex178] 

13. Ideal Quantum Gases I: Bosons [tsc14]

• Equation of state
• Reference values
• Isochores
• Coexistence of gas and condensate
• Isotherms
• Isobars
• Phase diagrams
• Entropy
• Internal energy
• Heat capacity

Exercises:

• Fundamental relations [tex113]
• Isochores [tex114]
• Isotherms and isobars [tex115]
• Entropy and internal energy [tex179]
• Heat capacity at high temperature [tex97]
• Heat capacity at low temperature [tex116]
• Isothermal compressibility [tex128]
• Isobaric expansivity [tex129]
• Speed of sound [tex130]
• Ultrarelativistic Bose-Einstein gas [tex98]
• Statistical mechanics of blackbody radiation [tex105]

Additional materials:

• Bose-Einstein functions [tsl36]
• Blackbody radiation [tln68]

14. Ideal Quantum Gases II: Fermions [tsc15]

• Equation of state
• Chemical potential
• Level occupancies
• Isochores
• Phase transition
• Isotherms
• Entropy
• Internal energy
• Heat capacity

Exercises:

• Chemical potential I [tex117]
• Chemical potential II [tex118]
• Statistical interaction pressure [tex119]
• Isotherm and adiabate [tex120]
• Ground-state energy [tex102]
• Heat capacity at high temperature [tex100]
• Heat capacity at low temperature [tex101]
• Stable white dwarf star [tex121]
• Unstable white dwarf star [tex122]

Additional materials:

• Fermi-Dirac functions [tsl42]

15. Nearly Free Electrons [tsc16]

• Thermionic emission (Richardson effect)
• Schottky effect
• Photoelectric emission (Hallwachs effect)
• Pauli paramagnetism (PPM)
• PPM analyzed in canonical ensemble
• PPM thermodynamic potentials and functions
• PPM response functions
• PPM magnetization curves (numerical analysis)
• PPM magnetization curves at T = 0 (exact results)
• PPM magnetization curves in D = 2 (exact analysis)
• PPM isothermal susceptibility at H = 0
• PPM correction to Langevin-Brillouin result at high T

Exercises:

• Paramagnetic FD gas I: pressure and entropy [tex161]
• Paramagnetic FD gas II: internal energy [tex162]
• Paramagnetic FD gas III: heat capacity C_VM [tex163]
• Paramagnetic FD gas IV: heat capacity C_VH [tex164]
• Paramagnetic FD gas V: isothermal susceptibility [tex165]
• Paramagnetic FD gas VI: isothermal compressibilities [tex166]
• Paramagnetic FD gas VII: isobaric expansivity [tex167]
• Paramagnetic FD gas VIII: magnetization curves at T > 0 [tex180]
• Paramagnetic FD gas IX: magnetization curves at T = 0 [tex181]
• Paramagnetic FD gas X: exact magnetization curve for D = 2 [tex182]
• Paramagnetic FD gas XI: isothermal susceptibility at H = 0 [tex183]
• Paramagnetic FD gas XII: Langevin-Brillouin limit at high T [tex184]

16. Thermodynamics of Phase Transitions III [tsc8]

• Ginzburg-Landau theory for secon-order phase transition
• Ginzburg-Landau theory for first-order phase transition
• Ornstein-Zernike theory for correlations
• Critical-point exponents
• Critical singularities of magnetic system
• Critical singularities of fluid system
• Inequalities of critical-point exponents
• Test of scaling laws
• Marginal dimensionality

Exercises:

• Order parameter of first-order Ginzburg-Landau transition [tex174]
• Critical singularities of van der Waals gas [tex175]

 17. Interacting Degrees of Freedom [tsc17]

• Nearly ideal classical gas

18. Ising model I [tsc18]

• Ising magnet
• Ising lattice gas
• Mapping between magnet and lattice gas
• Transfer matrix solution of 1D Ising magnet
• Expectation values via transfer matrix
• Correlation functions via transfer matrix
• Ising lattice gas in D=1
• Ideal lattice gas limit
• Ising lattice gas equation of state in D=1
• Ising lattice gas entropy in D=1
• Ising lattice gas internal energy in D=1

Exercises:

• Ideal lattice gas [tex172]
• Ising chain: transfer matrix solution I [tex185]
• Ising chain: transfer matrix solution II [tex189]
• Ising lattice gas in D=1: equation of state [tex194]
• Ising lattice gas in D=1: entropy I [tex195]
• Ising lattice gas in D=1: entropy II [tex201]
• Ising model in Bethe approximation [tex203] 
• …

Additional materials:

• Ising model and descendents [tln93]
• Exchange interaction [tln94]
• Metallic alloys [tln95]
• T=0 phase diagrams of Ising chains [tln96]
• Approximating the Ising model [tln97] 
• Equivalent-neighbor Ising model [tln98]
• …

19. Coordinate Bethe Ansatz [tsc19]

• NLS model
• Free bosons
• Impenetrable bosons and free fermions
• Two bosons with finite contact repulsion
• Three bosons subject to finite two-body contact repulsion
• General Bethe ansatz equations of the NLS model
• Trigonometric Bethe ansatz equations
• Iterative solutions
• Particles and holes
• Densities of particle momenta and hole momenta
• Ground state of the NLS model from Lieb-Liniger equation
• …

Exercises:

• Wave function of impenetrable bosons for N=2 and N=3 [tex190]
• Two-boson Bethe wave functions [tex191]
• Three-boson Bethe ansatz equations [tex192]
• Trigonometric Bethe ansatz equations for the NLS model [tex193]
• …

20. Statistical Interactions I: Combinatorics [tsc20]

• Fermions
• BosonsC
• Size L and XL particles
• Semions
• Particles with internal degrees of freedom
• Distinguishable species of particles in shared orbitals
• Hosts and caps
• Hosts, hybrids, and caps
• Hosts, hybrids, and tags
• Configurational entropy
• Examples with one species
• Example with two species
• Semions versus hosts and caps

Exercises:

• Configurational entropy of statistically interacting particles [tex186]
• Hosts and tags at level 1 [tex187]
• Hosts and tags at level 2 [tex188]

Additional materials

• Particles at two levels [tln91]
• Merging particle species [tln92]
• …

 21. Statistical Interactions II: Partition Functions [tsc21]

• Conversion of dynamical interactions into statistical interactions
• Partition function, average occupancies, and entropy
• …

22. Ergodic Theory [tsc22]

• …

Last updated 08/07/22