Equilibrium Statistical Physics
Free and easy access
to complete set of presentable lecture notes and exercises is
available on URI
Digital Commons (downloadable pdf files covering entire
sections). The most recent updates are available only
here.
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:
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 CV 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]
- Triple-point phase changes III [tex204]
Additional materials:
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:
Additional materials:
14. Ideal Quantum Gases II: Fermions [tsc15]
- Equation of state
- Chemical potential
- Level occupancies
- Isochores
- Phase transition
- Isotherms
- Entropy
- Internal energy
- Heat capacity
Exercises:
Additional materials:
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 CVM
[tex163]
- Paramagnetic FD gas IV: heat capacity CVH
[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
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 11/04/23