Equilibrium Statistical Physics

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1. Equilibrium Thermodynamics I: Introduction

Table of contents. [ttc]
Thermodynamics overview. [tln2]
Preliminary list of state variables. [tln1]
Physical constants. [tsl47]
Equations of state. [tln78] 
Equation of state for ideal gas and real fluid. [tsl12]
Classification of thermodynamic systems. Laws of thermodynamics. [tln10]
Thermodynamic processes (irreversible, quasi-static, adiabatic). [tln79] 
Fast heat. [tex143] 
Expansion and compression of nitrogen gas. [tex144] 
Bathtub icebreaker. [tex145] 
Exact differentials. Applications to internal energy and entropy. [tln14]
Exact and inexact differentials I. [tex5]
Exact and inexact differentials II. [tex146]
Exact and inexact differentials III. [tex168]  *

2. Equilibrium Thermodynamics II: Engines

Carnot engine. [tln11]
Maximum efficiency. [tln12]
Absolute temperature. [tln13]
Entropy change caused by expanding ideal gas. [tex1]
Carnot cycle of the classical ideal gas. [tex3]
Carnot cycle of an ideal paramagnet. [tex4]
Reversible processes in fluid systems. [tln15]
Adiabates of the classical ideal gas. [tex7]
Roads from 1 to 2: isothermal, isentropic, isochoric, isobaric. [tex25]
Room heater: electric radiator versus heat pump. [tex13]
Mayer's relation for the heat capacities of the classical ideal gas. [tex12]
Positive and negative heat capacities. [tex26]
Work extracted from finite heat reservoir in infinite environment. [tex9] 
Work extracted from finite heat reservoir in finite environment. [tex10] * 
Heating the air in a room. [tex2]
Gasoline engine. [tln65]
Idealized gasoline engine(Otto cycle). [tex8]
Diesel engine. [tln66]
Idealized Diesel engine. [tex16] 
Escher-Wyss gas turbine. [tln75]  *
Joule cycle. [tex108]
Stirling engine. [tln76] 
Idealized Stirling cycle. [tex131] 
Ideal-gas engine with two-step cycle I. [tex106]
Ideal-gas engine with two-step cycle II. [tex107]
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 

Fundamental equation of thermodynamics. [tln16]
Free energy. [tln3]
Legendre transform. [tln77] 
Thermodynamic potentials. [tln4]
Alternative set of thermodynamic potentials. [tln9]
Thermodynamic functions. [tln5]
Maxwell's relations. [tln17]
Free energy stored and retrieved. [tln18]
Retrievable and irretrievable energy put in heat reservoir. [tex6]
Useful relations between partial derivatives. [tln6]
Response functions (thermal, mechanical, magnetic). [tln7] (2)
Isothermal and adiabatic processes in fluid systems and magnetic systems. [tln8]
Conditions for thermal equilibrium. [tln19] 
Stability of thermal equilibrium. [tln20]
Jacobi transformation. [tln21]
Entropy of mixing. [tln25]
Osmotic pressure. [tln26]  

4. Equilibrium Thermodynamics IV: Applications

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]
Ideal gas heat capacity by design. [tex35]
Sound velocity in the classical ideal gas I. [tex18]
Sound velocity in the classical ideal gas II. [tex99]
Absolute temperature from measurements. [tex134]  
Polytropic process of classical ideal gas. [tex138]
Heavy piston. [tex141] 
Heavy piston II. [tex170] *
Isothermal atmosphere. [tex150] 
Adiabatic atmosphere. [tex151] 
Homogeneous atmosphere. [tex152] 
Van der Waals equation of state. [tln22]
Cooling gases: Joule effect (free expansion) and Joule-Thomson effect (throttling). [tln23]
Joule-Thomson inversion curves. [tsl1]
Heat capacities of the van der Waals gas. [tex27]
Internal energy and entropy of the van der Waals gas. [tex38]
Joule coefficient of the van der Waals gas. [tex31]
Joule-Thomson coefficient of the van der Waals gas. [tex32]
Assembling thermodynamic information. [tex29]
How not to modify the ideal gas equation of state. [tex11]
Reconstructing the equation of state of a fluid system. [tex42]
Reconstructing the equation of state of a gas. [tex43]
Effects of first virial correction on ideal gas properties. [tex33]
Entropy due to electronic spins in iron ammonium alum. [tsl2]
Adiabatic demagnetization. [tln24]
Thermodynamics of an ideal paramagnet I. [tex19]
Thermodynamics of an ideal paramagnet II. [tex20]
Thermodynamics of an ideal paramagnet III. [tex21]
Thermodynamics of a real paramagnet. [tex36]
Thermodynamics of a classical ideal paramagnetic gas I. [tex22]
Thermodynamics of a classical ideal paramagnetic gas II. [tex133]
Hydrostatic pressure. [tex132] 
Rubber band heat engine. [tex39]
Equation of state and adiabate of an elastic band. [tex40] 
Determining CV of condensed matter. [tex28]
Thermodynamics of blackbody radiation. [tex23]
Carnot cycle of thermal radiation. [tex24]

5. Thermodynamics of Phase Transitions I

Phase diagram of a "normal" substance. [tsl3]
Phase diagram of H2O. [tsl4]
Ferrimagnetic phases. [tsl49]
Liquid crystal phases. [tsl51]
Ordering of surfactant molecules. [tsl50]
Phase coexistence: Gibbs phase rule. [tln27]
Classification of phase transitions. [tln28]
Gibbs free energy and derivatives at discontinuous transition. [tsl7]
Gibbs free energy and derivatives at continuous transition. [tsl8]
Clausius-Clapeyron equation. [tln29]
Entropy of a supercooled liquid. [tex30]
Coexistence line of continuous phase transition. [tex37]
Latent Heat and response functions. [tex124]
Heat capacity of vapor in equilibrium with liquid phase. [tex41]
Discontinuous transition: change in internal energy. [tex123]
Dry ice. [tex125]
Abnormal phase behavior. [tex54]
Melting or freezing. [tex51]
Triple-point phase changes. [tex52]
Triple-point phase changes II. [tex156] *
Cooling down? Heating up? [tex153]
Phase coexistence of ammonia. [tex55]
Effects of heat input. [tex159] *

6. Thermodynamics of Phase Transitions II

Van der Waals equation of state with coexistence curve. [tsl10]
Law of corresponding states. [tln30]
Maxwell construction. [tln31]
Gibbs and Helmholtz free energies of the van der Waals fluid at T<Tc. [tsl11]
Condensation and evaporation. [tln32]
Dieterici equation of state. [tex34]
Helium liquids. [tln33]
Phase diagram of  4He. [tsl13]
Phase diagram of  3He. [tsl14]
Exotic properties of helium II. [tln34]
Superconducting transition. [tln35]
 Thermodynamics of a ferromagnet. [tsl5]
Structural transitions of iron. [tex53]
Latent heat and heat capacities at superconducting transition. [tex44]
Thermodynamics of the mean-field ferromagnet I. [tex45]
Thermodynamics of the mean-field ferromagnet II. [tex46]

7. Kinetic Theory I

Statistical uncertainty and information. [tln37]
Statistical concept of uncertainty. [tex47]
Statistical uncertainty and information. [tex48]
Information of sequenced messages. [tex61]
Kinetics of classical ideal gas. [tsl28] *
Pressure and mean square velocity in classical ideal gas. [tex49]
Maxwell velocity distribution. [tln38]
Maxwell velocity distribution (Maxwell's derivation). [tex50]
Maxwell distribution in D-dimensional space. [tex56]
Boltzmannn equation. [tln39]
Boltzmann's H-theorem. [tln40]
Energy distribution for N ideal gas atoms. [tex57] *
Maxwell velocity distribution (Boltzmann's derivation). [tex58]
Ideal-gas entropy and Boltzmann's H-function. [tex59]
H-theorem and irreversibility. [tln41]
Boltzmann's H-function simulated. [tsl27]
Maxwell distribution derived from minimizing the H-function. [tex60]
Doppler broadening of atomic spectral lines. [tex63]

8. Kinetic Theory II

Ideal gas atoms escaping from a container. [tex62]
Toward thermal equilibrium via particle transfer. [tex64]
Isotope separation via diffusion. [tex65]
Kinematic pressure and interaction pressure. [tln42]
Interaction pressure produced by Gaussian interparticle potential. [tex66] *
Kinetic forces and mobility. [tln43]
Average force of particle beam on heavy hard sphere. [tex68]
Mobility of a hard sphere in a dilute gas. [tex69]
Collision rate and mean free path. [tln44]
Collision rate in classical ideal gas. [tex70]
Mean free path of particle in classical ideal gas. [tex71]
Rate of chemical reaction A + A -> A_2 in gas phase. [tex67]
Effect of escaping particles on temperature of 1D ideal gas. [tex72]

9. Microcanonical Ensemble

Classical Hamiltonian system. [tln45]
Classical Liouville operator. [tln46]
Quantum Liouville operator. [tln47]
Gibbs entropy. [tln48]
Microcanonical ensemble. [tln49]
Classical ideal gas (microcanonical ensemble). [tex73]
Array of classical harmonic oscillators (microcanonical ensemble). [tex74] *
Quantum harmonic oscillators (microcanocal ensemble I). [tex75]
Quantum harmonic oscillators (microcanocal ensemble II). [tex126]
Quantum paramagnet (microcanonical ensemble). [tex127]
Entropy of mixing revisited. [tln50]

10. Canonical Ensemble I

Canonical ensemble. [tln51]
Classical ideal gas (canonical ensemble). [tex76]
Ultrarelativistic classical ideal gas (canonical idela gas). [tex77]
Ultrarelativistic classical ideal gas in two dimensions. [tex154] 
Array of classical harmonic oscillators (canonical ensemble). [tex78]
Irreversible decompression. [tex136]
Irreversible heat exchange. [tex137]
Reversible decompression. [tex139]
Reversible heat exchange. [tex140]
Ensemble averages. [tln52]
Classical virial theorem. [tln83] 
Systems of noninteracting particles. [tln54]
Further ensemble averages. [tln55]
Classical ideal gas in a uniform gravitational field. [tex79]
Gas pressure and density inside centrifuge. [tex135]
Relative momentum of two ideal gas particles. [tex80]
Partition function and density of states. [tln56]
Ideal gas partition function and density of states. [tex81]
Vibrational heat capacities of solids. [tln57]
Array of quantum harmonic oscillators (canonical ensemble). [tex82]
Vibrational heat capacities of solids (Debye theory). [tsl29]
Thermodynamic perturbation expansion. [tln80] 
Vibrational heat capacity of a solid. [tex83]
Anharmonic oscillator and thermodynamic perturbation. [tex104]

11. Canonical Ensemble II

Paramagnetism. [tln58]
Paramagnetic salts. [tsl30]
Fluctuations in a magnetic system. [tln53]
Fluctuations in a magnetic system. [tex109] *
Classical paramagnet (canonical ensemble). [tex84] *
Quantum paramagnet (two-level system). [tex85]
Quantum paramagnet (three-level system). [tex157] *
Quantum paramagnet (Brillouin function). [tex86]
Ising trimer. [tex142] * 
Negative temperatures. [tsl31]
Gases with internal degrees of freedom. [tln59]
Classical rotational free energy of NH3 gas. [tex87]
Classical rotational entropy of HCl and N2 gas. [tex88] *
Quantum rotational heat capacity of a gas at low temperature. [tex89]
Quantum rotational heat capacity of a gas at high temperature. [tex90]
Rotational and vibrational heat capacities. [tsl32]
Orthohydrogen and parahydrogen. [tln81]
Relativistic classical ideal gas (canonical partition function). [tex91]
Relativistic classical ideal gas (entropy and internal energy). [tex92]
Relativistic classical ideal gas (heat capacity). [tex93]
Relativistic classical ideal gas (heat capacity). [tsl34]

12. Grandcanonical Ensemble

Grandcanonical ensemble. [tln60]
Classical ideal gas (grandcanonical ensemble). [tex94]
Ultrarelativistic classical ideal gas (grandcanonical ensemble). [tex169] *
Density fluctuations and compressibility [tln61]
Density fluctuations in the grand canonical ensemble. [tex95] *
Density fluctuations and compressibility in the classical ideal gas. [tex96]
Energy fluctuations and thermal response functions. [tex103]
 Microscopic states of quantum ideal gases. [tln62]
Partition function of quantum ideal gases. [tln63]
Ideal quantum gases: grand potential and thermal averages. [tln64]
Ideal quantum gases: average level occupancies. [tsl35]
Occupation number fluctuations. [tex110]
Density of energy levels for ideal quantum gas. [tex111]
Maxwell-Boltzmann gas in D dimensions. [tex112]
Some fantasy gas. [tex171] *
Ideal lattice gas. [tex172] *

13. Ideal Quantum Gases I: Bosons

Bose-Einstein functions. [tsl36] *
Ideal Bose-Einstein gas: equation of state and internal energy. [tln67] *
BE gas in D dimensions I: fundamental relations. [tex113]
Reference values for T, V/N, and p. [tln71] *
Bose-Einstein condensation. [tsl38] *
Ideal Bose-Einstein gas: isochores. [tsl39]
BE gas in D dimensions II: isochore. [tex114]
BE gas in D dimensions III: isotherm and isobar. [tex115]
Bose-Einstein gas: isotherms. [tsl40]
Bose-Einstein gas: isobars. [tsl48]
Bose-Einstein gas: phase diagram. [tln72]
Bose-Einstein heat capacity. [tsl41]
BE gas in D dimensions IV: heat capacity at high temperature. [tex97]
BE gas in D dimensions V: heat capacity at low temperature. [tex116] *
BE gas in D dimensions VI: isothermal compressibility. [tex128]
BE gas in D dimensions VII: isobaric expansivity. [tex129]
BE gas in D dimensions VIII: speed of sound. [tex130]
Ultrarelativistic Bose-Einstein gas. [tex98]
Blackbody radiation. [tln68]
Statistical mechanics of blackbody radiation. [tex105]

14. Ideal Quantum Gases II: Fermions

Fermi-Dirac functions. [tsl42] *
Ideal Fermi-Dirac gas: equation of state and internal energy. [tln69]
Ideal Fermi-Dirac gas: chemical potential. [tsl43] *
FD gas in D dimensions: chemical potential I. [tex117]
FD gas in D dimensions: chemical potential II. [tex118] *
Ideal Fermi-dirac gas: average level occupancy. [tsl44] *
Ideal Fermi-Dirac gas: isochores I. [tsl46]
FD gas in D dimensions: statistical interaction pressure. [tex119] *
Ideal Fermi-Dirac gas: isotherms. [tln70]
FD gas in D dimensions: isotherm and adiabate. [tex120]
FD gas in D dimensions: ground-state energy. [tex102]
Ideal Fermi-Dirac gas: heat capacity. [tsl45] *
FD gas in D dimensions: heat capacity at high temperature. [tex100]
FD gas in D dimensions: heat capacity at low temperature. [tex101] *
Ideal Fermi-Dirac gas: isochores II. [tln73] *
Ideal Fermi-Dirac gas: phase diagram in infinite dimensions. [tln74]
Stable white dwarf. [tex121]
Unstable white dwarf. [tex122]

15. Ideal Quantum Gases III: Pauli Paramagnetism

Pauli paramagnetism I: electron gas from spin-polarized FD gases. [tln85] * 
Pauli paramagnetism II: canonical ensemble. [tln86] *
Pauli paramagnetism III: grandcanonical ensemble. [tln87] *
Pauli paramagnetism IV: parametric thermodynamic quantities. [tln   ] * 

16. Thermodynamics of Phase Transitions III

Mean-field ferromagnet. [tln84] *

Acknowledgments

I am grateful to Dr. Geoffrey Potter for preparing the graphs to the ideal quantum gas part of these lecture notes. The quality and accuracy of these lecture notes and exercises has greatly benefited from the questions and comments of countless students and correspondents.

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Last updated 12/30/19