Statistical Physics: Linking the Microscopic with the Macroscopic World

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Contents
A Statistical physics
B Thermodynamics
	B.1 Ideal classical gases
		B.1.1 Van der Waals-gases
		B.1.2 Phenomenological temperature measurements
	B.2 Zeroth law of thermodynamics
		B.2.1 Exact and closed differentials
	B.3 First law of thermodynamics
		B.3.1 Isochoric changes of state
		B.3.2 Adiabatic changes of state
		B.3.3 Isobaric changes of state
		B.3.4 Adiabatic index of a gas
		B.3.5 Entropy
	B.4 Second law of thermodynamics
		B.4.1 Conversion from thermal to mechanical energy
		B.4.2 Generalised Carnot-engines
		B.4.3 Entropy and energy conversion
		B.4.4 Irreversible processes and entropy increase in closed systems
		B.4.5 Concave functions and Legendre transforms
	B.5 Third law of thermodynamics
C Probability theory
	C.1 Kolmogorov axioms for probability
	C.2 Laplace probability
	C.3 Conditional probabilities and Bayes’ law
	C.4 Random variables
	C.5 Characteristic function and moment generating function
	C.6 Information entropies
D microcanonical ensemble
	D.1 Virial theorem
	D.2 Energy conservation
	D.3 Legendre transforms and the Hamilton-function
	D.4 Hamilton-equations of motion
E Phase space dynamics
	E.1 Phase space Γ
	E.2 Phase space density ρ
	E.3 Phase space volume: surface and volume
	E.4 Microcanonical ensemble
	E.5 Equipartition theorem
	E.6 Entropy for Hamiltonian systems
	E.7 Adiabatic invariance of entropy
		E.7.1 Example: adiabatic invariance in a string pendulum
	E.8 Entropy as phase space volume at fixed energy
	E.9 Gibbs-paradox
	E.10 Thermal wavelength λth
F Canonical ensemble
	F.1 Marginalisation procedure
	F.2 Canonical ensemble
	F.3 Equipartition theorem
	F.4 Canonical partition function
	F.5 Helmholtz-free energy F
	F.6 Gibbs-enthalpy
G Macrocanonical ensemble
	G.1 Macrocanonical ensemble
	G.2 Macrocanonical potential J and macrocanonical partition Z
	G.3 Chemical potential μ and fugacity z
H Quantum statistics
	H.1 Averages in classical and quantum physics
	H.2 Quantum mechanical partitions
		H.2.1 Construction of phase space densities
		H.2.2 Canonical ensemble in quantum statistics
		H.2.3 Canonical averages of arbitrary observables
		H.2.4 Macrocanonical ensembles in quantum statistics
	H.3 Symmetries of wave functions of many particles
	H.4 Non-interacting systems
	H.5 Macrocanonical ensembles in quantum statistics
		H.5.1 Ultrarelativistic ideal Bose-gas
		H.5.2 Planck-spectrum
	H.6 Fluctuations in quantum and classical statistics
	H.7 Third law of thermodynamics and entropy at absolute zero
	H.8 Fermionic statistics and the exclusion principle
		H.8.1 Density of states for non-relativistic fermions
		H.8.2 Fermi-gases
		H.8.3 Degenerate Fermi-gases in the limit T → 0
	H.9 Ideal non-relativistic Bose-gas
I Phase transitions
	I.1 Van der Waals-gases
	I.2 Ising-model for ferromagnetism
		I.2.1 Transfer matrix method for the Ising-model
		I.2.2 Isothermal susceptibility χ
	I.3 Metropolis-Hastings algorithm and the Ising model in 2d
J Stochastic differential equations
	J.1 Brownian motion
		J.1.1 Langevin-equation
		J.1.2 Random walks
	J.2 Fokker-Planck-equation and evolution of distributions
	J.3 Spectral decomposition of random forces ση(t)
	J.4 Irreversible processes
		J.4.1 Irreversibility in statistical physics
		J.4.2 Reversibility in classical mechanics
X Mathematical supplement
	X.1 Relative information entropies
	X.2 Volume and area of the unit sphere
Y Physical supplement
Acknowledgements
Index
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