708Notes2018/708LectureNotes.toc at master · dvasques83/708Notes2018 · GitHub

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\contentsline {section}{\numberline {1}Thermodynamics - A Refresher}{2}
\contentsline {subsection}{\numberline {1.1}The Four Laws of Thermodynamics}{3}
\contentsline {subsection}{\numberline {1.2}Zeroth Law}{3}
\contentsline {subsection}{\numberline {1.3}First Law}{4}
\contentsline {subsection}{\numberline {1.4}Third Law}{4}
\contentsline {subsection}{\numberline {1.5}Heat Capacity}{6}
\contentsline {subsection}{\numberline {1.6}Intensive \& Extensive Variables}{6}
\contentsline {subsection}{\numberline {1.7}The Fundamental Hypothesis of Thermodynamics}{7}
\contentsline {subsection}{\numberline {1.8}The Central Problem of Thermodynamics}{7}
\contentsline {subsection}{\numberline {1.9}Suggested Reading}{9}
\contentsline {section}{\numberline {2}Probability}{9}
\contentsline {subsection}{\numberline {2.1}Probability generating functions}{10}
\contentsline {subsection}{\numberline {2.2}Properties of generating functions}{10}
\contentsline {subsection}{\numberline {2.3}The binomial distribution}{11}
\contentsline {subsection}{\numberline {2.4}The Poisson distribution}{12}
\contentsline {subsection}{\numberline {2.5}The expected long-run outcome of a binary process}{12}
\contentsline {subsection}{\numberline {2.6}Brownian motion}{13}
\contentsline {section}{\numberline {3}Statistical postulates}{14}
\contentsline {subsection}{\numberline {3.1}An ideal gas}{14}
\contentsline {subsection}{\numberline {3.2}Maxwell's postulates}{15}
\contentsline {subsection}{\numberline {3.3}Equation of state}{15}
\contentsline {section}{\numberline {4}Phase space, Hamiltonian dynamics, and the fundamental postulate of statistical mechanics}{16}
\contentsline {subsection}{\numberline {4.1}Phase space}{16}
\contentsline {subsection}{\numberline {4.2}Observables}{18}
\contentsline {subsection}{\numberline {4.3}The fundamental postulate of statistical mechanics}{18}
\contentsline {subsection}{\numberline {4.4}Symplectic transformations and phase space area preservation}{19}
\contentsline {subsection}{\numberline {4.5}Liouville's theorem and preservation of phase space volume}{20}
\contentsline {subsection}{\numberline {4.6}Recommended reading}{21}
\contentsline {section}{\numberline {5}Molecular dynamics and simulating Hamiltonian Systems}{21}
\contentsline {subsection}{\numberline {5.1}Numerical integration}{23}
\contentsline {subsection}{\numberline {5.2}Recommended reading}{26}
\contentsline {section}{\numberline {6}Ensembles}{26}
\contentsline {subsection}{\numberline {6.1}The canonical ensemble}{27}
\contentsline {subsection}{\numberline {6.2}Other ensembles}{29}
\contentsline {subsubsection}{\numberline {6.2.1}The grand-canonical ensemble}{29}
\contentsline {subsubsection}{\numberline {6.2.2}The $P-T$ ensemble}{30}
\contentsline {subsection}{\numberline {6.3}Information theory and the Gibbs Formula for entropy}{30}
\contentsline {subsection}{\numberline {6.4}Calculating things with ensembles}{31}
\contentsline {subsubsection}{\numberline {6.4.1}Internal energy}{32}
\contentsline {subsubsection}{\numberline {6.4.2}Specific heat}{32}
\contentsline {subsubsection}{\numberline {6.4.3}Entropy}{33}
\contentsline {subsubsection}{\numberline {6.4.4}Simple harmonic oscillator}{33}
\contentsline {subsection}{\numberline {6.5}Recommended reading}{34}
\contentsline {section}{\numberline {7}Quantum statistical mechanics}{34}
\contentsline {subsection}{\numberline {7.1}A refresher: bra-ket notation}{35}
\contentsline {subsection}{\numberline {7.2}Quantum micro-canonical ensemble}{36}
\contentsline {subsection}{\numberline {7.3}Quantum canonical ensemble}{37}
\contentsline {subsection}{\numberline {7.4}Quantum grand-canonical ensemble}{38}
\contentsline {subsection}{\numberline {7.5}Systems of indistinguishable particles}{38}
\contentsline {subsection}{\numberline {7.6}Fermionic and bosonic partition functions}{40}
\contentsline {subsubsection}{\numberline {7.6.1}Bose-Einstein Statistics}{41}
\contentsline {subsubsection}{\numberline {7.6.2}Fermi-Dirac statistics}{41}
\contentsline {subsubsection}{\numberline {7.6.3}Maxwell-Boltzman statistics}{41}
\contentsline {subsection}{\numberline {7.7}An example: the quantum harmonic oscillator}{42}
\contentsline {subsection}{\numberline {7.8}Recommended reading}{43}
\contentsline {section}{\numberline {8}Complex networks}{43}
\contentsline {subsection}{\numberline {8.1}An introduction to graph theory \& networks}{44}
\contentsline {subsection}{\numberline {8.2}Real-world networks: some examples}{45}
\contentsline {subsection}{\numberline {8.3}Properties of networks}{46}
\contentsline {subsection}{\numberline {8.4}Network growth models}{49}
\contentsline {subsection}{\numberline {8.5}Exponential random graph models}{50}
\contentsline {subsection}{\numberline {8.6}Recommended reading}{51}