Flashcards
Statistical Mechanics
Cards in this deck (21)
- 1What is \(\Omega\) for \(k\) indistinguishable elements from \(n\) objects with replacement?\[\Omega=n^k\]
- 2What is \(\Omega\) for \(k\) distinguishable elements from \(n\) objects without replacement?\[\Omega=\frac{n!}{(n-k)!}\]
- 3What is \(\Omega\) for \(k\) indistinguishable elements from \(n\) objects without replacement?\[\Omega={n\choose k}=\frac{n!}{k!(n-k)!}\]
- 4Sterling Approximation\[ln(N!)=N\,ln(N)-N+O(ln(N))\]
- 5The Microcanonical Interconnect between Statistical Mechanics and ThermodynamicsEntropy:
\[S=k_bln(\Omega)\] - 6The Canonical Partition Function for Classical Systems\[Z_{CM}=\frac{1}{h^N}\int\cdots\int\prod_{i=1}^Ndp_idq_ie^{-H(p,q)\beta}\]
- 7Microcanonical Partition Function\[Z=\sum_j\omega_je^{\beta E_j}\]
- 8Canonical Partition Function\[Z_{can}(N,V,T)=\frac{1}{N!}\frac{1}{h^{3N}}\int\prod_{i=1}^{3N}dp_idq_ie^{-H(p,q)\beta}\]
- 9The Canonical Interconnect between Statistical Mechanics and ThermodynamicsHelmholtz Free Energy:
\[F(N,V,T)=-kT\,ln(Z_{can}(N,V,T))\] - 10The Grand Canonical Partition Function\[Z_G=e^{\lambda Z_{can}}\]
where \(\lambda=e^{\mu/kT}\) is the "fugacity" - 11The Grand Canonical Interconnect between Statistical Mechanics and ThermodynamicsThe Grand Potential:
\[\Phi(T,V,\mu)=-pV=-kT\,ln(Z_G)\] - 12Characteristic of Microcanonical EnsemblesEvery state is equally probable.
- 13Characteristic of Canonical EnsemblesEnergy is allowed to fluctuate.
- 14Characteristic of Grand Canonical EnsemblesEnergy and particle number are allowed to fluctuate.
- 15The Canonical Equation for Entropy\[S=\frac{\partial}{\partial T}(T\,ln(Z_{can}))\]
- 16The Canonical Equation for Internal Energy\[U=k_bT^2\frac{\partial}{\partial T}ln(Z_{can})\]
- 17The Grand Canonical Equation for Total Particle Number\[N=k_bT\frac{\partial}{\partial\mu}ln(Z_G)\]
- 18The Grand Canonical Equation for Fluctuations in Particle Number\[(\Delta N)^2=\frac{\partial^2}{\partial(\mu\beta)^2}ln(Z_G)\]
- 19/[dE/]/[dE=T\;dS-P\;dV+\mu\;dN/]
- 20\[dF\]\[dF=-S\;dT-P\;dV+\mu\;dN\]
\[F=E-T\,S\] - 21\[C_V\]\[T\frac{\partial S}{\partial T}\Big|_{V,N}=\frac{\partial E}{\partial T}\Big|_{V,N}\]