
[NOTES/CM05001] Cyclic coordinates and constants of motionNode id: 6176pageCyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion. 

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[NOTES/SM06001] Equilibrium Conditions for Open SystemsNode id: 6174page$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ 

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[NOTES/CM02007] Conservation of EnergyNode id: 6044page If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_kL\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved.


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[NOTES/SM06002] Gibbs DistributionNode id: 6175page$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ 

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[NOTES/SM06004] Grand Canonical Ensemble  Pressure and Chemical PotentialNode id: 6172pageUsing thermodynamics to express TdS in terms of Gibbs energy and comparing TdS expression in terms of the grand canonical partition function we obtain pressure and chemical potential function in terms of grand canonical partition function. 

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[NOTES/SM06005] Entropy and Energy in Terms of Grand Canonical Partition FunctionNode id: 6171pageIn this article various thermodynamic functions are expressed in terms of the grand canonical partition function. $\newcommand{\pp}[2][]{\frac{\partila #1}{\partial #2}}\newcommand{\Zca}{\mathcal Z}\newcommand{\Label}[1]{\label{#1}}$ 

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[NOTES/SM06006] Mean Occupation Number for Identical Fermions and BosonsNode id: 6173page$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\Zca}{\mathcal Z}$
Using occupation number representation for identical particles, the grand canonical partition function is expressed in terms of single particle partition function. This is then used to discuss cases of identical fermions and identical bosons. Mean occupation number for fermions and bosons is obtained. 

240407 12:04:52 
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[NOTES/SM06003] Grand Canonical EnsembleNode id: 6170pageIn this article various thermodynamic functions are expressed in terms of the grand canonical partition function. $\newcommand{\Zca}{\mathcal Z}$ 

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[NOTES/SM04018] Boltzmann Entropy from Canonical Partition FunctionNode id: 6169pageWe derive the Boltzmann relation for entropy using the canonical partition function. This derivation uses the fact that the number of micro states \(\Omega(E)\) as a function energy has a sharp peak around the mean energy. 

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[NOTES/SM04019] Properties of an Ideal GasNode id: 6168pageSeveral properties such as internal energy, entropy etc. of perfect gases are calculated using the canonical partition function. 

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[NOTES/SM04009] The Imperfect GasNode id: 6167pageThe ideal gas equation \(pV=NkT\) is good approximation for low densities. In this section a scheme of obtaining corrections to the ideal gas equation is discussed. 

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[NOTES/SM04015] Equilibrium of a System with a Heat ReservoirNode id: 6165pageFor a system in equilibrium with a heat bath at temperature \(T\) the energy is not constant. We start with the Boltzmann relation \(S=k\log \Omega\). It will be shown that if of a microstate has energy \(E_r\), its probability \(p_r\) is proportional to \(e^{\beta E_r}\) and is given by \[p_r = \frac{\exp(\beta E_r)}{Z} \] where \(Z\) is function of \(T,V,N\), called the canonical partition function and is given by \[Z = \sum_{\text{MS }r} \exp(\beta E_r).\] $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\pp}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2 #1}{d #2^2}}$ 

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[NOTES/SM04005] Applications of Canonical Ensemble to ParamanetismNode id: 6166pageIn this section the method of canonical ensemble is applied to paramagnetism. For a paramagnetic substance the variation of paramagnetic susceptibility with temperature is derived. 

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[NOTES/SM04010] Classical Theory of Specific Heat of GasesNode id: 6164pageIn this lecture we derive law of equipartition of energy under the assumption that the energy is quadratic function of some variable such as coordinates and momenta. As an application, the classical theory of specific heat of gases is given. 

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[NOTES/SM04011] Specific Heat of Diatomic Gases  Quantum EffectsNode id: 6163pageThe quantum effect of vibrations on specific heat of diatomic gases is presented. The results are in agreement with experimentally observed facts. 

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[NOTES/SM04014] Partition Function of an Ideal GasNode id: 6162pageIn this section the classical canonical partition function of an ideal gas is computed. 

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[NOTES/SM04008] `Distribution of Molecules under GravityNode id: 6161pageDistribution function of molecules in presence of gravity is as function of height is derived. 

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[NOTES/SM04007] Applications of Maxwell's DistributionNode id: 6160pageThe equation of state of a perfect gas is derived using Maxwell distribution of velocities of molecules in a perfect gas. The Maxwell distribution can be verified by means of an experiment on effusion of a gas through a hole. This also lead to determination of the Boltzmann constant 

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[NOTES/SM04006] Maxwell Distribution of speeds in an ideal gasNode id: 6159pageFor an ideal gas Maxwell's distribution of velocities is obtained using canonical ensemble. 

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[NOTES/SM04013] Internal energy in terms of canonical partition functionNode id: 6158pageIt is shown that the internal energy, can be computed from the canonical partition function using
\begin{align*}U=\frac{\partial}{\partial\beta} \log Z.\end{align*} 

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