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[NOTES/CM-05001] Cyclic coordinates and constants of motion

Node id: 6176page

Cyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion. 

kapoor's picture 24-04-07 16:04:01 n

[NOTES/SM-06001] Equilibrium Conditions for Open Systems

Node id: 6174page
kapoor's picture 24-04-07 16:04:13 n

[NOTES/CM-02007] Conservation of Energy

Node id: 6044page


If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_k-L\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved.

kapoor's picture 24-04-07 15:04:35 n

[NOTES/SM-06002] Gibbs Distribution

Node id: 6175page
kapoor's picture 24-04-07 13:04:11 n

[NOTES/SM-06004] Grand Canonical Ensemble --- Pressure and Chemical Potential

Node id: 6172page

Using 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.

kapoor's picture 24-04-07 12:04:54 n

[NOTES/SM-06005] Entropy and Energy in Terms of Grand Canonical Partition Function

Node id: 6171page

In this article various thermodynamic functions are expressed in terms of the grand canonical partition function.

kapoor's picture 24-04-07 12:04:13 n

[NOTES/SM-06006] Mean Occupation Number for Identical Fermions and Bosons

Node id: 6173page

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.

kapoor's picture 24-04-07 12:04:52 n

[NOTES/SM-06003] Grand Canonical Ensemble

Node id: 6170page

In this article various thermodynamic functions are expressed in terms of the grand canonical partition function.

kapoor's picture 24-04-07 12:04:27 n

[NOTES/SM-04018] Boltzmann Entropy from Canonical Partition Function

Node id: 6169page

We 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.

kapoor's picture 24-04-06 19:04:11 n

[NOTES/SM-04019] Properties of an Ideal Gas

Node id: 6168page

Several properties such as internal energy, entropy etc. of perfect gases are calculated using the canonical partition function.

kapoor's picture 24-04-06 17:04:28 n

[NOTES/SM-04009] The Imperfect Gas

Node id: 6167page

The 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.

kapoor's picture 24-04-05 23:04:46 n

[NOTES/SM-04015] Equilibrium of a System with a Heat Reservoir

Node id: 6165page

For 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
\[Z = \sum_{\text{MS }r} \exp(-\beta E_r).\]

kapoor's picture 24-04-05 13:04:19 n

[NOTES/SM-04005] Applications of Canonical Ensemble to Paramanetism

Node id: 6166page

In this section the method of canonical ensemble is applied to paramagnetism. For a paramagnetic substance the variation of paramagnetic susceptibility with temperature is derived.

kapoor's picture 24-04-05 13:04:43 n

[NOTES/SM-04010] Classical Theory of Specific Heat of Gases

Node id: 6164page

In 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.

kapoor's picture 24-04-05 07:04:53 n

[NOTES/SM-04011] Specific Heat of Diatomic Gases --- Quantum Effects

Node id: 6163page

The quantum effect of  vibrations on specific heat of diatomic gases is presented. The results are in agreement with experimentally observed facts.

kapoor's picture 24-04-05 07:04:53 n

[NOTES/SM-04014] Partition Function of an Ideal Gas

Node id: 6162page

In this section the classical canonical partition function of an ideal gas is computed.

kapoor's picture 24-04-04 16:04:55 n

[NOTES/SM-04008] `Distribution of Molecules under Gravity

Node id: 6161page

Distribution function of molecules in presence of gravity is as function of height is derived. 

kapoor's picture 24-04-04 13:04:46 n

[NOTES/SM-04007] Applications of Maxwell's Distribution

Node id: 6160page

The 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

kapoor's picture 24-04-04 12:04:48 n

[NOTES/SM-04006] Maxwell Distribution of speeds in an ideal gas

Node id: 6159page

For an ideal gas Maxwell's distribution of velocities is obtained using canonical ensemble.

kapoor's picture 24-04-04 12:04:24 n

[NOTES/SM-04013] Internal energy in terms of canonical partition function

Node id: 6158page

It 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*}

kapoor's picture 24-04-04 11:04:43 n

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