Microcanonical Ensemble

A Microcanonical Ensemble is a collection of a large number of isolated systems, each having the same fixed values of:

These systems are completely isolated from their surroundings, meaning they do not exchange energy or particles. This ensemble is the most fundamental and simplest of all statistical ensembles and is governed by the laws of classical or quantum mechanics.

Basic Assumptions

- All accessible microstates of the system consistent with the given \( E, V, N \) are equally probable.
- The system is in equilibrium.

Phase Space and Density of States

In classical statistical mechanics, the number of accessible microstates within a small energy range \( [E, E + \delta E] \) is given by:

\[ \Omega(E, V, N) = \int_{H(q,p) \in [E, E + \delta E]} d^{3N}q \, d^{3N}p \]

where \( H(q, p) \) is the Hamiltonian of the system and \( \Omega \) is the density of states.

Entropy

The entropy in the microcanonical ensemble is defined as:

\[ S = k_B \ln \Omega(E, V, N) \]

where \( k_B \) is the Boltzmann constant. Entropy is a measure of the number of accessible microstates.

Temperature

The temperature is obtained by differentiating entropy with respect to energy:

\[ \frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{V, N} \]

Properties of Microcanonical Ensemble

Applications

- Studying ideal gases in isolated containers
- Understanding entropy and fundamental thermodynamic relations
- Forming the basis for the derivation of canonical and grand canonical ensembles

Thermodynamic Parameters in Microcanonical Ensemble

Entropy \( S \)

In the microcanonical ensemble, the entropy is defined as:

\[ S(E, V, N) = k_B \ln \Omega(E, V, N) \]

where \( \Omega \) is the number of accessible microstates in the energy shell \( [E, E + \delta E] \), and \( k_B \) is the Boltzmann constant.

Temperature \( T \)

The temperature is derived from the entropy using the thermodynamic relation:

\[ \frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{V, N} \]

This expression shows how temperature relates to the change of entropy with energy at constant volume and particle number.

Pressure \( P \)

Pressure is obtained from the derivative of entropy with respect to volume:

\[ \frac{P}{T} = \left( \frac{\partial S}{\partial V} \right)_{E, N} \] \[\Rightarrow P = T \left( \frac{\partial S}{\partial V} \right)_{E, N} \]

This reflects how the entropy changes when volume changes, keeping energy and particle number constant.

Chemical Potential \( \mu \)

The chemical potential is related to the change of entropy with respect to particle number:

\[ \frac{-\mu}{T} = \left( \frac{\partial S}{\partial N} \right)_{E, V} \] \[ \Rightarrow \mu = -T \left( \frac{\partial S}{\partial N} \right)_{E, V} \]

This tells how entropy varies when the number of particles is changed.

First Law of Thermodynamics

The total differential of entropy is:

\[ dS = \left( \frac{\partial S}{\partial E} \right) dE + \left( \frac{\partial S}{\partial V} \right) dV + \left( \frac{\partial S}{\partial N} \right) dN \]

Substituting the expressions above, we get:

\[ dE = T dS - P dV + \mu dN \]

This is the first law of thermodynamics in terms of thermodynamic parameters derived from the microcanonical ensemble.