Microcanonical Ensemble
A Microcanonical Ensemble is a collection of a large number of isolated systems, each having the same fixed values of:
- Energy \( E \)
- Volume \( V \)
- Number of particles \( N \)
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
- Isolated system: No exchange of energy or particles with the surroundings.
- Fixed variables: Energy \( E \), Volume \( V \), and Particle Number \( N \) are constant.
- Equal probability: All accessible microstates are equally probable.
- Entropy is additive: For independent systems, total entropy is the sum of individual entropies.
- Foundation for other ensembles: Canonical and Grand Canonical ensembles are derived from the microcanonical ensemble by allowing limited exchanges.
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.