Review#
Checklist of Key Concepts#
Section 2.1: Introduction to Statistical Mechanics#
Microscopic–Macroscopic Connection
Macroscopic (thermodynamic) properties can be understood as statistical averages of microscopic properties.
“Expected value” (ensemble average) is the central idea: \(\langle X \rangle = \sum X_i\,p_i\).
Arithmetic Average vs. Expected Value
Arithmetic average: \(\bar{X} = \frac{1}{M}\sum X_i\).
Expected value: \(\langle X \rangle = \sum X_i\,p_i\), where \(p_i\) is the probability of microstate \(i\).
Microstates and Ensembles
Microcanonical ensemble: \((N, V, E)\) — all accessible microstates have the same \(E\).
Canonical ensemble: \((N, V, T)\) — system can exchange energy with a heat bath.
Grand canonical ensemble: \((\mu, V, T)\) — system exchanges both energy and particles with the reservoir.
Fundamental Postulate (Microcanonical)
For an isolated system, each accessible microstate is equally probable.
Section 2.2: Canonical Ensemble#
Closed System
Exchanges energy (heat) with surroundings; cannot exchange matter.
Boltzmann Factor and Partition Function
Probability of microstate \(i\):
\[ p_i = \frac{e^{-\beta E_i}}{Q}, \quad \beta = \frac{1}{k_{\text{B}} T}. \]Partition function \(Q\): normalizes probabilities; sum of Boltzmann factors over all states:
\[ Q = \sum_{i} e^{-\beta E_i}. \]
Two-State System
Simplest nontrivial example: energies \(E_1\) and \(E_2\).
Partition function: \(Q = e^{-\beta E_1} + e^{-\beta E_2}\).
Probabilities:
\[ p_1 = \frac{1}{1 + e^{-\beta\,\Delta E}}, \quad p_2 = 1 - p_1, \quad \Delta E = E_2 - E_1. \]
Interpretation of \(Q\)
“Effective number” of thermally accessible microstates; as \(T \to 0\), few states are accessible; as \(T \to \infty\), many states are accessible.
Section 2.3: Ensemble Averages#
Internal Energy
\(U \equiv \langle E \rangle\).
In canonical ensemble:
\[ U = \frac{1}{Q}\sum_i E_i\, e^{-\beta E_i} = -\left(\frac{\partial \ln Q}{\partial \beta}\right)_{N,V}. \]
Heat Capacity (\(C_V\))
\(C_V = \bigl(\frac{\partial U}{\partial T}\bigr)_{N,V}\).
Also relates to energy fluctuations:
\[ \sigma_E^2 \;=\; \langle(E - \langle E\rangle)^2\rangle \;=\; k_{\text{B}}\,T^2\,C_V. \]
Pressure (brief introduction)
Later formal definition involves \(\bigl(\frac{\partial U}{\partial V}\bigr)_{S}\).
In canonical ensemble, \(\displaystyle P \;=\; k_{\text{B}}\,T\,\bigl(\frac{\partial\ln Q}{\partial V}\bigr)_{N,T}\).
2. Checklist of Most Important Equations#
Expected Value of a General Variable
\[ \langle X \rangle \;=\;\sum_i X_i\,p_i \quad\text{or}\quad \int X(\omega)\,p(\omega)\,d\omega. \]Applicability: general definition in statistical mechanics/probability theory.
Microcanonical Ensemble (Fundamental Postulate)
\[ p_i \;=\;\frac{1}{M} \quad\text{(for all accessible microstates)}. \]Applicability: isolated system with fixed \((N, V, E)\).
Canonical Ensemble Probability
\[ p_i \;=\;\frac{e^{-\beta\,E_i}}{Q}, \quad \beta = \frac{1}{k_{\text{B}}\,T}, \]Applicability: closed system in thermal equilibrium with a heat bath at temperature \(T\).
Canonical Partition Function
\[ Q \;=\;\sum_i e^{-\beta\,E_i}. \]Applicability: all systems that can be described by \((N, V, T)\); sum or integral runs over all microstates.
Internal Energy in the Canonical Ensemble
\[ U \;=\;\langle E\rangle \;=\;\frac{1}{Q}\,\sum_i E_i\,e^{-\beta E_i} \;=\;-\left(\frac{\partial \ln Q}{\partial \beta}\right)_{N,V}. \]Applicability: same as above (canonical ensemble).
Heat Capacity at Constant Volume
\[ C_V \;=\; \left(\frac{\partial U}{\partial T}\right)_{N,V}. \]Applicability: measures how internal energy changes with \(T\) when \(N, V\) are constant.
Energy Fluctuations
\[ \sigma_E^2 \;=\;\langle E^2\rangle - \langle E\rangle^2 \;=\; k_{\text{B}}\,T^2\,C_V. \]Applicability: canonical ensemble; relates variance in energy to the heat capacity.
Pressure (Canonical)
\[ P \;=\; k_{\text{B}}\,T \,\left(\frac{\partial \ln Q}{\partial V}\right)_{N,T}. \]Applicability: canonical ensemble; a precursor to more rigorous derivation using thermodynamic potentials.