2.1 Statistical Thermodynamics

Statistical thermodynamics bridges the gap between microscopic particle behavior and macroscopic properties. It uses probability distributions and partition functions to describe how individual molecules contribute to overall system characteristics.

This approach allows us to derive equations of state, understand phase transitions, and calculate thermodynamic properties. However, it has limitations, like assuming equilibrium and neglecting quantum effects in some cases.

Fundamentals of Statistical Thermodynamics

Principles of statistical mechanics

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• Connects microscopic properties of a system to its macroscopic thermodynamic behavior
  • Based on the idea that macroscopic properties arise from the collective behavior of constituent particles (molecules, atoms)
• Boltzmann distribution describes the probability $P_i$ of a system being in a particular microstate with energy $E_i$
  • $P_i = \frac{e^{-E_i/kT}}{Q}$, where $k$ is the Boltzmann constant, $T$ is the absolute temperature, and $Q$ is the partition function
  • Allows calculation of average properties by summing over all microstates weighted by their probabilities
• Partition function $Q$ is a sum over all possible microstates of the system
  • $Q = \sum_i e^{-E_i/kT}$
  • Serves as a normalization factor in the Boltzmann distribution
  • Encodes information about the system's thermodynamic properties
• Thermodynamic properties can be derived from the partition function using mathematical relationships
  • Internal energy: $U = -\frac{\partial \ln Q}{\partial \beta}$, where $\beta = \frac{1}{kT}$
  • Helmholtz free energy: $A = -kT \ln Q$, a measure of the useful work obtainable from the system
  • Entropy: $S = k \ln Q + \frac{U}{T}$, a measure of the system's disorder or randomness

Partition functions for systems

• Microcanonical ensemble describes an isolated system with fixed number of particles $N$, volume $V$, and energy $E$
  • Partition function: $Q = \Omega(N, V, E)$, where $\Omega$ is the number of microstates with energy $E$
  • All accessible microstates are equally probable
• Canonical ensemble describes a system in contact with a heat bath, with fixed $N$, $V$, and temperature $T$
  • Partition function: $Q = \sum_i e^{-E_i/kT}$
  • System exchanges energy with the heat bath, leading to a Boltzmann distribution of energy levels
• Grand canonical ensemble describes a system in contact with a heat bath and particle reservoir, with fixed chemical potential $\mu$, $V$, and $T$
  • Partition function: $\Xi = \sum_{N=0}^\infty e^{\beta \mu N} Q(N, V, T)$, where $\mu$ is the chemical potential
  • System exchanges both energy and particles with the reservoir
• For an ideal gas, the single-particle partition function $q$ can be factored into contributions from different degrees of freedom
  • $q = q_\text{trans} q_\text{rot} q_\text{vib}$, representing translational, rotational, and vibrational motion
  • Total partition function: $Q = \frac{q^N}{N!}$ for indistinguishable particles (bosons or fermions), accounting for quantum statistics

Applications and Limitations

Microscopic vs macroscopic properties

• Equations of state relating pressure $P$, volume $V$, and temperature $T$ can be derived from the partition function
  • For an ideal gas: $PV = NkT$, obtained from $P = -\left(\frac{\partial A}{\partial V}\right)_{N,T}$
  • More complex equations of state (van der Waals, Redlich-Kwong) account for intermolecular interactions
• Heat capacity $C_V$ measures the system's response to temperature changes at constant volume
  • $C_V = \left(\frac{\partial U}{\partial T}\right)_V = \frac{\partial}{\partial T}\left(-\frac{\partial \ln Q}{\partial \beta}\right)_V$
  • Related to the system's energy fluctuations and microscopic degrees of freedom
• Phase transitions (melting, vaporization) can be understood in terms of changes in the partition function
  • Discontinuities or divergences in derivatives of $\ln Q$ indicate phase transitions
  • Order parameters (density, magnetization) characterize the different phases

Limitations of thermodynamic models

• Assumption of thermodynamic equilibrium
  • Statistical thermodynamics assumes the system has reached a state of maximum entropy
  • Non-equilibrium processes (transport, relaxation) require more advanced techniques (kinetic theory, master equations)
• Classical approximation assumes particles are distinguishable and obey Maxwell-Boltzmann statistics
  • Quantum effects become important at low temperatures or high densities (Bose-Einstein, Fermi-Dirac statistics)
  • Quantum statistical mechanics is necessary for accurate descriptions of metals, semiconductors, superfluids
• Ideal gas model assumes no intermolecular interactions
  • Real gases and liquids require more complex models to account for attractive and repulsive forces
  • Virial expansion, perturbation theory, and molecular simulations can improve accuracy
• Statistical thermodynamics typically assumes large systems in the thermodynamic limit ($N \to \infty$, $V \to \infty$, $N/V$ constant)
  • Small systems (nanoscale, biological) may exhibit deviations from bulk behavior due to surface effects, fluctuations
  • Finite-size corrections and mesoscopic models (lattice gases, Ising model) can capture these effects