9.3 Canonical and grand canonical ensembles

Statistical thermodynamics bridges microscopic and macroscopic worlds. Canonical and grand canonical ensembles are key tools for understanding system behavior at fixed temperature. They allow us to calculate average properties and predict equilibrium states.

These ensembles differ in what they keep constant. The canonical ensemble fixes particle number and volume, while the grand canonical ensemble allows particle exchange. Both are crucial for modeling real-world systems and connecting to experimental observations.

Ensembles

Types of Ensembles

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• Canonical ensemble represents a system in thermal equilibrium with a heat bath at a fixed temperature, allowing energy exchange while keeping the number of particles and volume constant
• Grand canonical ensemble describes an open system that can exchange both energy and particles with a reservoir, characterized by a fixed chemical potential, temperature, and volume
• Microcanonical ensemble represents an isolated system with constant energy, volume, and number of particles, where all accessible microstates are equally probable (statistical weight)

Thermodynamic Limit and Ensemble Equivalence

• Thermodynamic limit refers to the behavior of a system as it approaches an infinite size (number of particles and volume) while keeping the density constant
• In the thermodynamic limit, the specific properties of a system (per particle) become independent of the size and the choice of ensemble
• Different ensembles (canonical, grand canonical, microcanonical) yield equivalent results for average thermodynamic properties in the thermodynamic limit, known as ensemble equivalence

Thermodynamic Potentials

Gibbs and Helmholtz Free Energies

• Gibbs free energy is a thermodynamic potential that measures the maximum reversible work that can be extracted from a system at constant temperature and pressure
  • Defined as $G = U - TS + PV$, where $U$ is the internal energy, $T$ is the temperature, $S$ is the entropy, $P$ is the pressure, and $V$ is the volume
  • Minimized at equilibrium for systems at constant temperature and pressure (spontaneous processes)
• Helmholtz free energy is a thermodynamic potential that measures the maximum reversible work that can be extracted from a system at constant temperature and volume
  • Defined as $F = U - TS$, where $U$ is the internal energy, $T$ is the temperature, and $S$ is the entropy
  • Minimized at equilibrium for systems at constant temperature and volume (spontaneous processes)

Chemical Potential

• Chemical potential is the change in a system's free energy (Gibbs or Helmholtz) with respect to the change in the number of particles at constant temperature and pressure or volume
  • Defined as $\mu = \left(\frac{\partial G}{\partial N}\right)_{T,P}$ or $\mu = \left(\frac{\partial F}{\partial N}\right)_{T,V}$, where $G$ is the Gibbs free energy, $F$ is the Helmholtz free energy, and $N$ is the number of particles
• Represents the energy required to add or remove a particle from a system at equilibrium
• In a multi-component system, each component has its own chemical potential, and equilibrium is reached when the chemical potentials of each component are equal across all phases

System Properties

Fluctuations and Thermodynamic Limit

• Fluctuations are random deviations of a system's properties (energy, number of particles) from their average values due to the system's finite size and thermal motion
  • Examples include density fluctuations in a gas or magnetization fluctuations in a ferromagnet near its critical temperature
• The magnitude of fluctuations typically scales as the inverse square root of the system size (number of particles or volume)
• In the thermodynamic limit (infinite system size), the relative magnitude of fluctuations becomes negligible compared to the average values of the system's properties
  • This allows for the use of average thermodynamic quantities to describe the system's behavior accurately

Intensive and Extensive Properties

• Intensive properties are independent of the system size (number of particles or volume), such as temperature, pressure, and chemical potential
  • Intensive properties are the same for a system and its subsystems at equilibrium
• Extensive properties scale with the system size (number of particles or volume), such as energy, entropy, and Gibbs or Helmholtz free energy
  • Extensive properties are additive for subsystems at equilibrium
• In the thermodynamic limit, the ratio of two extensive properties becomes an intensive property, such as the specific heat capacity (heat capacity per particle) or the specific volume (volume per particle)