9.1 Fundamentals of statistical mechanics

Statistical mechanics bridges microscopic and macroscopic worlds in thermodynamics. It explains how individual particle behaviors create observable properties like temperature and pressure. This fundamental approach helps us understand complex systems and their thermodynamic behaviors.

Microstates, macrostates, and ensembles are key concepts in statistical mechanics. We'll explore probability distributions, the equipartition theorem, and entropy. These ideas form the basis for understanding how atomic-level interactions lead to large-scale thermodynamic phenomena.

Microstates and Macrostates

Microstates and Macrostates in Statistical Mechanics

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• Microstates represent the specific microscopic configurations or arrangements of particles in a system
  • Each microstate corresponds to a unique set of positions and momenta for all particles
  • The number of microstates depends on the number of particles and the available energy levels
• Macrostates describe the macroscopic properties of a system (temperature, pressure, volume)
  • A macrostate can be realized by a large number of microstates
  • Macrostates are observable and measurable, while microstates are not directly accessible
• The relationship between microstates and macrostates is fundamental to statistical mechanics
  • The macroscopic behavior of a system emerges from the collective properties of its microstates

Ensemble and Phase Space

• An ensemble is a collection of all possible microstates of a system that share the same macroscopic properties
  • Ensembles are used to calculate average properties and thermodynamic quantities
  • Different types of ensembles include microcanonical, canonical, and grand canonical ensembles
• Phase space is a mathematical representation of all possible states of a system
  • Each point in phase space corresponds to a specific microstate
  • The dimensions of phase space depend on the number of particles and degrees of freedom (position and momentum coordinates)
• Statistical weight is the number of microstates that correspond to a particular macrostate
  • Macrostates with higher statistical weights are more probable
  • The statistical weight is related to the entropy of the system (Boltzmann's principle)

Probability and Distributions

Probability Density and Boltzmann Distribution

• Probability density describes the likelihood of finding a system in a particular state or configuration
  • In classical mechanics, probability density is a function of position and momentum coordinates
  • In quantum mechanics, probability density is given by the square of the wavefunction amplitude
• The Boltzmann distribution gives the probability of a system being in a specific microstate with energy $E_i$ at temperature $T$
  • The probability is proportional to $e^{-E_i/kT}$, where $k$ is the Boltzmann constant
  • The Boltzmann distribution is a fundamental result of statistical mechanics and applies to many systems in thermal equilibrium

Equipartition Theorem

• The equipartition theorem states that, in thermal equilibrium, each degree of freedom that appears quadratically in the system's energy has an average energy of $\frac{1}{2}kT$
  • Degrees of freedom include translational, rotational, and vibrational modes
  • The theorem helps explain the specific heat capacities of gases and solids
• The equipartition theorem is a consequence of the Boltzmann distribution
  • It assumes that the system is in thermal equilibrium and the energy levels are continuous
• Deviations from equipartition can occur in quantum systems or when the energy levels are discrete (quantized vibrations in solids)

Fundamental Principles

Ergodic Hypothesis

• The ergodic hypothesis states that, over long periods, the time average of a system's properties is equal to the ensemble average
  • In other words, a system will eventually explore all accessible microstates given enough time
  • This allows the use of ensembles to calculate thermodynamic properties instead of following the system's evolution over time
• The ergodic hypothesis is a fundamental assumption in statistical mechanics
  • It justifies the use of statistical methods to describe the behavior of macroscopic systems
• Ergodicity breaks down in systems with broken ergodicity (glasses, spin glasses) or when there are constraints that prevent the system from exploring all microstates

Entropy in Statistical Mechanics

• Entropy is a measure of the number of microstates accessible to a system
  • It quantifies the disorder or randomness of a system
  • In statistical mechanics, entropy is defined as $S = k \ln \Omega$, where $\Omega$ is the number of microstates
• The second law of thermodynamics states that the entropy of an isolated system always increases or remains constant
  • This is a consequence of the system evolving towards the most probable macrostate, which has the highest number of microstates
• Entropy plays a crucial role in determining the direction of spontaneous processes and the stability of thermodynamic systems
  • Systems tend to evolve towards states of higher entropy (maximum disorder)
  • The change in entropy determines the feasibility and spontaneity of thermodynamic processes (Clausius inequality)