Ensenmble Theory

Macrostate & Microstate

• Ensemble: a collection of systems with all possible microstate that are consistent with macrostate.
• Different choices of macro different statistical ensemble:
  1. isolated system: Microcanomical Ensemble
  2. allow energy exchange: Canomical Ensemble
  3. allow energy & particle exchange: Grand Canomical Ensemble ## Description of microstates

1. Classical system of N particles: spans a -dimensional phase space.
2. Quantum system: (-particle wave function)
3. Classical systems of discrete variables: label as (e.g. Ising magnet) ## Consider a collection of microstates

• Infinitesmall volumn element: , quantities of representative points in :
• So we can define the density of representative points:
• Observable is the ensemble average:
• Equalbrium:

Hamiltorian

• , Equation of motion:
• Continuity equaltion: Extended: In the consequence of:

Liouville's Theorem: Under Hamiltoriun dynamics, density is a constant along the flow.

Microcanomical Ensemble

• Microstate={all representive points lacated on a constant E hypersurface}

Assumption of equal a priori probability: =total number of microstates

• e.g. Ideal Gas:

Surface Area: Volumn: Solid Angle:

- Consider:

• Saddle point integration expand near :

Sterling's approximation An approximation for , ： in that So we can makeor

• Consider a isolated systems, . Total number of microstates Extremum: And: Identify:

• For ideal Gas: but it‘s not a extensivity. Mixing Entropy: Two distinct gases, remove partition, gases mix But for two same gases of the same density: Gibbs Paradox

  • Macroscopically: nothing happened()
  • Microscopically: the actual paticles changed
  • Realism: particles are identical.
  • Now is a extensivity.
  • Phase space for indentical particles: Properties of ideal gas from M.E. : thermal de Broglie wavelength. When , for Classical ideal gas. Maxwell-Boltzmann distribution: possibility of a particle being :

• Ising magnet: of microstates : : , : so :

Justifiction of ensemble average

• Standard textbook argument:
  1. observable
  2. timescale of microscopic motion. So . >Ergodic Hypothesis: , system visits every region in phase space, time spent in each region volume of the region. So
• Caution:
  1. Proceed only for few-body or single-body systems.
  2. Timescale . Irrelevant for any realistic physical system.
  3. Time-average
  4. Ergodicity comes in a levels. e.g. Mixing

Canomical Ensemble

Macrostate , allow energy exchange.

Total energy of 1+2 fixed M.E

- Probability of system1 in a particular state (quantum state), with energy : total n of microstates for 1+2: And , Expand around : - Probability of system1 having energy : : numbers of microstate with energy E. Barely used for calculation in practice. - Phyical meaning of Z: in which minimize . - Suggest - Thermodynamic Quantities: 1. 2. 3. (Using the differential of )() 1. e.g. Information entropy

Grand Canomical Ensemble

1. Microstate () allow for both energy and particle exchange. Consider the system contact with a bath: total number of microstates: Expand : So: Grand partition function: Define fugacity: So: Grand potential:
2. Thermodynamic function