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: $ =\int O(p,q)\rho(p,q,t) , d\Gamma$
• Equalbrium:

Hamiltorian

• , Equation of motion:
  

• Continuity equaltion:
  Extended: $$\frac{\partial \rho}{\partial t}+\sum_{i=1}^{3N}\left[ \rho\left( \frac{\partial \dot{q}{i}}{\partial q{i}}+\frac{\partial \dot{p}{i}}{\partial p{i}} \right) + \dot{q}{i}\frac{\partial \rho}{\partial q{i}}+ \dot{p}{i}\frac{\partial \rho}{\partial p{i}}\right]=\frac{\partial \rho}{\partial t}+\sum_{i=1}^{3N}\left[ \dot{q}{i}\frac{\partial \rho}{\partial q{i}}+ \dot{p}{i}\frac{\partial \rho}{\partial p{i}}\right]=\frac{d\rho}{dt}=0\frac{\partial \dot{q}{i}}{\partial q{i}}+\frac{\partial \dot{p}{i}}{\partial p{i}} =\frac{\partial }{\partial q_{i}}\left( \frac{\partial H}{\partial p_{i}} \right)-\frac{\partial }{\partial p_{i}}\left( \frac{\partial H}{\partial q_{i}} \right)=0$$

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 $$\Omega(E)=\int dE_{1}\Omega_{1}(E_{1}){}\Omega_{2}(E-E_{1}) , \approx \Omega_{1}(E^{}{1})\Omega{2}(E-E^{}{1}),E{1}^{*}\text{ maximize }\Omega_{1}\Omega{2}\frac{d\Omega_{1}}{dE_{1}}\Omega_{2}- \frac{d\Omega_{2}}{dE_{2}}\Omega_{2}=0\frac{d\ln\Omega_{1}}{dE_{1}}=\frac{d\ln \Omega_{2}}{dE_{2}}\frac{\partial S}{\partial E}=\frac{1}{T}S(E,V,N)=k_{B}\ln \Omega(E,V,N)$$

• 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 $=\int O(p,q)\rho(p,q) , d\Gamma$

• 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. $$=\sum_n E_nP(E_n)=\frac{1}{Z}\sum_n E_ne^{-\beta E_n}=\frac{1}{Z}\left( -\frac{\partial }{\partial \beta} \right)\sum_n e^{-\beta E_n}=-\frac{\partial }{\partial \beta}\ln Z$$
    2. 
       
    3. 
       (Using the differential of )()
    4. 
       
       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
   $$E=-\left( \frac{\partial }{\partial \beta}\ln Q \right){z}\quad N=\left( \frac{\partial }{\partial (\beta \mu)}\ln Q \right){\beta}d\Psi=-SdT-pdV-Nd\mu$$