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Connection between partition functions

Connection between partition functions. Looking at the natural variables. From Q to X. An example on the equivalency among ensembles. N distinguishable particles, 2 possible states ( D E= e with E 1 =0)  {a j }={a 1 ,a 2 ,…a N } where a j =0, or 1 and therefore.

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Connection between partition functions

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  1. Connection between partition functions Looking at the natural variables

  2. From Q to X

  3. An example on the equivalency among ensembles N distinguishable particles, 2 possible states (DE=e with E1=0)  {aj}={a1,a2,…aN} where aj=0, or 1 and therefore Equivalency of ensembles Microcanonical ensemble: degeneracy of the mthlevel  number of ways to distribute m objects in a pool of N (i.e. distribute m quanta to obtain E total energy)

  4. Equivalency of ensembles II

  5. Summary of Ensembles Useful ensembles at least one extensive variable N,V,or E Generalized ensemble with only intensive properties, (m,p,T) but –kbTlnZ(m,p,T)=0  no fundamental function

  6. Fluctuation:spontaneous deviation of a mechanical variable from its mean… How much it deviates? Fluctuations Ergodic hypothesis <time>  <ensemble> rms fluctuation of X=X(t) is equivalent to sX

  7. What are the fluctuations in the canonical ensemble? Fluctuations in Canonical Ensemble

  8. Fluctuations in Canonical Ensemble II The spread of the fluctuations corresponds to the rate at which the energy changes with T For an ideal gas, Distribution of energies is like a delta function centered at <E>

  9. Fluctuations in Grand canonical

  10. Isothermal compressibility Isothermal compressibility

  11. Fluctuations in N For a canonical ensemble, even thought there are fluctuations, The energy is distributed uniformly. Each system is most likely to be found with energy <E> canonical ensemble equivalent to microcanonical (where E is constant) Fluctuations in N show that a grand canonical ensemble is most likely to be found with <N> particles  grand canonical canonical ensemble equivalent to canonical (where N is constant)

  12. W E E P(E) E*=<E> What is the probability of finding a particular value of E? P(E) P(E)

  13. Let’s count… How do we count states

  14. Canonical ensemble of DISTIGUISHABLE particles/quasi-particles: a,b,…n. Distinguishable particles

  15. Imagine a system with N=1000 degrees of freedom (1000 quasi particles) Example for distinguishable particles Each particle can be in one of 5 microstates  There are 51000 states to be sampled (and counted!!!) Using the factorization due to equal-but-distinguishable particles, we only need to enumerate 5 states to evaluate q  conversion of one N-body problem to N, 1-body problems

  16. If the particles are INDISTINGUISHABLE Indistinguishable particles FERMIONS : All indices j; k,…, l must be different. Hence summations over indices depend on each other. BOSONS : Indices j;k;…;lneed not all be different. Permutations like j; k;…;land k;j;…;lrefer to identical states and must occur only once in the summation.

  17. INDISTINGUISHABLE particles  Boltzman Statistics A particular (and common) case: T,d number of available energy states >> N  each an every particle is in a different state Boltzman Statistics we have to consider those distribution that are equivalent, that is There are N! of these combinations which can be subtracted from the pool of microstates by dividing by N!

  18. Boltzman number of 1-particle state >>number of particles Singel particle energy from Bolztman How many 1-particle states? Remember the sphere used to explain degeneracy? number of 1-particle states with an energy lower than e =number of lattice points enclosed by the sphere in the positive octant:

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