BME418 Ch. 6 (2)
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The partition function for state 1: RNA polymerase molecules (P) to bind to nonspecific sites (NNS) (one molecule per site):

The partition function for state 2: one RNA polymerase molecule to bind to the promoter (equivalent to P1 RNA polymerase molecules bound to NNS nonspecific sites AND one on promoter):

The total partition function for the p RNA polymerase molecules when they are bound to nonspecific or promoter sites:

The total number of states available to the combined universe of system and reservoir when the system is in a particular microstate with energy Es(1) is:

The probability for finding the system in state i with energy Es(i) is (Boltzmann distribution): (With the partition function)

The probability of the macrostate X equation (include State, Weight chart):

G(X)=? (In terms of Entropy & simplification of W, p(X)=?):

Boltzmann distribution in terms of total energy to be shared among N particles:

The entropy of ideal gas reflects the freedom to rearrange....
The entropy of ideal gas reflects the freedom to rearrange both molecular positions and velocities (energy)

For the momentum in x direction px, the contribution to the kinetic energy is:

Using the Boltzmann distribution for a system with energy Ei:

Average energy of a molecule in gas expressed in terms of momentum:

Converting Average energy of a molecule in gas into continuous variable:

Average energy of a molecule in gas: