P(a, p)=h-Mexp(-H(, p)/kT)Q where H is the classical Hamiltonian, h is Planck's constant, and the classical partition function Q is Q=hMexp(-H(q, p)k) dq dp Notice that the boltzmann formula does not say that only those states of a given energy can be populated; it gives non-zero probabilities for populating all states from the lowest to the highest. However, it does say that states of higher energy E, are disfavored by the exp(e /kT)factor, but if states of higher energy have larger degeneracies Q2, (which they usually do), the overall population of such states may not be low. That is, there is a competition between state degeneracy Q,, which tends to grow as the state's energy grows, and exp (E, /kT)which decreases with increasing energy. If the number of particles N is huge, the degeneracy Q2 grows as a high power (lets denote this power as K)of e because the degeneracy is related to the number of ways the energy can be distributed among the n molecules. In fact, K grows at least as fast as N. As a result of Q2 growing as E, the product function P(E)=E exp(-E/kT)has the form shown in Fig. 7.1 (for K=10) PAGE 3PAGE 3 P(q,p) = h-M exp (- H(q, p)/kT)/Q, where H is the classical Hamiltonian, h is Planck's constant, and the classical partition function Q is Q = h-M ò exp (- H(q, p)/kT) dq dp . Notice that the Boltzmann formula does not say that only those states of a given energy can be populated; it gives non-zero probabilities for populating all states from the lowest to the highest. However, it does say that states of higher energy Ej are disfavored by the exp (- Ej /kT) factor, but if states of higher energy have larger degeneracies Wj (which they usually do), the overall population of such states may not be low. That is, there is a competition between state degeneracy Wj , which tends to grow as the state's energy grows, and exp (-Ej /kT) which decreases with increasing energy. If the number of particles N is huge, the degeneracy W grows as a high power (let’s denote this power as K) of E because the degeneracy is related to the number of ways the energy can be distributed among the N molecules. In fact, K grows at least as fast as N. As a result of W growing as EK , the product function P(E) = EK exp(-E/kT) has the form shown in Fig. 7.1 (for K=10)