Statistical Mechanics - Physics at Oregon State University

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142 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS. X ⇒ 1 h3N d X (7.9) If the particles are identical, however, any permutation of the coordinates gives the same state, and we have to replace the sum by ⇒ 1 N!h3N d X (7.10) As a result, the classical partition function is equal to 1 Z = N!h3N d Xe −βH( X) and the density matrix is X ρ( X) = 1 Z e−βH( X) (7.11) (7.12) 7.2 Classical formulation of statistical mechanical properties. Entropy. The entropy follows from the information-theoretical definition 6.29 and we find S(T, V, N) = − kB ZN!h 3N d Xe −βH( X) 1 log Z e−βH( X) = kB ZN!h 3N d Xe −βH( X) log(Z) + βH( X) = kB log Z + kBβU (7.13) Note that the volume V is included in this formula, it simply limits the integrations over the coordinates xi. As expected, our equation leads to F = U − T S and our definitions are equivalent to the old definitions of the entropy. This was already shown explicitly in the previous chapter for the canonical ensemble. Density of states. The micro-canonical ensemble can be treated in a similar way. In this case the density matrix is equal to ρ( X) = Cδ(U − H( X)) (7.14)
7.2. CLASSICAL FORMULATION OF STATISTICAL MECHANICAL PROPERTIES.143 where the normalization constant C follows from Trρ = 1, or C −1 1 = D(U, V, N) = N!h3N d Xδ(U − H( X)) (7.15) Note that in classical mechanics there are no problems with discrete levels, the set of possible values for H( X) is continuous. If the particles are not identical, the factor N! should be omitted. The function D(U) is called the density of states. This density of states also relates the micro-canonical and the canonical ensemble via Z = 1 N!h3N which leads to d Xe −βH( X) = 1 N!h3N d X Z(T, V, N) = duD(u, V, N)e −βu duδ(u − H( X))e −βu (7.16) (7.17) The partition function is the Laplace transform of the density of states. In many cases it is easier to evaluate Z and then this relation can be inverted to obtain the density of states D. One important technical detail is that the energy integral is bounded at the lower end. There is a ground state energy, and a minimum energy level. We can always shift our energy scale so that the minimum energy is zero. In that case we can write D(U, V, N) = 1 2πı Grand partition function. c+ı∞ c−ı∞ dte st Z(t, V, N) (7.18) The grand canonical ensemble follows from the density matrix defined in the much larger space of states with all possible numbers of particles. The grand partition function now involves a sum over all possible combinations of particles and is equal to Z(T, µ, V ) = 1 N!h3N d Xe −β(H( X,N)−µN) (7.19) This leads to N Z(T, µ, V ) = N e µN k B T Z(T, N, V ) (7.20) which is exactly the same relation as we derived in before. This shows that our classical formalism is consistent with the quantum statistical approach. In deriving classical statistical mechanics from its quantum analogue, the factors N! and h 3N come in naturally. It is also possible to postulate classical statistical mechanics directly, but then these factors have to be introduced in some ad-hoc fashion.
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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10 CHAPTER 1. FOUNDATION OF STATIST
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24 CHAPTER 2. THE CANONICAL ENSEMBL
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44 CHAPTER 3. VARIABLE NUMBER OF PA
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68 CHAPTER 4. STATISTICS OF INDEPEN
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90 CHAPTER 5. FERMIONS AND BOSONS
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192 CHAPTER 9. GENERAL METHODS: CRI
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230 APPENDIX A. SOLUTIONS TO SELECT
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Statistical Mechanics - Physics at Oregon State University

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