lecture notes on statistical mechanics - Scott Pratt - Michigan State ...

Some students find a nmemonic useful for deriving some of the Maxwell relations. Seehttp://en.wikipedia.org/wiki/Thermodynamic square. However, this is more cute than useful. Asyou will see with some of the more difficult problems at the end of this chapter.1.11 FluctuationsStatistical averages of higher order terms in the energy or the density can also be extracted from thepartition function. For instance, in the grand canonical ensemble (using β = 1/T and α = −µ/Tas the variables),∂ 2 β ln Z(β, α) = ∂ β∂ β ZZ= 1 Z ∂2 βZ −( 1Z ∂ βZ) 2(1.58)= ⟨ E 2 ⟩ − ⟨E ⟩ 2= ⟨ (E − ⟨E⟩) 2⟩ .Thus, the second derivative of ln Z gives the fluctuation of the energy from its average value.Similarly, one can see that∂ β ∂ α ln Z = ⟨(E − ⟨E⟩)(Q − ⟨Q⟩)⟩ (1.59)∂ β ∂ α ln Z = ⟨ (Q − ⟨Q⟩) 2⟩ . (1.60)Fluctuations play a critical role in numerous areas of physics, most notably in critical phenomena.1.12 Problems1. Using the methods of Lagrange multipliers, find x and y that minimize the following function,f(x, y) = 3x 2 − 4xy + y 2 ,subject to the constraint,3x + y = 0.2. Consider 2 identical bosons (A given level can have an arbitrary number of particles) in a2-level system, where the energies are 0 and ϵ. In terms of ϵ and the temperature T , calculate:(a) The partition function Z C(b) The average energy ⟨E⟩. Also, give ⟨E⟩ in the T = 0, ∞ limits.(c) The entropy S. Also give S in the T = 0, ∞ limits.(d) Now, connect the system to a particle bath with chemical potential µ. Calculate Z GC (µ, T ).Find the average number of particles, ⟨N⟩ as a function of µ and T . Also, give theT = 0, ∞ limits.Hint: For a grand-canonical partition function of non-interacting particles, one can state14