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1. Show that in the canonical ensemble we have CV = T −2 h(E − hEi) 2 i (1) which shows that the heat capacity is related to the magnitude of fluctuations of the energy of a system about its mean value. 2. Consider a one-dimensional classical gas of distinguishable particles moving in a single-particle potential Ui = κx 2 i , so that the energy of the system is E = ∑ N i (p 2 i /2m + m 2 ω2x 2 i ) with ω = √ 2κ/m. Assume that the system can move anywhere in the 2N-dimensional phase space. (a) Calculate the volume of phase space with energy below some energy E, and use this to calculate the number of states with energy at or below E, Σ(E), assuming the fiducial phase space volume is h for this onedimensional system. You may need the result Z ∞ −∞ dx1… Z ∞ −∞ dxMΘ(R 2 − M ∑ i=1 x 2 i ) = π M/2 Γ(M/2 + 1) R M, (2) which is just the M-dimensional volume of an M-sphere of radius R. Here Θ(x) = ( 1 for x ≥ 0 0 for x < 0 (3) is the Heaviside step function. (b) Calculate the entropy of the gas in the large N limit working in the microcanonical ensemble. (c) Write down the energy of the gas in terms of the temperature using your result from part (b). (d) Calculate the Helmholtz free energy, entropy and energy of the gas using the canonical ensemble. These should be the same as the results you found in the microcanonical ensemble. (e) Calculate the Helmholtz free energy from the canonical partition function for a system of N one-dimensional quantum harmonic oscillators with single-particle energies ei = h¯ 2ω(2ni + 1 2 ) (with ni = {0, 1, 2, …, ∞}) and verify that you arrive at the same result as for the classical expression in the low density limit aside from a zero-point energy offset. 1 3. If the “free volume” V¯ of a classical gas is defined by the equation V¯ N = Z d 3 r1…d 3 rN exp[β(hUi − U(~q))] where hUi is the average potential energy of the system and U(~r1, …,~rN) = ∑i a ∞ forrij < a where a is the radius of the spheres and rij is the distance between particles i and j.) 4. Consider a low-density, relativistic gas of particles moving in one-dimension (i.e. e = p p 2 + m2 ). Show that h p 2 e i = T both by the equipartition theorem and by integration over the phase-space density in the canonical ensemble. 2
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