Homework4Solution - Homework 4 Problem 1(see Pathria 3.15...

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Homework 4 Problem 1. (see Pathria, 3.15) Consider classical gas of N relativistic indistinguishable particles in volume V each described by the Hamiltonian H(p,q)=pc . Using canonical ensemble theory calculate 1. Partition function of the gas Q(N,V,T) 2. Helmholtz free energy A(N,V,T) 3. Internal energy U(N,V,T) 4. Entropy of the gas S(N,V,T) 5. Density of states for the gas g(E) . Verify that indeed S=k*ln[g] Now you should really appreciate how easy is to work with canonical distribution compared to microcanonical one as we have done in Problem 3 of Homework 2. Solution 1. Partition function for a single particle is given by /33 2 / 3 1 33 3 3 0 14 4 (,) 2 ( ) pc kT pc kT VV Q V T e d pd q p dpe kT hh h c π −− === ∫∫ As a result () 3 1 8 1 (,,) [ (,) ] !! N N N V kT QNVT QVT NN h c == 2. Helmholtz free energy is ( , , ) ln ( , , ) ln8 3 ln ln ln( ) 3 ln( ) (1 ln8 ) kT ANVT kT QNVT N kT V N N kT N N hc Vk T NkT NkT NkT Nh c =− + + 3. Internal Energy is , / 3 1/ NV AT UNVT kNT T ⎛⎞ ⎜⎟ ⎝⎠ 4. Entropy is , l n 3 l n ( 4l n 8) VN AV k T S N V T Nk Nk Nk TN h c = + + + By the way, expressing it via the energy we obtain l n n ( n 3 VU S N V E Nk Nk Nk h c =+ + +
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which is identical to the solution of problem 3 in homework 2. 5. Density of states for the gas '' 33 3 31 1 / 3 ' 3 3 ' 11 ( 8 ) () (,,) 22 ! ( ) 1( 8 ) ( ( 8 )3 2 3 2 !( ) !( ) (3 1)! !( ) (3 )! ! (3 )! NE ii E NN N x N N N N i N N i Ve gE e QNV d d N h c VE e V N E V N E dx i N hc x N hc N N hc N N hc N β ββ π ππ +∞ −∞ −− === ⎛⎞ == = ⎜⎟ ⎝⎠ ∫∫ which is identical to the result (divided by h 3N
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Homework4Solution - Homework 4 Problem 1(see Pathria 3.15...

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