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Unformatted text preview: 1 Homework 3 (due October 20) Problem 3-1 (4 pts.) Consider a dielectric ideal gas in a non-uniform electric f eld. The linear polarizability of each molecule is (i.e. d = E ). a) Based on the results of the previous homework, write the Helmholtz Free energy as a functional over local density ( r ) , and local electric f eld E ( r ) , i.e. F = Z f , E, T d 3 r Hint: do not forget the translational entropy term! b) Let the density in a zero- f eld region be . By requiring the chemical potential of the gas to be constant everywhere, f nd E . c) Find the F uctuation of the polarizttion P within a small region of volume V , assuming that the particles can be exchanged with the rest of the system. d) Find the correlation between the number of particles and themagnetization in the region of volume V : h ( P h P i ) ( N h N i ) i Hint: derive the general expression for this correlator , in terms of derivatives of the Grand potential. Solution a) F = Z T (log 1) E 2 2 !...
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