statistical mechanics topic. please don't write wrong answer..

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statistical mechanics topic. please don't write wrong
answer..

Statistical Mechanics Topic Please Don T Write Wrong Answer 1 (60.94 KiB) Viewed 41 times

2π a-b w(f)= K(0) for 0 <f≤ a 2л 2π a-b 25K(0) { a+b = rf} for <fs ab 2л a+b 0 for <f<∞0. 27 Verify that the function w(f) satisfies the requirement (15.5.16). [Note that, in the limit b→ 0, we recover the situation pertaining to equations (15.5.18).] 5.15. (a) Show that the mean square value of the variable Y(t), defined by the formula u+1 Y(t) = y(u)du, = [y(u)d where y(u) is a statistically stationary variable with power spectrum w(f), is given by (Y2(t)) = 1 272 zu {1-cos(27 ft)}df; and, accordingly, w(f) = 4rff (Y² (1)) sin(27 ft)dt 21 (Y² (t)) cos(27 ft)dt. a2 at² For details, see MacDonald (1962), Section 2.2.1. A comparison of the last result with equation (15.5.14) suggests that 1 3² Ky(s) = (Y² (s)). 2 as² 651 (b) Apply the foregoing analysis to the motion of a Brownian particle, taking y to be the v the particle and Y its displacement. 5.16. Show that the power spectra wy(f) and wA (f) of the fluctuating variables v(t) and A(t) tha. in the Langevin equation (15.3.5) are connected by the relation 7² wy(f) = WA(f); 1+(2лft)²¹ a+b 2μ