2b. 11 Now suppose the harmonic oscillators were capped at energy 0', Le. they are finite systems of dimension 451' corresponding to the states l0), ll), , Id 1). Thus the total energy is capped at 0 S E S N(d 1), or 0 5 u g d 1.You can think of them as spin-% particles with an external magnetic field if you prefer. For 0' = 2, repeat the above derivation. What happens to T as u > %? What happens for u > %? Answer: Ford = 2, Z = l + 6"} u__6an_ 1 =>_lnlu _ as e+1 _ u . 1 s = ln(2e_E' +'u = 1n(1u)+uln \" u 2d. Let's optimize f under the constraints. Show that in the d-dimensional unconstrained space of distributions {n; ), f is convex-up, (f" is negative-definite, i.e. all its eigenvalues are negative, everywhere) in the physically allowed region, so that there is a unique global maximum. Show that f" must therefore be convex-up in the (d - 2)-dimensional constrained space of distributions, so that there is a unique global maximum in the constrained space. Show that { n* }, where the constrained maximum of f occurs, is in the domain of integration. Use Lagrange optimisation to find this constrained maximum. Define S to be the Lagrange multiplier of the constraint fixing _; in; = u, and show that - BK d-1 n* = Z Z = >e-BK K= 0 Find u as a function of (at fixed d). Since it is difficult to invert u(B) to find B(u), find s and T as functions of B, u(B) and Z(B) Answer: Use two Lagrange multipliers a and B. Define T = N In ( N) - E N. In ( N.) + 0 N - EN. + B E - EN.E. =-N E n; In ( 1 ) + a N - N E n +B E - NE ME . Now, the maximisation requires - = 0. This gives - [In(n;) + 1] - [a] - [BE;] = 0, where we have used dx [x In x] = Inx + 1 This yields the solution2j. Suppose you have N1 > N2 >> 1 particles in systems 1 and 2, which have dimension d1, d2. How much bigger should N1 be than N2 for system 1 to act as a bath? Write a function to numerically solve for the equilibrium , given B1, B2, d1, d2, N7 For -1 = 2 = 1, and some small value for d1 (chosen by you), plot Beg as a function of N1 N2 for a variety of d2. How much bigger should N1 be than N2 for us to be able to use the canonical ensemble approach for system 2, for B > 0 and B