[試題] 106-2 蔡政達 統計物理(一) 期中考2
課程名稱︰統計物理(一) Statistical Physics(I)
課程性質︰必修／物理研究所；選修／天文物理所、應用物理所
課程教師︰蔡政達
開課學院：理學院
開課系所︰物理研究所、天文物理所、應用物理所
考試日期（年月日）︰2018年5月14日
考試時限（分鐘）：120分鐘
試題 :
Statistical Physics(I) Midterm Examination(2)
1.[20 points]Show that, for a classical ideal gas,
S Q_1 ╭ \partial ln Q_1 ╮
── = ln── + T│ ──────── │
Nk N ╰ \partial T ╯P.
2.[20 points]Determine the work done on a gas and the amount of heat absorbed
by it during a compression from volume V_1 to volume V_2, following yhe law
PV^n = const.
3.[20 points]Show that the entropy of a system in the grand canonical ensemble
can be written as
S = -k Σ P_{r,s} ln P_{r,s},
r,s
where P_{r,s} is given by
exp(-αN_r - βE_s)
P_{r,s} = ─────────── .
Σ exp(-αN_r - βE_s)
r,s
4.[20 points]In the thermodynamics limit (when the extensive properties become
infinitely large, while the intensive ones remain constant), the q-potential
of the system may be calculated by taking only the largest term in the sum
∞
Σ Z^{N_r} Q_{N_r}(V,T) .
N_r = 0
Verify this statement and interpret the result physically.
5.[20 points]Show that expression
U-A ╭ \partial q ╮
S = ─── = kT │ ───── │ - Nk lnz + kq
T ╰ \partial T ╯z,V
for the entropy of a system in the grand canonical ensemble can also be
written as
╭ \partial ╮
S = k │ ───── (Tq) │ .
╰ \partial T ╯μ,V
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