課程名稱︰統計物理(一) Statistical Physics(I)
課程性質︰必修／物理學研究所；選修／應用物理研究所、天文物理研究所
課程教師︰蔡政達
開課學院：理學院
開課系所︰物理學系、物理學研究所、應用物理研究所、天文物理研究所
考試日期（年月日）︰2018年4月9日
考試時限（分鐘）：120分鐘
試題 :
1. Assuming that the entropy S and the statistical number Ω of a physical
system are related through an arbitrary function form
S = f(Ω),
show that the additive character of S and the multiplicative character of
Ω necessarily that the function f(Ω) be of the form
S = k lnΩ.
2. A mole of argon and a mole of helium are contained in vessels of equal
volume. If argon is at 300 K, what should the temperature of helium be so
that the two have the same entropy?
3. The generalized coordinates of a simple pendulum are the angle displacement
θ and the angle momentum $ml^2$\dot{θ}. Study, both mathematically and
graphically, the nature of the corresponding trajecories in the phase space
of the system, and show that the area A enclosed by a trajecory is equal to
the product of the total energy E and the time period τ of the pendulum.
4. Derive (i) an asymptotic expression for the number of ways in which a given
energy E can be distributed among a set of N one-dimensional harmonic
oscillators, the energe eigenvalues of the oscillators being
1 h
(n + ─) ──ω ; n = 0,1,2,...,
2 2π
and (ii) the corresponding expression for the "volume" of the relevant
region of the phase space of this system. Establish the corespondence
betaeen the two results, showing the conversion factor $ω_0$ is precisly
$h^N$.
5. Making the use of the fact that the Helmholtz free energy A(N,V,T) of a
thermodynamic system is an extensuve property of the system, show that
\partial A \partial A
N(──────) + V(──────) = A.
\partial N V,T \partial V N,T
[Note that this result impies the well-known relationship:
Nμ = A + PV (= G).]