课程名称︰统计物理(一) Statistical Physics(I)
课程性质︰必修／物理学研究所；选修／应用物理研究所、天文物理研究所
课程教师︰蔡政达
开课学院：理学院
开课系所︰物理学系、物理学研究所、应用物理研究所、天文物理研究所
考试日期（年月日）︰2018年4月9日
考试时限（分钟）：120分钟
试题 :
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).]