[試題] 100下 王名儒 統計物理一 期中考
課程名稱︰統計物理(一)
課程性質︰必修
課程教師︰王名儒
開課學院：理學院
開課系所︰物理所
考試日期（年月日）︰101.4.18
考試時限（分鐘）：150min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
[1]Applying the concept of grand canonical ensemble, write down the grand
partition function for an ideal mono-atomic gas (particle mass m) system
in a box with volume V and in thermal equilibrium with a heat bath at
temperature T. (5%) Find the average number of particles, pressure,
Helmholtz free energy, and entropy of this system. (20%)
[2]Determine the partition function for an extremely relativistic gas
consisting of N identical particles with almost zero mass which obey the
energy-momentum relationship ε = pc, where c stands for the speed of
light. (5%) Note that the system occupies a volume of V and is in thermal
equilibrium at temperature T. Determine the energy density of this system
as a function of pressure. (5%) What's the specific heat and the chemical
potential of this system? (10%) Using the inverse Laplace transform method
to derive an expression for the density of states, g(E), of this system.
(5%)
[3]Write down the entropy (S = klnΩ) of an isolated paramagnetic salt with
total N atoms under the magnetic field H, assuming the value of the
magnetic moment of each atom can be either +μ or -μ, and the total
internal energy of the salt is E. (5%) Find the probabilities P+1 and P-1
for a single atom with spin parallel and anti-parallel to the magnetic
field, respectively. (5%) Show that the entropy is also equal to
-Nk( P+1*ln(P+1) + P-1*ln(P-1)). (5%) Find the temperature of this salt.
Can it be negative? (5%) What is the spin covariance, <σ1σ2> - <σ1><σ2>
between two atoms in this salt? The value of σ can be either +1 (parallel)
or -1 (anti-parallel) and < > stands for ensemble average. (5%)
(此題中，P+1 的 "+1" 為下標，P-1 的 "-1" 為下標)
[4]The following question is based on a toy model for a rubber band. Consider
the rubber band as a long polymer chain consisting of N molecules of length
l linked together end-to-end. The orientation between two molecules can be
either zero degree or 180 degree and there is no energy difference between
the two different orientations. Assuming this long polymer chain is hanged
vertically at one end and holds an object with mass m at the other end,
determine the equilibrium length of this rubber band at temperature T. (10%)
You need to determine the entropy as a function of the total length, L, of
this polymer chain first in order to find the equilibrium length. (10%)
The gravitational field on earth is denoted by g. If the temperature is
increased, what will happen to the length of this polymer chain? (5%)
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