课程名称︰统计物理(一) 课程性质︰必修 课程教师︰王名儒 开课学院：理学院 开课系所︰物理所 考试日期（年月日）︰101.4.18 考试时限（分钟）：150min 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : [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%) (此题中，P+1 的 "+1" 为下标，P-1 的 "-1" 为下标) [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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