第一次作业

热力学与统计物理 姓名：郭忠智 学号：2014301020087 班级：材料班

Problem 1.1. Approach to equilibrium

Suppose that we divide a box into three equal parts and place N particles in the middle third of the box.6 The particles are placed at random with the constraint that no two particles can be closer than the length parameter σ. This constraint prevents the initial force between any two particles from being too big, which would lead to the breakdown of the numerical method used to solve the differential equations. The velocity of each particle is assigned at random and then the velocity of the center of mass is set to zero. At t = 0, we remove the “barriers” between the three parts and watch the particles move according to Newton’s equations of motion. We say that the removal of the barrier corresponds to the removal of an internal constraint. What do you think will happen? Program ApproachToEquilibriumThreePartitions implements this simulation.7 Double click on the jar file to open the program, and click the Start button to begin the simulation. The program shows the motion of the particles in the box and plots the number of particles in the left (n1), center (n2), and right (n3) part of the box as a function of time. The input parameter is N, the number of particles initially in the center cell, so that n1 = 0, n2 = N, and n3 = 0 at t = 0. Give your answers to the following questions before you do the simulation.
(a) Does the system appear to show a direction of time for N = 6?
答：是的。
(b) What is the nature of the time dependence of n1, n2, and n3 as a function of the time t for N = 27? Does the system appear to show a direction of time? Choose various values of N that are multiples of 3 up to N = 270. Is the direction of time better defined for larger N?
答：当N=27时，由于初始时刻所有粒子全部集中在中间的区域，两边没有粒子，每两个粒子之间都会有力的作用，在本例子中是Lennard-Jones potential给出的相互作用，n1, n2, 和 n3会随时间的改变而改变，因为在粒子集合成的一团粒子间粒子会因为相互作用力而运动、碰撞，大量的相互作用会使粒子更倾向于往没有阻碍作用的空间移动，大量粒子表现出来的行为就是扩散。N=27时也是如此，同样的，只要有粒子间的相互作用存在，以上行为就应该发生，只是N的数值越大，时间的方向性会越好。
(c) The time shown in the plots is in terms of $σ(m/ϵ)^{1/2}$, where σ and ǫ are the length and energy parameters of the Lennard-Jones potential in (1.1) and m is the mass of a particle. Verify that this combination has units of time. For argon σ = 3.4× 10−10 m, ϵ = 1.65 × 10−21 J, and m = 6.69× 10−26 kg. What is the value of $σ(m/ϵ)^{1/2}$ for argon? How much real time has elapsed if the program shows that t = 100?
答：$σ(m/ϵ)^{1/2}$=$3.4 × 10^{-21} m × [(6.69×10^{-26} kg)/(1.65×10^{-21} J)]^{1/2}$ =$2.16 ×10^{-12} s$
t = $100 × 2.16 ×10^{-12} s$= $2.16 ×10^{-10} s$

(d) To better understand the direction of time make a video8 of the motion of the positions of 270 particles starting from t = 0. Run the simulation until the particles are approximately equally divided between the three regions. Run the video both forward and backward. Can you tell by just looking at the video which direction is forward? Repeat for increasing values of N. Does your conclusion about the direction of time become more certain for larger N?
答：通过观看三个部分粒子数接近相等时的录像，能分辨前进与后退。说明随着粒子数的变大，三个部分的粒子数随时间的改变越来越连续。即在达到平衡之前，N越大，时间的方向性越明显。
Repeat part (d) but start the video after the particles are distributed equally among the three regions, say at t = 20 as in Figure 1.2. Is the direction of time obvious now? Repeat for various values of N.
Figure 1.2

答：当系统达到三个部分粒子数近似相等以后，就很难分辨录像中的前进和后退了，也就是说，这个时候系统没有了时间的方向性，无论N怎么增大，都是如此。
(e) *After n1, n2, and n3 first become approximately equal for N = 270, reverse the time andcontinue the simulation. Reversing the time is equivalent to letting t → −t and changing the signs of all the velocities. Do the particles return to the middle third of the box? Do the simulation again, but let the system evolve longer before the time is reversed. What happens now? Are your results consistent with the fact that Newton’s equations of motion are deterministic and time reversal invariant so that reversing the velocities should exactly retrace the original particle trajectories? (See Problem 1.9 for a discussion of the extreme sensitivity of the trajectories to very small errors.)
答：在时间过了15个单位后进行翻转操作，得到如下曲线：

这表明，系统已经不能回到初始状态，于牛顿运动定律给出的预测并不一样。
(f) Program ApproachToEquilibriumTwoPartitions initially divides the box into two partitions rather than three. Run the program and verify that the simulation shows similar qualitative behavior. Explain the use of toroidal boundary conditions.
答：运行所得结果如下：

这说明了模拟和真实情况是相似的。
模拟中用到的周期性边界条件是模拟实现成功的关键。这模拟了我们观察的那一堆粒子所处的环境，要知道它们并没有被束缚在一个容器里，它们的周围仍然存在着大量的粒子，周期性边界条件保证了这一点，所以能实现模拟。

Problem 1.2. Particles in a box

(a) Before you run the program describe what you think will be the qualitative behavior of n(t), the time-dependence of the number of particles on the left side of the box.
答：n(t)在t = 0 时刻与N相等，随着t 的增大，n(t)会在0 < t < t_0时间段内几乎一直递减，n(t)增大的几率很小；在t > $t_0$ 时间会在N/2上下浮动，其中$t_0$>0。
(b) Run the program and describe the behavior of n(t) for various values of N. Does the system approach equilibrium? How would you characterize equilibrium? In what sense is equilibrium better defined as N becomes larger? Does your definition of equilibrium depend on how the particles were initially distributed between the two halves of the box?
答：N越大，初始态离平衡态越远，即要达到平衡态所用时间越长。
平衡态的特征：当到达一定时间后，左边部分的粒子数的平均值接近于N/2，此时系统达到平衡态。
平衡态与系统的初始状态无关。
(c) When the system is in equilibrium, does the number of particles on the left-hand side remain a constant? If not, how would you describe the nature of equilibrium?
答：不是；平衡态是系统的统计结果，表征平衡态的物理量是统计量，不能以某一个时刻的值来进行判断，必须是该物理量的统计值才能表征平衡态。
(d) If $N \ge32$, does the system return to its initial state during the time you have patience to watch the system?
答：不会。
(e) How does $\bar n$, the mean number of particles on the left-hand side, depend on N after the system has reached equilibrium? For simplicity, the program computes various averages from the time t = 0. Why would such a calculation not yield the correct equilibrium average values? Use the Zero Averages button to reset the averages.
答：$\bar{n}≈N/2$；$\bar{n}$是一个统计量，系统不可能达到绝对的平衡态，所以统计平均值也和N/2严格相等。
(f) Define the quantity σ by the relation9

$$σ^2= \bar {(n-\bar n )^2 }$$ (1.2)
What does σ measure? What would be its value if n were constant? How does σ depend on N? How does the ratio σ/n depend on N? We say that σ is a measure of the fluctuations of n about its mean, and σ/n is a measure of the relative fluctuations of n.
答：σ表征n(t)偏离平均值的程度，称之为方差；
如果n(t)是连续的， $$σ^2 = 1/T ∫_{t_0}^{t_1} (n-\bar n )^2 dt,$$
$$n ̅ = 1/T\int_{t_0}^{t_1}\bar{n}(t)dt,$$
$$T = t_1 – t_0 ;$$
N越大，σ越大，$σ⁄\bar{n}$不变。

Problem 1.3. Counting microstates

(a) Calculate the number of possible microstates for each macrostate n for N = 8 particles. What is the probability that n = 8? What is the probability that n = 4? It is possible to count the number of microstates for each n by hand if you have enough patience, but because there are a total of 28 = 256 microstates, this counting would be very tedious. An alternative is to obtain an expression for the number of ways that n particles out of N can be in the left half of the box. Motivate such an expression by enumerating the possible microstates for smaller values of N until you see a pattern.
答：N = 8 时，总共有2^8 = 256 种微观态。n = 8 的概率为1/256 ；n = 4 的概率为$C_8^4/2^8$= 70/256。
(b) The macrostate with n = N/2 is much more probable than the macrostate with n = N. Why?
答：因为n = N 只包含一种微观态，这时所有粒子只有一种选择，而n = N/2时，粒子可以有更多方式来选择待在右边还是左边，这时待在左边和待在右边均能满足条件，这时有$C_N^{N/2}$种选择方式，比n = N是的一种选择方式多，所以n = N/2时概率更大。

Problem 1.4. Different phases

(a) Program LJ2DMD simulates an isolated system of N particles interacting via the Lennard-Jones potential. Choose N = 144 and L = 18 so that the density ρ = N/L2 ≈ 0.44. The initial positions are chosen at random except that no two particles are allowed to be closer than the length σ. Run the simulation and satisfy yourself that this choice of density and resultant total energy corresponds to a gas. What are your criteria?
答：选择粒子数为256时，挑选初始分子速度，直到每个分子速度为0.99999时，粒子的运动看起来才像气体。通过观察到稳定的压强、稳定的温度、以及粒子运动时不会成团和没有观察到较大的空白区域这几点来进行判断。
(b) Slowly lower the total energy of the system. (The total energy is lowered by rescaling the velocities of the particles.) If you are patient, you will be able to observe “liquid-like” regions. How are they different from “gas-like” regions?
答：看起来像液体的区域的粒子会一直聚集在一起，偶尔会有几个粒子脱离，但数量很少；与前者不同，想气体一样的粒子运动速率较大，整个区域中不会出现粒子成团和较大空白区的现象。
(c) If you decrease the total energy further, you will observe the system in a state roughly corresponding to a solid. What are your criteria for a solid? Explain why the solid that we obtain in this way will not be a perfect crystalline solid.
答：固体：粒子会聚集成一团或者分裂的几团，团不再发生位移，每个粒子在其平衡位置振动但不会离开次位置，气压为0。
我们得到的固体是从杂乱无章的“气体”凝聚而来，没有完整的晶体结构。
(d) Describe the motion of the individual particles in the gas, liquid, and solid phases.
答：气体：分子运动速度快，分子间距离大，分子无固定位置；
液体：分子运动速度比气体分子慢，分子间距离比气体小，无固定位置但许多分子会聚集在一起；
固体：分子聚集成团，分子的运动仅仅在其平衡位置上振动，分子间距离小
(e) Conjecture why a system of particles interacting via the Lennard-Jones potential in (1.1) can exist in different phases. Is it necessary for the potential to have an attractive part for the system to have a liquid phase? Is the attractive part necessary for there to be a solid phase? Describe a simulation that would help you answer this question.
答：势能的吸引部分给粒子凝聚在一起提供了可能性，在一定范围内，粒子间距离越小，系统能量越低，这也就是符合Lennard-Jones potential相互作用的粒子能存在于不同的相的原因。

Problem 1.5. Nature of temperature

(a) Summarize what you know about temperature. What reasons do you have for thinking that it has something to do with energy?
答：温度是表示物体冷热程度的物理量，微观上来讲是物体分子热运动的剧烈程度。国际单位为热力学温标 (K)。目前国际上用得较多的其他温标有华氏温标 (°F)、摄氏温标 (°C)和国际实用温标。从分子运动论观点看，温度是物体分子运动平均动能的标志。温度是大量分子热运动的集体表现，含有统计意义。对于个别分子来说，温度是没有意义的。根据某个可观察现象(如水银柱的膨胀)，按照几种任意标度之一所测得的冷热程度。
从生理上的感觉来看，两个由同种材料构成的体积相同的物体，总是温度高的那个物体的能量高，因为由$E_k = 3NkT/2$(单原子分子，双原子分子$E_k = 5NkT/2$),物体温度越高，其分子的平均运动动能越高。
(b) If you add energy to a pot of boiling water, does the temperature of the water change?
答：当给沸水继续增加能量时，其温度不会升高，此时沸水吸收的能量会等量的释放出来。
(c) Discuss what happens to the temperature of a hot cup of coffee. What happens, if anything, to the temperature of its surroundings?
答：当把一杯热咖啡放置于桌面上，热咖啡的温度会逐渐降低，其初始的能量会被其周围的环境吸收，直到咖啡温度和周围环境温度变得一致，咖啡和其周围环境的能量交换达到平衡。

Problem 1.6. Identification of the temperature

(a) Consider two systems of particles A and B which interact via the Lennard-Jones potential in (1.1). Use Program LJThermalEquilibrium. Both systems are in a square box with linear dimension L = 12. In this case toroidal boundary conditions are not used and the particles also interact with fixed particles (with infinite mass) that make up the walls and the partition between them. Initially, the two systems are isolated from each other and from their surroundings. We take NA = 81, ǫAA = 1.0, σAA = 1.0, NB = 64, ǫBB = 1.5, and σBB = 1.2. Run the simulation and monitor the kinetic energy and potential energy until each system appears to reach equilibrium. What is the mean potential and kinetic energy of each system? Is the total energy of each system fixed (to within numerical error)?
答：运行程序后会看到左（红色）右（绿色）两部分的粒子的势能和动能都明显不同，因为彼此之间没有能量交换。平均势能是两个系统各自的粒子势能的平均值，平均动能是各自的粒子的动能的平均值，不是两个系统的总能量，也不是各自的总能量。

(b) Remove the barrier and let the particles in the two systems interact with one another.14 We choose ǫAB = 1.25 and σAB = 1.1. What quantity is exchanged between the two systems? (The volume of each system is fixed.)
答：让两个系统交换能量，去除挡板，运行程序后看到明显改变的量是每个粒子的动能，到达平衡后看到二者平均动能变得相等了，而两种粒子的势能均只是受到微小扰动，平均势能没有受到大的影响。

(c) After equilibrium has been established compare the average kinetic and potential energies of each system to their values before the two systems came into contact.
答：达到平衡时的平均动能的值居于初始时两个系统各自平均动能之间，而达到平衡时的各自平均势能依旧和初始状态的平均势能相差不大。

(d) We seek a quantity that is the same in both systems after equilibrium has been established. Are the average kinetic and potential energies the same? If not, think about what would happen if you doubled the number of particles and the area of each system. Would the temperature change? Does it make more sense to compare the average kinetic and potential energies or the average kinetic and potential energies per particle? What quantity becomes the same once the two systems are in equilibrium? Do any other quantities become approximately equal? What can you conclude about the possible identification of the temperature in this system?
答：可以看到达到平衡后系统的两种粒子的平均动能值变得相等，而对于每一个粒子来说，它们各自的动能都有所不同，但是都会在平均动能附近涨落。根据日常的生活经验，两个系统的温度在系统达到平衡时变得相同，故温度和粒子的平均动能之间是有必然的联系的。

Problem 1.7. Thermometers

(a) Describe some of the simple thermometers with which you are familiar.
答: 根据某个可观察现象，按照物理关系可以制得各种温度计。生活中常用的有水银温度计、酒精温度计。
(b) On what physical principles do these thermometers operate?
答：水银温度计和酒精温度计使用的物理原理是物体的热胀冷缩、和两物体互相接触会达到热平衡，以水银或者酒精的温度来衡量物体的温度。
(c) What requirements must a thermometer have to be useful?
答：一个温度计要变得有用，必须有参考温度点、根据物理关系计算得的可读刻度，还要将其应用在物理关系正确的条件下。

Problem 1.8. The demon and the ideal gas

Program IdealThermometerIdealGas simulates a demon that exchanges energy with an ideal gas of N particles in d spatial dimensions. Because the particles do not interact, the only coordinate of interest is the velocity of the particles. In this case the demon chooses a particle at random and changes each component of its velocity by an amount chosen at random between −∆ and +∆. For simplicity, we set the initial demon energy Ed = 0 and the initial velocity of each particle equal to +v0ˆ x, where v0 = (2E0/m)1/2/N. E0 is the desired total energy of the system, and m is the mass of the particles. We choose units such that m = 1; the energy and momentum are measured in dimensionless units (see Section 1.11).
(a) Run the simulation using the default parameters N = 40, E = 40, and d = 3. Does the mean energy of the demon approach a well-defined value after a sufficient number of energy exchanges with the system? One Monte Carlo step per particle (mcs) is equivalent to N trial changes.
答：是的。
(b) What is Ed, the mean energy of the demon, and E, the mean energy of the system? Compare the values of Ed and E/N.
答：$E_d = 0.658， E/N = 39.342/40 =0.98355.$ $E_d$是系统被温度计吸收的能量，E/N是每个粒子的平均动能，可以看出，$E_d小于 E/N。$
(c) Fix N = 40 and double the total energy of the system. (Remember that Ed = 0 initially.) Compare the values of Ed and E/N. How does their ratio change? Consider other values of N ≥ 40 and E and determine the relation between Ed and E/N.
答：$E_d = 1.308，E/N = 78.692/40 = 1.9673，E/N的值约为(b)中2倍。$
$N = 50，E=80：E_d=1.056，E/N=1.5789;$
$N = 60，E=80：E_d=0.885，E/N=1.3184$
$N = 70，E=80：E_d=0.756，E/N=1.3121$
$E_d≈3/2•E/N$
(d) You probably learned in high school physics or chemistry that the mean energy of an ideal gas in three dimensions is equal to 3/2 NkT, where T is the temperature of the gas, N is the number of particles, and k is a constant. Use this relation to determine the temperature of the ideal gas in parts (b) and (c). Our choice of dimensionless variables implies that we have chosen units such that k = 1. Is Ed proportional to the temperature of the gas?
答：$(b) : T =2/3•E/N = 0.6557 ,$ $(c) : T =2/3•E/N = 1.1315、1.0526、0.8789、0.7547$
可以看出，在忽略微小误差后得到$T = E_d$
(e) Suppose that the energy momentum relation of the particles is not ǫ = p2/2m, but is ǫ = cp, where c is a constant (which we take to be 1). Consider various values of N and E as you did in part (c). Is the dependence of Ed on E/N the same as you found in part (d)? We will find (see Problem 4.30) that Ed is still proportional to the temperature.
答：是的。
(f) *After the demon and the system have reached equilibrium, we can compute the histogram H(Ed)∆Ed, the number of times that the demon has energy between Ed and Ed + ∆Ed. The bin width ∆Ed is chosen by the program. This histogram is proportional to the probability p(Ed)∆E that the demon has energy between Ed and Ed + ∆E. What do you think is the nature of the dependence of p(Ed) on Ed? Is the demon more likely to have zero or nonzero energy?
答：遵循指数分布，E_d的值更趋向于0，或者说E_d=0的概率最大。
(g) Verify the exponential form of p(Ed) = Ae−βEd, where A and β are parameters.16 How does the value of 1/β compare to the value of Ed? We will find that the exponential form of p(Ed) is universal, that is, independent of the system with which the demon exchanges energy, and that 1/β is proportional to the temperature of the system.
答: 由$\int_0^\infty Ae^{-\beta x} dx=-A/\beta$，得$A=- \beta$.
所以$\bar{E}_d= \int_0^\infty Axe^{-\beta x}dx=-A/{\beta}^2=1/{\beta}$
(h) Discuss why the demon is an ideal thermometer.
答：因为demon从系统移走的能量小，并且移走的能量的值和系统温度值相等，通过读取demon的温度可以知道系统精确的温度值。

∗Problem 1.9. Irreversibility

Program SensitivityToInitialConditions simulates a system of N = 11 particles with the special initial condition described in the text. Confirm the results that we have discussed. Perturb the velocity of particle 6 and stop the simulation at time t after the change and reverse all the velocities. Confirm that if t is sufficiently short, the particles will return approximately to their initial state. What is the maximum value of t that will allow the particles in the system to return to their initial positions if t is replaced by −t (all velocities reversed)?
答：约为30个时间单位。

Problem 1.10.Independence of initial conditions

Show that if the number of particles on the left-hand side of the box at t = 0 is equal to n(0) rather than N/2, the solution of (1.7) is

$$n(t)=N/2-N/2[1-2n(0)/N] e{-2t/N}$$ (1.9)
Note that n(t) → N/2 as t → ∞ independent of the value of n(0).
答：方程
$$d\bar{n}(t)/dt=1-2\bar{n}(t)/N$$ (1.7)
的一般解为：
$$\bar{n}(t)=(N-Ce^{-2t/N})/2$$
代入初始条件 t = 0, n = 0 得：
$$n(t)=N/2-N/2[1-2n(0)/N] e{-2t/N}$$
即式(1.9)

Problem 1.11. Common derivatives

Calculate the derivative with respect to x of the following functions:

$$e^x, (e^3)^x, (e^a)^x, lnx, lnx2, ln3x, ln1/x, sinx, cosx, sin3x, cos2x.$$
解：
$$e^x , 3e^(3x) ,ae^(ax) ,1/x , 2/x ,1/3x , -1/x, cosx, -sinx, 3cos3x, -2sin2x$$

Problem 1.12. Common integrals

Calculate the following integrals:

$$\int_{1}^2 1/2x^2dx , \int_{1}^2 1/4xdx, \int_{1}^2 e^{3x}dx , and∫x^{-γ} dx$$
解： $$0.25, 0.1733, 127.781, (-x)^{-(1+γ}/(1+γ)+C$$

Problem 1.13. Partial derivatives

Calculate the partial derivatives of x^2 + xy + 3y^2 with respect to x and y.
解：

$$∂(x^2+ xy + 3y^2)/∂x = 2x + y$$
$$∂(x^2+ xy + 3y^2)/∂y = x + 6y$$

Problem 1.14. Taylor series approximations

Calculate the first three nonzero terms of the Taylor series approximations about x = 0 for the following functions:

$$e^{ax}, ln(1 + x),and(1 + x)^n.$$ (1.11)
where a and n are constants.
解：
$$e^{ax} ≈ 1 + ax + (a^2 x^2)/2$$
$$ln(1 + x) ≈ x/(1+x) - x^2/[2(1+x)]^2 + x^3/[3(1+x)]^3$$
$$(1+x)^n≈1+ nx/1!+(n(n-1) x^2)/2!$$