the homework of Statistical and Thermal Physics

Chapter 1

by Defeng kong

student number:2014301020081

This pdf is turned from zuoyebuluo Cmd markdown(https://www.zybuluo.com/mdeditor#516330)

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Problem 1.1.

(a) Does the system appear to show a direction of time for N = 6?

No,It's not very easy to find the direction of time.

(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?

Probability;Yes;Yes

(c) The time shown in the plots is in terms of σ(m/$\epsilon$)^1/2, where σ and $\epsilon$ 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, $\epsilon$ = 1.65 × 10⁻²¹ J, and m = 6.69× 10⁻²⁶ kg. What is the value of σ(m/$\epsilon$)^1/2 for argon? How much real time has elapsed if the program shows that t = 100?

σ(m/$\epsilon$)^1/2 =3.4 × 10⁻¹⁰(6.69× 10⁻²⁶/ 1.65 × 10⁻²¹)^1/2=2.16× 10^-13;t=2.16× 10^-13*100=2.16× 10^-10S.

(d) To better understand the direction of time make a video 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?

Yes;Yes

(e) 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.

Yes

(f) *After n1, n2, and n3 first become approximately equal for N = 270, reverse the time and continue 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.)

Yes，It's can come back; No;No

(g) 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.

Yes,it's similar with the situation of three.Because of the use of toroidal boundary,the box divided into two or three can be regarded as a infinite or a large space.For a large space,it has no boundary for a tiny particle.and for these small systems the fraction of particles near the walls of the container will be significant,the behavior of such a system will be dominated by surface efferts,so the toroidal boundary is reasonable.

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.

The n(t) on the left side of the box will be less and less.

(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?

Yes;The number of particles in two halves will become stable and equal;The range of number of particles become more narrow;No.

(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?

No,it should be a dynamic equilibrium.

(d) If N$\geq$32 does the system return to its initial state during the time you have patience to watch the system?

No

(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.

* $\bar{n}$ is near N/2 that has a small fluctuations.Because they have numerical differential.*

(f) Define the quantity σ by the relation $$\sigma ^{2}=\overline{\left ( n-\bar{n} \right )^{2}}$$

What does σ measure? What would be its value if n were constant? How does σ pend on N? How does the ratio σ/$\bar{n}$ depend on N? We say that σ is a measure of the fluctuations of n about its mean, and σ/$\bar{n}$ is a measure of the relative fluctuations of n.

The fluctuations of n about its mean ;0;from the simulation(I set the N=64;100;150)we can see when N is setted a larger number ,σ will become smaller,also σ/$\bar{n}$ will become smaller.

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.

1/2⁸ ;70/2⁸

(b) The macrostate with n = N/2 is much more probable than the macrostate with n = N. Why?

Because the probability that n=N/2 is higher than that n=N.

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?

The particles move at random.its trajectory is found everywhere in the box.

(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?

The particles arrange more closely and move slowlier.its trajectory is at random just in the small area.

(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.

The particles arrange more densely and move more slowly than 'liquid-like' regions ,it just likes a "vibration" in smaller range near itself;Because the particles of crystalline solid arrange more regularly than the 'solid-like'region.

(d) Describe the motion of the individual particles in the gas, liquid, and solid phases.

The particles move at random and arrange loosely in the gas phases;The particles arrange more closely and move slowlier in the liquid phases ,which still idicates a little out of order.The particles arrange more densely and move more slowly in the solid phases,which idicates the regular of arrangement.

(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.

Because of the difference of the distance between two particles 'r'.they have difference u_{L J};Yes;Yes;

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?

The temperature is the criteria to measure the hot and cold of objects,its expressive methods that I can find has degree centigrade and Fahrenheit;According to the common sense,the water of a pot become more hotter should need more firewood,in other word,more energy.

(b) If you add energy to a pot of boiling water, does the temperature of the water change?

No.

(c) Discuss what happens to the temperature of a hot cup of coffee. What happens, if anything, to the temperature of its surroundings

The hot cup of coffee will be colder and colder up to the temperature of its susrroundings,which changes hardly.

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 N_A = 81, $\epsilon$_AA= 1.0, σ_AA= 1.0, N_B = 64, $\epsilon$_BB= 1.5, and σ_BB=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)?

red:mean potential energy:-3.0 mean kinetic energy:0.08

green:mean potential energy:-4.4 mean kinetic energy:0.13; fixed,-509.31~-509.33

(b) Remove the barrier and let the particles in the two systems interact with one another.We choose $\epsilon$_AB= 1.25 and σ_AB = 1.1. What quantity is exchanged between the two systems? (The volume of each system is fixed.)

mean kinetic energy

(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.

red:kinetic energy:lesser;mean potential energy:more;

green:kinetic energy:more;mean potential energy:lesser;

(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.

Kinetic energies are the same while potential energies are not;No;No;Temperature;NO;Temperature is proportional to the averge kinetic energy per particle.

Problem 1.7. Thermometers

(a) Describe some of the simple thermometers with which you are familiar.

The mercurial thermometer made of mercury is common,which its range is between -10^。C ^。c.The minimum scale can be 0.1^。c or 1^。c.also we have other thermometers like semiconductor thermometer and Light thermometer

(b) On what physical principles do these thermometers operate?

Mercurial thermometer takes use of the physical priciple that The object can become bilges when meets heat and shrink when cold;The semiconductor thermometer takes use of the temperature characteristic of the semiconductor;

(c) What requirements must a thermometer have to be useful?

It can measure the temperture quickiy,have little error and appropriate range of temperture,and so on.

Problem 1.8. The demon and the ideal gas

(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.

Yes

(b) What is$\bar{E}$_d , the mean energy of the demon, and $\bar{E}$, the mean energy of the system? Compare the values of $\bar{E}$_d and$\bar{E}$ /N.

0.656;39.344;$\bar{E}$_d<$\bar{E}$ /N.1.5$\bar{E}$_d =$\bar{E}$ /N.

(c) Fix N = 40 and double the total energy of the system. (Remember that E_d = 0 initially.) Compare the values of$\bar{E}$_d and$\bar{E}$ /N.How does their ratio change? Consider other values of N ≥ 40 and E and determine the relation between $\bar{E}$_d and $\bar{E}$/N.

No change;1.5$\bar{E}$_d=$\bar{E}$ /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 3/2NkT, 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 $\bar{E}$ proportional to the temperature of the gas?

because $\bar{E}$=1.5N$\bar{E}$_d , $\bar{E}$=1.5NkTsoT=$\bar{E}$_d/k when k=1 $\bar{E}$_dis propotional to the temperature of the gas.

(e) Suppose that the energy momentum relation of the particles is not $\epsilon$ = p²/2m, but is $\epsilon$ = 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 $\bar{E}$_d and$\bar{E}$ /N the same as you found in part (d)? We will find (see Problem 4.30) that $\bar{E}$_d is still proportional to the temperature.

Yes

(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?

The propability gepend on the Ed.The time of Ed is proportional to the propability；Yes.

(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 energ, and that 1/β is proportional to the temperature of the system.

Value of 1/β is equal to value of Ed

(h) Discuss why the demon is an ideal thermometer.

From (g),we can see that the exponential form of p(Ed)is independent of the system with which the demon exchanges energy.so the demon is independent of the system.the demon can be an ideal thermometer.

∗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)?

the maximum value of t is about 20

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, the solution of (1.7) is

$$\bar{n}\left ( t \right )=\frac{N}{2}-\frac{N}{2}\left [ 1-\frac{2n\left ( 0 \right )}{N} \right ]e^{-2t/N} (1.9)$$

Note that n(t) → N/2 as t → ∞ independent of the value of n(0).

$$\frac{\mathrm{d} \bar{n}}{\mathrm{d} x}=1-\frac{2\bar{n}\left ( t \right )}{N}$$

$$\Rightarrow \frac{d\bar{n}}{-\frac{N}{2}+\bar{n}\left ( t \right )}=-\frac{2dt}{N}$$

$$\Rightarrow \bar{n}\left ( t \right )=Ae^{-2t/N}+\frac{N}{2}$$

$$\bar{n}\left ( 0 \right )=n\left ( 0 \right )\Rightarrow A= -\frac{N}{2}\left [ 1-\frac{2n\left ( 0 \right )}{N} \right ]$$

so $$\bar{n}\left ( t \right )=\frac{N}{2}-\frac{N}{2}\left [ 1-\frac{2n\left ( 0 \right )}{N} \right ]e^{-2t/N}$$

Problem 1.11. Common derivatives

Calculate the derivative with respect to x of the following functions: e^x, e^3x, e^ax, lnx, lnx², ln3x, ln1/x, sinx, cosx, sin3x, and cos2x.

(e^x)'= e^x, (e^3x)'=3e^3x, (e^ax)'= ae^ax,(lnx)'=1/x,(lnx²)'=2/x,(sinx)'=cosx,(cosx)'=-sinx,(sin3x)'=3cos3x,(cos2x)'=-2sin2x

Problem 1.12. Common integrals

Calculate the following integrals:

$\int_{1}^{2}\frac{dx}{2x^{2}}=-\frac{1}{2}\int_{1}^{2}d\left ( \frac{1}{x} \right )=-\frac{1}{2}\left ( \frac{1}{2}-1 \right )=\frac{1}{4}$

$\int_{1}^{2}\frac{dx}{4x}=4\int_{1}^{2}d\left (lnx \right )=4\left ( ln2-ln1 \right )=2.77$

$\int_{1}^{2}e^{3x}dx=\left ( \frac{1}{3} \right )\int_{1}^{2}d\left ( e^{3x} \right )=\left ( \frac{1}{3} \right )\left ( e^{6} -e^{3}\right )=127.78$

$\int x^{-\gamma }dx=\frac{1}{1-\gamma }x^{1-\gamma }$

Problem 1.13. Partial derivatives

Calculate the partial derivatives of x² + xy + 3y² with respect to x and y.

$$\frac{\partial\left ( x^{2}+xy+3y^{^{2}}\right )}{\partial x}=2x+y$$

$$\frac{\partial\left ( x^{2}+xy+3y^{^{2}}\right )}{\partial y}=x+6y$$

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.

$$e^{ax}=1+ae^{ax}x+\frac{a^{2}e^{ax}x^{2}}{2}+\cdot \cdot \cdot$$

$$ln\left ( 1+x\right )=\frac{x}{x+1}-\frac{x^{2}}{2\left ( x+1 \right )^{2}}+\frac{x^{3}}{3\left ( x+1 \right )^{3}}+\cdot \cdot \cdot$$

$$\left ( 1+x \right )^{n}=1+nx(1+x)^{n-1}+\frac{1}{2}n\left ( n-1 \right )x^{2}\left ( 1+x \right )^{n-2}+\cdot \cdot \cdot$$