## MATH110 Statistical calculation课程简介

This course introduces the logic and methods of statistics. We begin with a discussion of the role of statistics, introducing the concepts of internal and external validity. Common methods for describing the characteristics of individuals and educational outcomes are presented, including the use of graphs and summary measures such as the mean, median, standard deviation, and correlation coefficient. Because of the large natural variability among individuals, one must be able to determine whether or not an apparent difference or patterns present in the data seems to be merely a chance occurrence.
Probability concepts are introduced to help us in this effort. Probability then forms the basis of all of the inferential statistical procedures subsequently presented. At the end of the course, students will:

• identify basic statistical applications for educational research.
• explain how to implement quantitative approaches to educational research.
• identify types of statistical methods and strategies, and select data collection and analysis approaches for different research interests.
• integrate fundamental statistical theories and concepts with functions of SPSS programs in the context of an analysis project’s overall design.

## PREREQUISITES

Students have 3 to 9 months to complete 18 lessons, including a final research project. Each lesson contains a variety of items which include required chapter reading, a quiz on the reading assignment, at least one instructor video and supporting material for the lesson, and a lesson activity assignment. Lesson topics include:

1. Introduction to statistics
2. Frequency Distributions
3. Central Tendency
4. Variability
5. Z scores: Standardized distributions
6. Probability
7. Probability Sampling
8. Hypothesis Testing
9. $t$ Statistic
10. Independent Sample $t$ Test
11. Related Samples $t$ Test
12. Intro to Analysis of Variance
13. Repeated Measure Analysis of Variance
14. Two-factor Analysis of Variance
15. Correlations
16. Regression
17. Chi-square
18. Research Project

## MATH110 Statistical calculation HELP（EXAM HELP， ONLINE TUTOR）

For a Bose gas with fixed number of particles and at a given temperature, calculate the critical volume at which the Bose-Einstein condensation takes place. Repeat your analysis in $2 D$.

In three-dimensional space, the critical volume $\$ V_{-} c \$$for Bose-Einstein condensation (BEC) can be calculated using the formula:$$
V_c=\frac{N}{\zeta(3 / 2)}\left(\frac{h^2}{2 \pi m k_B T}\right)^{3 / 2}
$$where \ N \$$ is the number of particles, $\$ \backslash z e t a(3 / 2) \backslash$approx$2.612 \$$is the Riemann zeta function, \ \mathrm{~h} is Planck’s constant, \ \mathrm{~m} \$$ is the mass of the bosons, $\$ k_{-} B \$$is the Boltzmann constant, and \ T \$$ is the temperature.

In two-dimensional space, the critical area $\$ A_{-} c \$$can be calculated using a similar formula:$$
A_c=\frac{N}{\zeta(2)}\left(\frac{h^2}{2 \pi m k_B T}\right)
$$where \ \backslash z e t a(2)=\backslash f r a c{p i \wedge 2}{6} \$$.
Note that the critical volume/area is the minimum volume/area required for a Bose gas to undergo $\mathrm{BEC}$ at a given temperature.
Let’s assume a Bose gas with $\$ \mathrm{~N}=10^{\wedge} 6 \$$particles, \ \mathrm{~m}= 1.44 \backslash times 10^{\wedge}{-25} \ \mathrm{~kg} (mass of helium-4 atom), and \ T= 2.17 \ \mathrm{~K} (critical temperature for helium-4). We can then calculate the critical volume/area as follows: For a 3D Bose gas:$$
V_c=\frac{10^6}{2.612}\left(\frac{\left(6.626 \times 10^{-34}\right)^2}{2 \pi\left(1.44 \times 10^{-25}\right)\left(1.38 \times 10^{-23}\right)(2.17)}\right)^{3 / 2} \approx 2.16 \times 10^{-5} \mathrm{~m}^3
$$For a 2D Bose gas:$$
A_c=\frac{10^6}{\pi^2 / 6}\left(\frac{\left(6.626 \times 10^{-34}\right)^2}{2 \pi\left(1.44 \times 10^{-25}\right)\left(1.38 \times 10^{-23}\right)(2.17)}\right) \approx 1.22 \times 10^{-8} \mathrm{~m}^2
$$Therefore, for a Bose gas with \ N=10^{\wedge} 6 \$$ particles and $\$ T=2.17 \$\mathrm{~K}$, the critical volume for $\mathrm{BEC}$ is approximately $\$ 2.16\backslash$times$10^{\wedge}{-5} \$m \$ \wedge 3 \$$in 3 \mathrm{D} and the critical area for BEC is approximately \ 1.22 \backslash times 10^{\wedge}{-8} \ \mathrm{~m} \^{\wedge} 2 \$$ in $2 \mathrm{D}$.

For a dispersion relation $\varepsilon \propto|p|^\sigma$, what is the constraint on dimensionality of space $D$ for Bose-Einstein condensation to take place?

The condition for Bose-Einstein condensation (BEC) to occur is that the chemical potential $\$ \backslash m u \$$becomes equal to the ground-state energy of the system. In the low-temperature limit, this ground-state energy is proportional to \ N^{\wedge}{1 / D} \$$, where $\$ N \$$is the number of particles and \ D \$$ is the dimensionality of space.

For a dispersion relation \$|varepsilon \propto$|p|^{\wedge} \backslash$sigma\$, the number of particles in momentum space is given by:
$$N=\int \frac{d^D p}{(2 \pi)^D} \frac{1}{e^{(\varepsilon(p)-\mu) / k_B T}-1}$$
where $\$ T \$$is the temperature and \ \mathrm{k} B \$$ is the Boltzmann constant. In the low-temperature limit, we can approximate the Bose distribution function as:
$$\frac{1}{e^{(\varepsilon(p)-\mu) / k_B T}-1} \approx e^{-(\varepsilon(p)-\mu) / k_B T}$$
Using this approximation, we can rewrite the number of particles as:
$$N=\int \frac{d^D p}{(2 \pi)^D} e^{-(\varepsilon(p)-\mu) / k_B T}$$
Substituting \$|varepsilon \propto$|\mathrm{p}|^{\wedge} \backslash$sigma\$ and changing to spherical coordinates, we get:
$$N=\frac{V_D}{(2 \pi)^D} \int_0^{\infty} p^{D-1} e^{-\left(|p|^\sigma-\mu\right) / k_B T} d p$$
where $\$ V_{-} D \$$is the volume of a \ D \$$-dimensional sphere of radius $\$ R \$$:$$
V_D=\frac{\pi^{D / 2}}{\Gamma(D / 2+1)} R^D
$$For a fixed \ \backslash m u \$$, the integral can be evaluated by a saddle-point approximation. In the limit of low temperatures, the saddle-point is dominated by small values of $\$ \mathrm{p} \$$, so we can expand \|p|^{\wedge} \backslash sigma \$$ around $\$ p=0 \$$and keep only the leading term:$$
|p|^\sigma \approx p^\sigma \quad \text { for } p \rightarrow 0
$$Using this approximation, the integral becomes:$$
N \approx \frac{V_D}{(2 \pi)^D} \int_0^{\infty} p^{D-1} e^{-\left(p^\sigma-\mu\right) / k_B T} d p
$$Changing variables to \ \mathrm{x}=\mathrm{p}^{\wedge} \backslash sigma \$$ and using the gamma function, we get:
$$N \approx \frac{V_D}{\sigma(2 \pi)^D}\left(\frac{k_B T}{\mu}\right)^{D / \sigma} \Gamma\left(\frac{D}{\sigma}+1\right)$$
The ground-state energy in the low-temperature limit is then:
$$E_0=\frac{\mu}{\sigma}\left(\frac{N}{V_D}\right)^{1 / D}=\frac{\mu}{\sigma}\left(\frac{\sigma(2 \pi)^D}{V_D}\right)^{1 / D}\left(\frac{k_B T}{\mu}\right)^{1 / \sigma} \Gamma\left(\frac{D}{\sigma}+1\right)^{1 / D}$$
BEC occurs when the chemical potential $\$ \backslash$mu$\$$becomes equal to the ground-state energy \ E _0 \$$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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## EFR515 Statistical calculation课程简介

This course introduces the logic and methods of statistics. We begin with a discussion of the role of statistics, introducing the concepts of internal and external validity. Common methods for describing the characteristics of individuals and educational outcomes are presented, including the use of graphs and summary measures such as the mean, median, standard deviation, and correlation coefficient. Because of the large natural variability among individuals, one must be able to determine whether or not an apparent difference or patterns present in the data seems to be merely a chance occurrence.
Probability concepts are introduced to help us in this effort. Probability then forms the basis of all of the inferential statistical procedures subsequently presented. At the end of the course, students will:

• identify basic statistical applications for educational research.
• explain how to implement quantitative approaches to educational research.
• identify types of statistical methods and strategies, and select data collection and analysis approaches for different research interests.
• integrate fundamental statistical theories and concepts with functions of SPSS programs in the context of an analysis project’s overall design.

## PREREQUISITES

Students have 3 to 9 months to complete 18 lessons, including a final research project. Each lesson contains a variety of items which include required chapter reading, a quiz on the reading assignment, at least one instructor video and supporting material for the lesson, and a lesson activity assignment. Lesson topics include:

1. Introduction to statistics
2. Frequency Distributions
3. Central Tendency
4. Variability
5. Z scores: Standardized distributions
6. Probability
7. Probability Sampling
8. Hypothesis Testing
9. $t$ Statistic
10. Independent Sample $t$ Test
11. Related Samples $t$ Test
12. Intro to Analysis of Variance
13. Repeated Measure Analysis of Variance
14. Two-factor Analysis of Variance
15. Correlations
16. Regression
17. Chi-square
18. Research Project

## EFR515 Statistical calculation HELP（EXAM HELP， ONLINE TUTOR）

1. Calculate the Fermi energy and the total energy of a $1 D$, spin- $1 / 2$ Fermi gas at $T=0$. What is the relationship between the two? Repeat your calculation for a $2 D$ gas.

The Fermi energy is the energy of the highest occupied quantum state in a system of non-interacting fermions at zero temperature. For a one-dimensional Fermi gas with spin-1/2 particles, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m} \left(\frac{\pi n}{2}\right)^2EF​=2mℏ2​(2πn​)2

where $n$ is the one-dimensional number density of fermions and $m$ is their mass. At zero temperature, all quantum states up to the Fermi level are occupied, and all states above the Fermi level are unoccupied. The total energy of the system is given by the sum of the energies of all occupied quantum states:

E = \int_{-\infty}^{E_F} g(E) E \, dEE=∫−∞EF​​g(E)EdE

where $g(E)$ is the density of states. For a one-dimensional gas, the density of states is given by:

g(E) = \frac{m}{\pi\hbar^2}g(E)=πℏ2m​

Substituting this expression into the equation for the total energy and integrating, we obtain:

E = \frac{\hbar^2}{4m} \left(\frac{\pi n}{2}\right)^2E=4mℏ2​(2πn​)2

We can see that the total energy is proportional to the Fermi energy, with a factor of $\frac{1}{2}$. This is a general property of non-interacting fermion systems in any dimension.

For a two-dimensional Fermi gas, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m} \left(\pi n\right)EF​=2mℏ2​(πn)

The density of states for a two-dimensional gas is given by:

g(E) = \frac{m}{\pi\hbar^2}g(E)=πℏ2m​

Substituting this expression into the equation for the total energy and integrating, we obtain:

E = \frac{\pi}{2} n E_F = \frac{\hbar^2}{4m} \left(\pi n\right)^2E=2π​nEF​=4mℏ2​(πn)2

Again, we can see that the total energy is proportional to the Fermi energy, with a factor of $\frac{\pi}{2}$.

1. Calculate the sound velocity
$$u^2=\left(\frac{\partial P}{\partial \rho}\right){T=0}$$ of a spin-1/2 Fermi gas. Also, calculate $(\partial P / \partial \rho){T=0}$ for a Bose gas below the BoseEinstein temperature.

For a spin-1/2 Fermi gas, the pressure is given by the Fermi gas equation of state:

P = \frac{2}{3} E \cdot n,P=32​E⋅n,

where $E$ is the total energy of the gas, $n$ is the number density of fermions, and we have used the fact that the kinetic energy of a Fermi gas is $3/5$ of its total energy.

Taking the partial derivative of $P$ with respect to $\rho$ at $T=0$ gives:

\left(\frac{\partial P}{\partial \rho}\right)_{T=0} = \frac{2}{3} n \left(\frac{\partial E}{\partial \rho}\right)_{T=0}(∂ρ∂P​)T=0​=32​n(∂ρ∂E​)T=0​

To find $\left(\frac{\partial E}{\partial \rho}\right)_{T=0}$, we use the fact that for a non-interacting Fermi gas, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m}(3\pi^2n)^{2/3}EF​=2mℏ2​(3π2n)2/3

At $T=0$, the energy per particle is given by:

\frac{E}{N} = \frac{3}{5} E_FNE​=53​EF​

Taking the derivative of this expression with respect to $n$ gives:

\left(\frac{\partial E}{\partial n}\right)_{T=0} = \frac{2}{3} E_F \cdot \frac{1}{n}(∂n∂E​)T=0​=32​EF​⋅n1​

Substituting this expression into the equation for $\left(\frac{\partial P}{\partial \rho}\right)_{T=0}$, we obtain:

\left(\frac{\partial P}{\partial \rho}\right)_{T=0} = \frac{2}{5}\frac{E_F}{m}(∂ρ∂P​)T=0​=52​mEF​​

The sound velocity $u$ is defined as:

u^2 = \left(\frac{\partial P}{\partial \rho}\right)_{T=0}u2=(∂ρ∂P​)T=0​

Substituting the expression we found above for $\left(\frac{\partial P}{\partial \rho}\right)_{T=0}$, we obtain:

u^2 = \frac{2}{5}\frac{E_F}{m} = \frac{2}{5} \frac{\hbar^2}{2m^2}(3\pi^2n)^{2/3}u2=52​mEF​​=52​2m2ℏ2​(3π2n)2/3

For a Bose gas below the Bose-Einstein temperature, the pressure is given by the ideal Bose gas equation of state:

P = \frac{2}{3} E \cdot nP=32​E⋅n

where $E$ is the total energy of the gas, $n$ is the number density of bosons. The energy of an ideal Bose gas is given by:

E = \frac{3}{2} N k_B T \zeta\left(\frac{5}{2}\right)\left(\frac{m}{2\pi \hbar^2}\right)^{3/2}E=23​NkB​Tζ(25​)(2πℏ2m​)3/2

where $N$ is the total number of bosons, $k_B$ is Boltzmann’s constant, $T$ is the temperature, $\zeta(x)$ is the Riemann zeta function, and we have used the fact that the Bose gas has a kinetic energy equal to $3/2$ of its total energy.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.