物理代写|热力学代写thermodynamics代考|Why Statistical Thermodynamics?

Doug I. Jones

Doug I. Jones

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物理代写|热力学代写thermodynamics代考|Why Statistical Thermodynamics?

物理代写|热力学代写thermodynamics代考|Why Statistical Thermodynamics?

While the above classification scheme might please the engineering mind, it does little to acquaint you with the drama and excitement of both learning and applying statistical thermodynamics. Yes, you will eventually be able to calculate from atomic and molecular properties the thermodynamic properties of ideal gases, real gases, and metals. Examples might include equations of state, measurable properties such as specific heats and the internal energy, and also ephemeral properties such as the entropy and free energies. And yes, you will learn how to calculate various transport properties, such as the thermal conductivity and the diffusion coefficient. Furthermore, with respect to chemical reactions, you will eventually be able to determine equilibrium constants and estimate elementary rate coefficients.

While these pragmatic aspects of statistical thermodynamics are important, the real drama of the subject lies instead in its revelations about our natural world. As you work through this book, you will slowly appreciate the limitations of classical thermodynamics. In particular, the first, second, and third laws of thermodynamics should take on a whole new meaning for you. You will understand that volumetric work occurs because of microscopic energy transfers and that heat flow occurs because of redistributions in molecular population. You will realize that entropy rises in isolated systems because of a fundamental enhancement in molecular probabilities. You will also appreciate in a new way the important underlying link between absolute property values and crystalline behavior near absolute zero.

Perhaps more importantly, you will come to understand in a whole new light the real meaning of thermodynamic equilibrium and the crucial role that temperature plays in defining both thermal and chemical equilibrium. This new understanding of equilibrium will pave the path for laser-based applications of statistical thermodynamics to measurements of both temperature and species concentrations, as discussed in Chapter 11. Such optical measurements are extremely important to current research in all of the thermal sciences, including fluid mechanics, heat transfer, combustion, plasmas, and various aspects of nanotechnology and manufacturing.

物理代写|热力学代写thermodynamics代考|Probability: Definitions and Basic Concepts

Probability theory is concerned with predicting statistical outcomes. Simple examples of such outcomes include observing a head or tail when tossing a coin, or obtaining the numbers $1,2,3,4,5$, or 6 when throwing a die. For a fairly-weighted coin, we would, of course, expect to see a head for $1 / 2$ of a large number of tosses; similarly, using a fairlyweighted die, we would expect to get a four for $1 / 6$ of all throws. We can then say that the probability of observing a head on one toss of a fairly-weighted coin is $1 / 2$ and that for obtaining a four on one throw of a fairly-weighted die is $1 / 6$. This heuristic notion of probability can be given mathematical formality via the following definition:
Given $N_s$ mutually exclusive, equally likely points in sample space, with $N_e$ of these points corresponding to the random event $A$, then the probability $P(A)=N_e / N_s$.
Here, sample space designates the available $N_s$ occurrences while random event $A$ denotes the subset of sample space given by $N_e \leq N_s$. The phrase mutually exclusive indicates that no two outcomes can occur simultaneously in a single sample space; this criterion is obviously required if we are to convert our heuristic understanding of chance to a welldefined mathematical probability.

As a further example, for a standard deck of playing cards, we have 52 points in sample space, of which four represent aces. Hence, the probability of drawing a single ace from a well-mixed deck is $P(A)=4 / 52=1 / 13$, where the event $A$ designates the random drawing of an ace. Visually, the relation between event $A$ and sample space can be described by a so-called Venn diagram, as shown in Fig. 2.1. Here, sample points resulting in event $A$ fall within the area $A$, while those not resulting in event $A$ fall elsewhere in the surrounding box, whose total area represents the entire sample space. Hence, assuming a uniform point density, we find that the ratio of the cross-hatched area to the total area in Fig. 2.1 provides a visual representation of $P(A)$. Similarly, from the viewpoint of set theory, we observe that for a fairly-weighted die the random event of obtaining an even number $E={2,4,6}$ from within the entire sample space $S={1,2,3,4,5,6}$ clearly occurs with probability $P(A)=1 / 2$.

Our notion of probability becomes more complicated if we consider two different random events, $A$ and $B$, which can both occur within a given sample space. On this basis, we may define the compound probability, $P(A B)$, which represents events $A$ and $B$, and also the total probability, $P(A+B)$, which represents events $A$ or $B$ (including both). From the viewpoint of set theory, $P(A B)$ is called the intersection of $A$ and $B(A \cap B)$, while $P(A+B)$ is labeled the union of $A$ and $B(A \cup B)$. Pictorial displays of the (a) intersection and (b) union of $A$ and $B$ are given by the two Venn diagrams shown in Fig. 2.2.

If the events $A$ and $B$ are mutually exclusive, a single trial by definition permits no overlap in sample space. Therefore, $P(A B)=0$ so that
as displayed by the Venn diagram of Fig. 2.3(a). As an example, the probability of picking a king $(K)$ or a queen $(Q)$ from a single deck of playing cards is given by the total probability $P(K+Q)=P(K)+P(Q)=2 / 13$. In comparison, the probability of picking a king from one deck and a queen from a different deck is $P(K Q)=(1 / 13)^2$. In the latter case, we have two different sample spaces, as indicated by the Venn diagram of Fig. 2.3(b), so that the events are now mutually independent. Hence, in general, the compound probability becomes
P(A B)=P(A) \cdot P(B) .

物理代写|热力学代写thermodynamics代考|Why Statistical Thermodynamics?


物理代写|热力学代写thermodynamics代考|Why Statistical Thermodynamics?




物理代写|热力学代写thermodynamics代考|Probability: Definitions and Basic Concepts

概率论与预测统计结果有关。这类结果的简单例子包括:投掷硬币时观察正面或反面,或者得到数字$1,2,3,4,5$,或者投掷骰子时得到6。对于一枚均匀加权的硬币,我们当然会期望在大量投掷中看到头像$1 / 2$;类似地,使用均匀加权的骰子,我们期望在所有投掷中得到$1 / 6$为4。然后我们可以说,在一次均匀加权硬币投掷中观察到正面的概率是$1 / 2$,而在一次均匀加权骰子投掷中获得4的概率是$1 / 6$。这种启发式的概率概念可以通过以下定义给出数学形式:
给定样本空间中$N_s$互斥、等可能的点,其中$N_e$点对应随机事件$A$,则概率为$P(A)=N_e / N_s$。
这里,样本空间表示可用的$N_s$事件,而随机事件$A$表示$N_e \leq N_s$给出的样本空间的子集。短语互斥表示在单个样本空间中不可能同时发生两种结果;如果我们要将我们对机会的启发式理解转化为定义良好的数学概率,显然需要这个标准。

作为进一步的例子,对于一副标准纸牌,我们在样本空间中有52个点,其中4个代表a。因此,从混合良好的牌组中抽到一张a的概率为$P(A)=4 / 52=1 / 13$,其中事件$A$表示随机抽到一张a。从视觉上看,事件$A$与样本空间的关系可以用所谓的维恩图来描述,如图2.1所示。在这里,导致事件$A$的样本点落在$A$区域内,而不导致事件$A$的样本点落在周围框的其他地方,其总面积代表整个样本空间。因此,假设点密度均匀,我们发现图2.1中交叉孵化面积与总面积的比值提供了$P(A)$的可视化表示。同样,从集合论的观点来看,我们观察到,对于一个公平加权的骰子,从整个样本空间$S={1,2,3,4,5,6}$中获得偶数$E={2,4,6}$的随机事件显然以$P(A)=1 / 2$的概率发生。

如果我们考虑两个不同的随机事件,$A$和$B$,它们都可能发生在给定的样本空间中,我们的概率概念就会变得更加复杂。在此基础上,我们可以定义复合概率$P(A B)$,它表示事件$A$和$B$,也可以定义总概率$P(A+B)$,它表示事件$A$或$B$(包括两者)。从集合论的观点来看,$P(A B)$被称为$A$和$B(A \cap B)$的交集,而$P(A+B)$被标记为$A$和$B(A \cup B)$的并集。$A$和$B$的(a)交集和(b)并的图形显示由图2.2所示的两个维恩图给出。

如果事件$A$和$B$相互排斥,则根据定义,单次试验不允许在样本空间中重叠。因此,$P(A B)=0$使
如图2.3(a)的维恩图所示。例如,从一副扑克牌中取出国王$(K)$或王后$(Q)$的概率由总概率$P(K+Q)=P(K)+P(Q)=2 / 13$给出。相比之下,从一副牌中取出国王和从另一副牌中取出王后的概率是$P(K Q)=(1 / 13)^2$。在后一种情况下,我们有两个不同的样本空间,如图2.3(b)的维恩图所示,因此事件现在是相互独立的。因此,一般来说,复合概率变成
P(A B)=P(A) \cdot P(B) .

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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