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# 物理代写|热力学代写Thermodynamics代考|EGR248 Travels with Entropy

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## 物理代写|热力学代写Thermodynamics代考|Travels with Entropy

Generalization of the two-cities gravity model of Sect. 5.5.4 to the problem of $T$ persons who travel between $i=1, \ldots n$ ‘origin’ cities (each with $O_i$ departures) and $j=1, \ldots m$ ‘destination’ cities (each with $D_j$ arrivals) per unit time is straightforward (the problem where the same cities are both ‘origin’ and ‘destination’ is a particular case with $n=m$ ). If $T_{i j}$ is the number of travels from $i$ to $j$ per unit time and $d_{i j}$ is the distance between the $i$-th origin city and the $j$-th destination city, then the gravity model for each couple $(i, j)$ gives just:
$$T_{i j}=\frac{N_i N_j}{d_{i j}^\beta}$$
with the proviso that the following self-consistency relationships hold:
\begin{aligned} &\Sigma_{i j} T_{i j}=T \ &\Sigma_j T_{i j}=O_i \ &\Sigma_i T_{i j}=D_j \end{aligned}
The question is supposedly settled. However, a problem arises. Let us double the distance between two given cities, say $O_{i=1}$ and $D_{j=1}$. If only these two cities exist, then the two-city gravity model unambiguously predicts that $T=T_{11}$ gets multiplied by $2^{-\beta}$. But in case of more cities, people may just change destination in agreement with PLE, as there is plenty of available, alternative choices. How do people redistribute? The lack of unambiguous answer reflects the fact that many possible configurations are possible, i.e., the number of ways of distributing the travelling people over the $O_i$ ‘s and the $D_j$ ‘s is usually $\gg 1$, all of them satisfying the self-consistency relationships above. In other words, even if our description of this transport network relies on the knowledge of the travels between just one origin and one destination, this knowledge is not enough for us to grasp the behaviour of the network.

All the same, PLE is still helpful. To start with, let us compute the number $W=$ $W\left(T_{11}, T_{12}, \ldots T_{n m}\right)$ of possible travellers’ combinations when the number of travels per unit time from the $i$-th origin city to the $j$-th destination city is $T_{i j}$. According to Ref. [65], we write
$$W=\frac{T !}{\Pi_{i j} T_{i j} !}$$

where $n ! \equiv 1 \cdot 2 \cdot 3 \cdot \ldots n$ (with $0 !=1$ by definition) and $\Pi_{i j}$ is the product over all $i$ and $j .{ }^{49}$ If a measure $c_{i j}$ of the effort ${ }^{50}$ a travel from the $i$-th origin city to the $j$-th destination city calls from a traveller is available, then the total effort $C$ for all travels per unit time is
$$C=\Sigma_{i j} T_{i j} c_{i j}$$

## 物理代写|热力学代写Thermodynamics代考|Muffled Intuitions

In discontinuous systems, LNET implies that the total amount of entropy produced per unit time by all irreversible processes in the system is twice the Rayleigh’s dissipation function of Sect. 4.1.3: $\frac{d S}{d t}=2 f$. Moreover, if the thermodynamic forces are kept fixed then the least dissipation principle holds: $\frac{d S}{d t}-f=\max$. These two relationships imply that a stable steady state corresponds to a constrained maximum of Rayleigh’s dissipation function, the constraint being given by fixed thermodynamic forces. Depending on the problem, LNET leads also to a maximum entropy production principle ( ‘MEPP’), not just to a MinEP. Many researchers have been vigorously working towards the establishment of MEPP beyond the domain of LNET. In continuous systems, LNET deals with the entropy production $P$ due to some (not all!) irreversible processes occurring within the bulk of the system. Even if LNET does not apply, there are variational principles which minimize the entropy production due to selected irreversible processes in the bulk for particular problems, like Kirchhoff’s principle of Sect. 5.3.1, Korteweg-Helmholtz’ principle of Sect. 5.3.7 and Chandrasekhar’s principle of Sect. 5.3.11.

Remarkably, researchers in many different fields independently posit that MEPP describes relaxed states of many systems far from thermodynamic equilibrium and beyond the domain of LNET. In a system where a MEPP holds, a relaxed state far from thermodynamic equilibrium corresponds to a maximum of the amount of entropy produced per unit time. Remarkably, most examples of MEPP available in the literature usually involve irreversible phenomena occurring on the boundary of the system $[43,69]$. In contrast, MinEP in LNET is usually related to dissipative phenomena occurring in the bulk of the system. Bejan himself claims [47] that the constructal law is in full agreement with Malkus’ findings discussed in Sect. 5.3.12 [41] when it comes to turbulence in fluids. Words in pure constructal jargon like Nature takes the easiest and most accessible paths and, hence, processes are accomplished very quickly in a minimum time are found in the work of a staunch supporter of MEPP [70]. According to Reis [71], both MEPP and MinEP are particular corollaries of Bejan’s constructal law of Sect. 5.4. According to Liu [35], if Bejan is right then relaxation minimizes all obstacles on the path of the amount of heat $Q$ flowing per unit time across a physical system entering from one hotter boundary at temperature $T_h$ and coming out through the opposite cooler boundary at temperature $T_l<T_h$, and maximizes therefore $Q$; if $T_l$ and $T_h$ are fixed, then the amount $Q\left(\frac{1}{T_l}-\frac{1}{T_h}\right)$ of entropy of the Universe produced per unit time is maximized. All the same, no generally accepted proof of MEPP is yet available ${ }^{53}$; see e.g. [73].

## 物理代写|热力学代写Thermodynamics代考|Travels with Entropy

$$T_{i j}=\frac{N_i N_j}{d_{i j}^\beta}$$

$$\Sigma_{i j} T_{i j}=T \quad \Sigma_j T_{i j}=O_i \Sigma_i T_{i j}=D_j$$

$$W=\frac{T !}{\Pi_{i j} T_{i j} !}$$

$$C=\Sigma_{i j} T_{i j} c_{i j}$$

## 物理代写|热力学代写Thermodynamics代考|Muffled Intuitions

$\frac{d S}{d t}=2 f$. 此外，如果热力学力保持固定，则最小耗散原埋成立: $\frac{d S}{d t}-f=\max$. 这两个关系意味着稳定的稳态对应于瑞利耗 散函数的约束最大值，该约束由固定的热力学力给出。根据问题，LNET 还导致最大墒生产原则（”MEPP”），而不仅仅是
MinEP。许多研究人员一直在积极致力于建立超越 LNET 领域的 MEPP。在连续系统中，LNET 处理樀产生 $P$ 由于系统主体内发生了

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