### 数学代写|信息论代写information theory代考|FEO3350

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## 数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

In this section we consider a special case of a system of interacting particles. We start with an ideal gas-i.e. system for which we can neglect all intermolecular interactions. Strictly speaking, such a system does not exist. However, if the gas is very dilute such that the average intermolecular distance is very large the system behaves as if there are no interactions among the particle.

Next, we increase the density of the particles. At first we shall find that pairinteractions affect the thermodynamics of the system. Increasing further the density, triplets, quadruplets, and so on interactions, will also affect the behavior of the system. In the following we provide a very brief description of the first order deviation from ideal gas; systems for which one must take into account pair-interactions but neglect triplet and higher order interactions. The reader who is not interested in the details of the derivation can go directly to the result in Eq. (2.51) and the following analysis of the MI.

We start with the general configurational PF of the system, Eq. (2.31) which we rewrite in the form:
$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$
where $U_{i j}$ is the pair potential between particles $i$ and $j$. It is assumed that the total potential energy is pairwise additive.
Define the so-called Mayer $f$-function, by:
$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$
We can rewrite $Z_N$ as:
$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$
Neglecting all terms beyond the first sum, we obtain:
$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

## 数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

In this section, we shall discuss the entropy-changes associated with phase transitions. Here, by entropy we mean thermodynamic entropy, the units of which are cal/(deg $\mathrm{mol}$ ). However, as we have seen in Chap. 5 of Ben-Naim [1]. The entropy is up to a multiplicative constant an SMI defined on the distribution of locations and velocities (or momenta) of all particles in the system at equilibrium. To convert from entropy to SMI one has to divide the entropy by the factor $k_B \log _e 2$, where $k_B$ is the Boltzmann constant, and $\log _e 2$ is the natural $\log$ arithm of 2 , which we denote by $\ln 2$. Once we do this conversion from entropy to SMI we obtain the SMI in units of bits. In this section we shall discuss mainly the transitions between gases, liquids and solids. Figure 2.9 shows a typical phase diagram of a one-component system. For more details on phase diagrams, see Ben-Naim and Casadei [8].

It is well-known that solid has a lower entropy than liquid, and liquid has a lower entropy of a gas. These facts are usually interpreted in terms of order-disorder. This interpretation of entropy is invalid; more on this in Ben-Naim [6]. Although, it is true that a solid is viewed as more ordered than liquid, it is difficult to argue that a liquid is more ordered or less ordered than a gas.

In the following we shall interpret entropy as an SMI, and different entropies in terms of different MI due to different intermolecular interactions. We shall discuss changes of phases at constant temperature. Therefore, all changes in SMI (hence, in entropy) will be due to locational distributions; no changes in the momenta distribution.

The line SG in Fig. 2.9 is the line along in which solid and gas coexist. The slope of this curve is given by:
$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$
In the process of sublimation ( $s$, the entropy-change and the volume change for both are always positive. We denoted by $\Delta V_s$ the change in the volume of one mole of the substance, when it is transferred from the solid to the gaseous phase. This volume change is always positive. The reason is that a mole of the substance occupies a much larger volume in the gaseous phase than in the liquid phase (at the same temperature and pressure).

The entropy-change $\Delta S_s$ is also positive. This entropy-change is traditionally interpreted in terms of transition from an ordered phase (solid) to a disordered (gaseous) phase. However, the more correct interpretation is that the entropy-change is due to two factors; the huge increase in the accessible volume available to each particle and the decrease in the extent of the intermolecular interaction. Note that the slope of the SG curve is quite small (but positive) due to the large $\Delta V_s$.

# 信息论代写

## 数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$

$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$

$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$

$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

## 数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$

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## MATLAB代写

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