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• Statistical Computing 统计计算
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物理代写|统计物理代写Statistical Physics of Matter代考|The Grand Canonical Ensemble Method

The adsorbate is in reality an open system that can exchange not only its energy but also the adsorbed particles with the background (Fig. 8.2). One may naturally consider the grand canonical ensemble theory in which the chemical potential is given instead of the adsorbed particle number, which can fluctuate. For a pedagogical reason, we redo the calculation of the earlier section using this theory. The grand can nonical partition function of the adsorbate is
\begin{aligned} Z_G(\mu, M, T) &=\sum_{\mathcal{N}=0}^M e^{\beta \mathcal{N} \mu} Z(\mathcal{N}, M, T) \ &=\sum_{\mathcal{N}=0}^M \frac{M !}{(M-\mathcal{N}) ! \mathcal{N} !} e^{\beta \mathcal{N}(\epsilon+\mu)} \end{aligned}
where $\mathcal{N}$ is the number of the adsorbed particles. Equation $(8.9)$ is just the binomial expansion of
$$Z(\mu, M, T)=\left(1+e^{\beta(\epsilon+\mu)}\right)^M .$$
This could also have been obtained using the Hamiltonian $\mathcal{H}=-\sum_{i=1}^M \in n_i$ where $n_i$ is either 1 (the site is occupied) or 0 (the site is empty):
\begin{aligned} Z_G(\mu, M, T) &=\sum_M e^{-\beta(\mathcal{H}-\mu \mathcal{N})}=\sum_{n_i=0}^1 \exp \left(\beta \sum_{i=1}^M(\epsilon+\mu) n_i\right) \ &=\prod_{i=1}^M \sum_{n_i=0}^1 e^{\beta(\epsilon+\mu) n_i}=\left(1+e^{\beta(\epsilon+\mu)}\right)^M \end{aligned}

物理代写|统计物理代写Statistical Physics of Matter代考|Effects of the Interactions

We now include the attraction between the neighboring adsorbed particles. Using occupation number representation, the Hamiltonian is
$$\mathcal{H}=-\sum_{i=1}^M \epsilon n_i-\frac{1}{2} \sum_{\langle i j\rangle} b n_i n_j$$

were $\langle i j\rangle$ denotes every pair of particles that mutually attract, and $b$ is the strength of the bond energy. The model can be mapped into the Ising model for ferromagnetism with non-vanishing magnetic field as shown next. For the one-dimensional problem, the exact solution is well known, and will be studied next. For the present problem, which is two-dimensional, the exact solution is not available in general, so an approximation is sought.

To study the effect of the interaction within the canonical ensemble theory, we introduce the Bragg-Williams approximation, according to which the internal energy is first approximated by
\begin{aligned} E &=\langle\mathcal{H}\rangle \approx-\sum_{i=1}^M \epsilon \theta-\frac{1}{2} \sum_{\langle i j\rangle} b \theta^2 \ &=-M\left(\epsilon \theta+\frac{1}{2} q b \theta^2\right) \end{aligned}
where $q$ is the coordination number. For the two dimensional cubic lattice (Fig. 8.1), $q=4$. The approximation may be naturally called the mean field approximation (MFA) in that the fluctuating variable $n_i$ is replaced by its mean $\theta=\left\langle n_i\right\rangle=N / M$. Using the mixing entropy given by (8.3),
$$S=-k_B M{\theta \ln \theta+(1-\theta) \ln (1-\theta)},$$
the free energy is given by
\begin{aligned} F(\theta, M, T) &=E-T S \ &=M\left[-\epsilon \theta-\frac{1}{2} q b \theta^2+k_B T{\theta \ln \theta+(1-\theta) \ln (1-\theta)}\right], \end{aligned}
from which we can obtain the chemical potential
$$\mu=\frac{\partial F\left(\frac{N}{M}, M, T\right)}{\partial N}=\frac{\partial F(\theta, M, T)}{M \partial \theta}=-\epsilon-q b \theta+k_B T \ln (\theta /(1-\theta)),$$
$$\theta=\frac{1}{\left{e^{\beta(-\epsilon-q b \theta-\mu)}+1\right}} .$$

统计物理代考

物理代写|统计物理代写Statistical Physics of Matter代考|The Grand Canonical Ensemble Method

$$Z_G(\mu, M, T)=\sum_{\mathcal{N}=0}^M e^{\beta \mathcal{N} \mu} Z(\mathcal{N}, M, T) \quad=\sum_{\mathcal{N}=0}^M \frac{M !}{(M-\mathcal{N}) ! \mathcal{N} !} e^{\beta \mathcal{N}(\epsilon+\mu)}$$

$$Z(\mu, M, T)=\left(1+e^{\beta(\epsilon+\mu)}\right)^M .$$

$$Z_G(\mu, M, T)=\sum_M e^{-\beta(\mathcal{H}-\mu \mathcal{N})}=\sum_{n_i=0}^1 \exp \left(\beta \sum_{i=1}^M(\epsilon+\mu) n_i\right) \quad=\prod_{i=1}^M \sum_{n_i=0}^1 e^{\beta(\epsilon+\mu) n_i}=\left(1+e^{\beta(\epsilon+\mu)}\right)^M$$

物理代写|统计物理代写Statistical Physics of Matter代考|Effects of the Interactions

$$\mathcal{H}=-\sum_{i=1}^M \epsilon n_i-\frac{1}{2} \sum_{\langle i j\rangle} b n_i n_j$$

$$E=\langle\mathcal{H}\rangle \approx-\sum_{i=1}^M \epsilon \theta-\frac{1}{2} \sum_{\langle i\rangle\rangle} b \theta^2 \quad=-M\left(\epsilon \theta+\frac{1}{2} q b \theta^2\right)$$

$$S=-k_B M \theta \ln \theta+(1-\theta) \ln (1-\theta),$$

$$F(\theta, M, T)=E-T S \quad=M\left[-\epsilon \theta-\frac{1}{2} q b \theta^2+k_B T \theta \ln \theta+(1-\theta) \ln (1-\theta)\right],$$

$$\mu=\frac{\partial F\left(\frac{N}{M}, M, T\right)}{\partial N}=\frac{\partial F(\theta, M, T)}{M \partial \theta}=-\epsilon-q b \theta+k_B T \ln (\theta /(1-\theta)),$$

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