# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYSICS7546

#### Doug I. Jones

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Grand Canonical Distribution and Thermodynamics

The distribution of an underlying microstate $\mathcal{M}$ of the system with the energy $\mathcal{H}{\mathcal{M}}$ and particle number $\mathcal{N}$ is derived using logic similar to that for the canonical ensemble:
$$P{\mathcal{M}}=\frac{e^{-\beta(\mathcal{H}{\mathcal{M}}-\mu \mathcal{N})}}{Z_{G}\left(T, \mu, X_{i}\right)}$$
where
\begin{aligned} Z_{G}\left(T, \mu, X_{i}\right) &=\sum_{\mathcal{M}} e^{-\beta(\mathcal{H}{\mathcal{M}}-\mu \mathcal{N}{\mathcal{M}})}=\sum_{\mathcal{N}=0}^{\infty} \sum_{\mathcal{M} / \mathcal{N}} e^{-\beta(\mathcal{H}{\mathcal{M}}-\mu \mathcal{N})} \ &=\sum_{\mathcal{N}=0}^{\infty} e^{-\beta \mu \mathcal{N}} Z_{\mathcal{N}} \end{aligned}
is the grand canonical partition function. Here $\sum_{\mathcal{M} / \mathcal{N}}$ is the summation over the microstates of the system with $\mathcal{N}$ given, of which the canonical partition function is $Z_{\mathcal{N}}$
The average number of particles in the system is given as
$$N=\langle\mathcal{N}\rangle=\frac{\sum_{\mathcal{M}} \mathcal{N}{\mathcal{M}} e^{-\beta(\mathcal{H}{\mathcal{M}}-\mu \mathcal{N}{\mathcal{M}})}}{\sum_{\mathcal{M}} e^{-\beta(\mathcal{H}{\mathcal{M}}-\mu \mathcal{N}{\mathcal{M}})}}=\frac{\partial Z_{G}}{Z_{G} \partial(\beta \mu)}$$
The grand canonical ensemble theory is useful for systems in which the number of particless variess, i.e., for ‘open systems’. Thé fluctuation in the numbèr of particless in the system about the mean $\langle\mathcal{N}\rangle=N$ is
\begin{aligned} \left\langle(\Delta \mathcal{N})^{2}\right\rangle &=\left\langle\mathcal{N}^{2}\right\rangle-\left\langle\mathcal{N}^{-2}\right\rangle \ &=\frac{\partial^{2} Z_{G}}{Z_{G} \partial(\beta \mu)^{2}}-\left[\frac{\partial Z_{G}}{Z_{G} \partial(\beta \mu)}\right]^{2}=\frac{\partial^{2} \ln Z_{G}}{\partial(\beta \mu)^{2}} \ &=\frac{\partial N}{\beta \partial \mu}, \end{aligned}
where (3.63) is used. Because $\partial N / \beta \partial \mu$ is an extensive quantity, the rms deviation $\bar{\Delta} \mathcal{N}=\left\langle(\Delta \mathcal{N})^{2}\right\rangle^{1 / 2}$ scales as $N^{1 / 2}$. Consider that $N$ is very large. Then, one can show the distribution over the number of the particles is very sharp Gaussian around $\mathcal{N}=N$, which dominates the partition sum.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Ligand Binding on Proteins with Interaction

As an example to show the utility of the grand canonical ensemble theory, we consider systems of molecules or ligands (such as $\mathrm{O}{2}$ ) that can bind on two identical, but distinguishable sites in a protein (e.g., myoglobin, hemoglobin) (Fig. 3.11). How does the average number of bound ligands depend on their ambient concentrations? Compared with a similar problem of two-state molecular binding treated in Sect. 3.1, there is an important difference: earlier, the system of interest was a biopolymer with fixed $N$ binding sites, whereas the system in question here is the bound ligands, whose number $\mathcal{N}$ can vary. In this case the grand partition function is expressed as $$Z{G}=\sum_{\mathcal{N}=0} e^{\beta \mu \mathcal{N}} Z_{\mathcal{N}}=Z_{0}(0,0)+z Z_{1}(1,0)+z Z_{1}(0,1)+z^{2} Z_{2}(1,1)$$
where $Z_{m+n}(m, n)$ is the canonical partition function with $m$ and $n$ ligands bound on two sites, and $z=e^{\beta \mu}$ is the fugacity of a ligand. If the energy in the bound state is $-\epsilon(<0)$, and the interaction energy is $\varphi, Z_{0}(0,0)=1, Z_{1}(1,0)=Z_{1}(0,1)=e^{\beta \epsilon}$, and $Z_{2}(1,1)=e^{\beta(2 \epsilon-\varphi)}$, so $Z_{G}$ is given as
$$Z_{G}=1+2 z e^{\beta \epsilon}+z^{2} e^{\beta(2 \epsilon-\varphi)} .$$
Using (3.63), the coverage per site is
$$\theta=\frac{1}{2}\langle\mathcal{N}\rangle=\frac{1}{2} z \frac{\partial}{\partial z} \ln Z_{G}=\frac{z\left{e^{\beta \epsilon}+z e^{\beta(2 \epsilon-\varphi)}\right}}{1+2 z e^{\beta \epsilon}+z^{2} e^{\beta(2 \epsilon-\varphi)}} .$$
If $\varphi=0$ so that two sites are independent of each other, the coverage is
$$\theta-\frac{z e^{\beta \epsilon}}{1+z e^{\beta \epsilon}}-\frac{1}{e^{-\beta(\epsilon+\mu)}+1}$$

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Grand Canonical Distribution and Thermodynamics

$$P \mathcal{M}=\frac{e^{-\beta(\mathcal{H M}-\mu \mathcal{N})}}{Z_{G}\left(T, \mu, X_{i}\right)}$$

$$Z_{G}\left(T, \mu, X_{i}\right)=\sum_{\mathcal{M}} e^{-\beta(\mathcal{H} \mathcal{M}-\mu \mathcal{N} \mathcal{M})}=\sum_{\mathcal{N}=0}^{\infty} \sum_{\mathcal{M} / \mathcal{N}} e^{-\beta(\mathcal{H} \mathcal{M}-\mu \mathcal{N})} \quad=\sum_{\mathcal{N}=0}^{\infty} e^{-\beta \mu \mathcal{N}}$$

$$N=\langle\mathcal{N}\rangle=\frac{\sum_{\mathcal{M}} \mathcal{N M} e^{-\beta(\mathcal{H M}-\mu \mathcal{N} \mathcal{M})}}{\sum_{\mathcal{M}} e^{-\beta(\mathcal{H} \mathcal{M}-\mu \mathcal{N} \mathcal{M})}}=\frac{\partial Z_{G}}{Z_{G} \partial(\beta \mu)}$$

$$\left\langle(\Delta \mathcal{N})^{2}\right\rangle=\left\langle\mathcal{N}^{2}\right\rangle-\left\langle\mathcal{N}^{-2}\right\rangle \quad=\frac{\partial^{2} Z_{G}}{Z_{G} \partial(\beta \mu)^{2}}-\left[\frac{\partial Z_{G}}{Z_{G} \partial(\beta \mu)}\right]^{2}=\frac{\partial^{2} \ln Z_{G}}{\partial(\beta \mu)^{2}}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Ligand Binding on Proteins with Interaction

$$Z G=\sum_{\mathcal{N}=0} e^{\beta \mu \mathcal{N}} Z_{\mathcal{N}}=Z_{0}(0,0)+z Z_{1}(1,0)+z Z_{1}(0,1)+z^{2} Z_{2}(1,1)$$

$\varphi, Z_{0}(0,0)=1, Z_{1}(1,0)=Z_{1}(0,1)=e^{\beta \epsilon}$ ，和 $Z_{2}(1,1)=e^{\beta(2 \epsilon-\varphi)}$ ，所以 $Z_{G}$ 给 出为
$$Z_{G}=1+2 z e^{\beta \epsilon}+z^{2} e^{\beta(2 \epsilon-\varphi)} .$$

Itheta= \frac ${1}{2} \backslash \mid a n g l e \backslash$ mathcal ${\mathrm{N}} \backslash$ rangle= $\backslash$ frac ${1}{2}$ z $\backslash$ frac ${$ partial} $} \backslash$ partial Z} $\backslash \cap Z_{-}{$

$$\theta-\frac{z e^{\beta \epsilon}}{1+z e^{\beta \epsilon}}-\frac{1}{e^{-\beta(\epsilon+\mu)}+1}$$

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