Tag Archives: Statistical Physics代写

统计物理学 Statistical Physics PX366

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这是一份warwick华威大学PX366的成功案例

统计物理学 Statistical Physics PX366


When we also define a characteristic temperature, $\Theta_{\mathrm{E}}$, the Einstein-temperature, via
$$
\Theta_{\mathrm{E}} \equiv \frac{h v_{\mathrm{E}}}{k_{\mathrm{B}}}
$$
We get the following expressions for the molar energy of the crystal and it’s specific heat.
and
$$
\begin{gathered}
\widetilde{E}^{\mathrm{E}}=\frac{3}{2} R \Theta_{\mathrm{E}}+\frac{3 R \Theta_{\mathrm{E}}}{\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1} \
\widetilde{C}{V}^{\mathrm{E}}=\left(\frac{\partial \widetilde{E}^{\mathrm{E}}}{\partial T}\right){V}=3 R \frac{\left(\frac{\Theta_{\mathrm{E}}}{T}\right)^{2} \exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)}{\left[\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1\right]^{2}}
\end{gathered}
$$

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PX366 COURSE NOTES ：

The density of frequencies, $N(v) \mathrm{d} v$, can be deduced from the density of states
$$
N(k) \mathrm{d} k=\frac{k^{2}}{2 \pi^{2}} V \mathrm{~d} k \quad \text { with } k=\frac{2 \pi}{\lambda} \quad \text { and } v \lambda=c
$$
Hence, with $k=\frac{2 \pi}{c} v$ and $\mathrm{d} k=\frac{2 \pi}{c} \mathrm{~d} v$
$$
N(v) \mathrm{d} v=\frac{4 \pi^{2} v^{2}}{2 \pi^{2} c^{2}} V \frac{2 \pi}{c} \mathrm{~d} v=\frac{4 \pi V v^{2}}{c^{3}} \mathrm{~d} v
$$










物理统计 Statistical Physics PHYS393/MATH327

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这是一份liverpool利物浦大学PHYS393/MATH327的成功案例

物理统计 Statistical Physics PHYS393/MATH327

$$
Z_{1}^{\mathrm{tr}}=\frac{V}{h^{3}}\left(2 \pi m k_{\mathrm{B}} T\right)^{\frac{3}{2}}
$$
The probability, $P_{s}$, of a particle being in a particular state $s$ (with energy $\varepsilon_{s}^{\mathrm{tr}}$ ) of translational motion is given by the Boltzmann distribution
$$
P_{s}=\frac{\exp \left(-\frac{\varepsilon_{s}^{\mathrm{tr}}}{k_{\mathrm{B}} T}\right)}{Z_{1}^{\mathrm{tr}}}
$$
When there are $N$ particles, the mean occupation number, $\bar{n}{s}$, of energy levels (average number of particles in state $s$ ) can be expressed as $$ \bar{n}{s}=N P_{s}
$$
Now we use the definition of the semi-classical system. In particular the fact, that the number of particles is much smaller than the number of available energy levels hence
$$
\bar{n}_{s} \ll<1 \text { for all } s
$$

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PHYS393/MATH327 COURSE NOTES ：

most probable speed (max. of $P(v))$
$$
v_{\mathrm{m} . \mathrm{p} .}=\sqrt{\frac{2 k_{\mathrm{B}} T}{m}}
$$
mean speed:
$$
\langle v\rangle=\int_{0}^{\infty} v P(v) \mathrm{d} v=\sqrt{\frac{8 k_{\mathrm{B}} T}{\pi m}}
$$
second moment:
$$
\left\langle v^{2}\right\rangle=\int_{0}^{\infty} v^{2} P(v) \mathrm{d} v=3 \frac{k_{\mathrm{B}} T}{m}
$$
note: this is equivalent to $\bar{E}=\frac{1}{2} m\left\langle v^{2}\right\rangle=\frac{3}{2} k_{\mathrm{B}} T$
and the rms (root-mean-square) speed: $v_{r m s}=\sqrt{\left\langle v^{2}\right\rangle}=\sqrt{\frac{3 k_{\mathrm{B}} T}{m}}$