91影视

The expression for the average energy $\stackrel{\mathbf{炉}}{\mathbf{E}}$of a system of one-dimensional harmonic oscillators is found to be:

$\stackrel{\mathbf{炉}}{\mathbf{E}}\mathbf{=}\frac{{\displaystyle 鈭�{\mathrm{ENAe}}^{鈭�E/k_{B}T}(1/\mathrm{鈩�}{蝇}_0)\mathrm{dE}}}{{\displaystyle 鈭�{\mathrm{NAe}}^{鈭�E/k_{B}T}(1/\mathrm{鈩�}{蝇}_0)\mathrm{dE}}}$

Simplify the expression further yields:

$\stackrel{炉}{\mathrm{E}}=\frac{{\displaystyle 鈭�{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}{{\displaystyle 鈭�{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}$

In the limit where ${\mathrm{k}}_{\mathrm{B}}\mathrm{T}>>\mathrm{鈩�}{\mathrm{蝇}}_0$, the integration is carried out from 0 to $鈭�$.

Thus:$\stackrel{炉}{\mathrm{E}}=\frac{{\displaystyle {鈭�}_0^{\mathrm{鈭�}}{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}{{\displaystyle {鈭�}_0^{\mathrm{鈭�}}{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}$

First evaluate the numerator. Consider:

${鈭�}_0^{\mathrm{鈭�}}{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}$

Use integration by parts, perform the change of variables:

Let:$\mathrm{u}=\mathrm{E};\mathrm{dv}=鈭�{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}$

Then solve further:

$\mathrm{du}=\mathrm{dE};\text{鈥�}\mathrm{v}=鈭�{\mathrm{k}}_{\mathrm{B}}{\mathrm{Te}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}$

Use integration by parts, recall that:

$鈭�\mathrm{udv}=\mathrm{uv}鈭�鈭�\mathrm{v}\mathrm{du}$

If:

$鈭�\mathrm{udv}=鈭�\mathrm{E}{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}$

then:

$\begin{array}{c}鈭�{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}=鈭�{\mathrm{k}}_{\mathrm{B}}{\mathrm{TEe}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}鈭�鈭�鈭�{\mathrm{k}}_{\mathrm{B}}{\mathrm{Te}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}\\ =鈭�{\mathrm{k}}_{\mathrm{B}}{\mathrm{TEe}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}+{\mathrm{k}}_{\mathrm{B}}\mathrm{T}鈭�{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}\end{array}$

Solve the integral:

$鈭�{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}=鈭�{\mathrm{k}}_{\mathrm{B}}{\mathrm{TEe}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}鈭�{\left({\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)}^2{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}$

Let us now evaluate the definite integral:

$\begin{array}{c}{鈭�}_0^{\mathrm{鈭�}}{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}=鈭�{{\mathrm{k}}_{\mathrm{B}}{\mathrm{TEe}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}|}_0^{\mathrm{鈭�}}鈭�{{\left({\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)}^2{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}|}_0^{\mathrm{鈭�}}\\ =0鈭�{\left({\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)}^2(鈭�1)\\ ={\left({\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)}^2\end{array}$

Now consider the denominator. Consider the definite integral:

${鈭�}_0^{\mathrm{鈭�}}{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}$

Evaluate the problem:

$\begin{array}{c}{鈭�}_0^{\mathrm{鈭�}}{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}={(鈭�{\mathrm{k}}_{\mathrm{B}}\mathrm{T}){\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}|}_0^{\mathrm{鈭�}}\\ =鈭�{\mathrm{k}}_{\mathrm{B}}\mathrm{T}(鈭�1)\\ ={\mathrm{k}}_{\mathrm{B}}\mathrm{T}\end{array}$

Take their quotient, the average energy for a system of harmonic oscillators is thus:

$\begin{array}{c}\stackrel{炉}{\mathrm{E}}=\frac{{\displaystyle {鈭�}_0^{\mathrm{鈭�}}{\mathrm{Ee}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}{{\displaystyle {鈭�}_0^{\mathrm{鈭�}}{\mathrm{e}}^{鈭�\mathrm{E}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\mathrm{dE}}}\\ =\frac{{\left({\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)}^2}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\\ ={\mathrm{k}}_{\mathrm{B}}\mathrm{T}\end{array}$