| 69 |
|
($E_{\text{bath}}+E_{\gamma}$) remain constant. $\Omega(E)$, where $E$ |
| 70 |
|
is the total energy of both systems, can be represented as |
| 71 |
|
\begin{equation} |
| 72 |
< |
\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}) |
| 72 |
> |
\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}). |
| 73 |
|
\label{introEq:SM1} |
| 74 |
|
\end{equation} |
| 75 |
|
Or additively as |
| 76 |
|
\begin{equation} |
| 77 |
< |
\ln \Omega(E) = \ln \Omega(E_{\gamma}) + \ln \Omega(E - E_{\gamma}) |
| 77 |
> |
\ln \Omega(E) = \ln \Omega(E_{\gamma}) + \ln \Omega(E - E_{\gamma}). |
| 78 |
|
\label{introEq:SM2} |
| 79 |
|
\end{equation} |
| 80 |
|
|
| 86 |
|
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
| 87 |
|
+ |
| 88 |
|
\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
| 89 |
< |
\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}} |
| 89 |
> |
\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}}, |
| 90 |
|
\label{introEq:SM3} |
| 91 |
|
\end{equation} |
| 92 |
< |
Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
| 92 |
> |
where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
| 93 |
|
$\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}}$ is |
| 94 |
|
$-1$. Eq.~\ref{introEq:SM3} becomes |
| 95 |
|
\begin{equation} |
| 96 |
|
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} = |
| 97 |
< |
\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
| 97 |
> |
\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}}. |
| 98 |
|
\label{introEq:SM4} |
| 99 |
|
\end{equation} |
| 100 |
|
|
| 105 |
|
process is the partitioning of energy among the two systems. This |
| 106 |
|
allows the following definition of entropy: |
| 107 |
|
\begin{equation} |
| 108 |
< |
S = k_B \ln \Omega(E) |
| 108 |
> |
S = k_B \ln \Omega(E), |
| 109 |
|
\label{introEq:SM5} |
| 110 |
|
\end{equation} |
| 111 |
< |
Where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
| 112 |
< |
also define the temperature of the system using the Maxwell relation |
| 111 |
> |
where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
| 112 |
> |
also define the temperature of the system using the Maxwell relation, |
| 113 |
|
\begin{equation} |
| 114 |
< |
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
| 114 |
> |
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V}. |
| 115 |
|
\label{introEq:SM6} |
| 116 |
|
\end{equation} |
| 117 |
|
The temperature in the system $\gamma$ is then |
| 118 |
|
\begin{equation} |
| 119 |
|
\beta( E_{\gamma} ) = \frac{1}{k_B T} = |
| 120 |
< |
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
| 120 |
> |
\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}}. |
| 121 |
|
\label{introEq:SM7} |
| 122 |
|
\end{equation} |
| 123 |
|
Applying this to Eq.~\ref{introEq:SM4} gives the following |
| 124 |
|
\begin{equation} |
| 125 |
< |
\beta( E_{\gamma} ) = \beta( E_{\text{bath}} ) |
| 125 |
> |
\beta( E_{\gamma} ) = \beta( E_{\text{bath}} ). |
| 126 |
|
\label{introEq:SM8} |
| 127 |
|
\end{equation} |
| 128 |
|
Eq.~\ref{introEq:SM8} shows that the partitioning of energy between |
| 140 |
|
to the total energy of both systems and the fluctuations in |
| 141 |
|
$E_{\gamma}$: |
| 142 |
|
\begin{equation} |
| 143 |
< |
\Omega( E_{\gamma} ) = \Omega( E - E_{\gamma} ) |
| 143 |
> |
\Omega( E_{\gamma} ) = \Omega( E - E_{\gamma} ). |
| 144 |
|
\label{introEq:SM9} |
| 145 |
|
\end{equation} |
| 146 |
|
As for the expectation value, it can be defined |
| 147 |
|
\begin{equation} |
| 148 |
|
\langle A \rangle = |
| 149 |
|
\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
| 150 |
< |
P_{\gamma} A(\boldsymbol{\Gamma}) |
| 150 |
> |
P_{\gamma} A(\boldsymbol{\Gamma}), |
| 151 |
|
\label{introEq:SM10} |
| 152 |
|
\end{equation} |
| 153 |
< |
Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
| 153 |
> |
where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
| 154 |
|
an integration over all accessible points in phase space, $P_{\gamma}$ |
| 155 |
|
is the probability of being in a given phase state and |
| 156 |
|
$A(\boldsymbol{\Gamma})$ is an observable that is a function of the |
