next up previous
Next: Partition function and the Up: Boltzmann statistics Previous: The zeroth law of

Boltzmann factor

Consider now a system in contact with a heat bath, or reservoir. System 1 is the one we are interested in, and we want to find the probability P(E1) that it has energy E1. We assume that system 2 is much larger than 1, so that E1<< E=E1+E2. Another way of putting this is to say that the heat capacity C2 of system 2 is very large. We assume that all compatible microstates are equally likely. We have

\begin{displaymath}P(E_1)dE_1={{g_1(E_1)g_2(E-E_1) dE_1}\over{
\int dE_1g_1(E_1)g_2(E-E_1)}}\end{displaymath}

From the definition of entropy

\begin{displaymath}g_2=\frac{1}{\delta E}\exp\left[{{S_2(E-E_1)}\over{k_B}}\right]\end{displaymath}

We expand in a Taylor series

 \begin{displaymath}
S_2(E-E_1)=S_2(E)-E_1{{\partial S_2}\over{\partial
E}}+{{E_1^2}\over{2}}{{\partial^2S_2}\over{\partial
E^2}}+\cdots\end{displaymath} (13)

With T the temperature of the heat bath and C its heat capacity, the partial derivatives are given by

 \begin{displaymath}
{{\partial S_2}\over{\partial E}}={{1}\over{T}}\end{displaymath} (14)


 \begin{displaymath}
{{\partial^2S_2}\over{\partial E^2}}={{\partial
{{1}\over{T...
...1}\over{T^2}}{{\partial
T}\over{\partial E}}=-{{1}\over{T^2C}}\end{displaymath} (15)

Since E1<<TC we neglect the last term in ([*]) giving

\begin{displaymath}g_2=const. \exp\left[{{-E_1}\over{k_BT}}\right]
=const. e^{-\beta {E_1}}\end{displaymath}

where we define $\beta =1/(k_BT)$ . We conclude

 \begin{displaymath}
P(E_1)= const. g_1(E_1)e^{-\beta {E_1}}\end{displaymath} (16)

The factor $e^{-\beta {E_1}}$ is the Boltzmann factor. When a system is in contact with a heat bath at a certain temperature, all possible microstates of the system are no longer equally likely. Instead, the Boltzmann factor acts as a weight factor biasing the distribution towards states with lower energy.
next up previous
Next: Partition function and the Up: Boltzmann statistics Previous: The zeroth law of
Birger Bergersen
1998-09-14