Overview
Temperature is a concept with which everyone is familiar, however, defining its meaning in a physical sense is not as straighforward. The starting place in defining temperature is the observation that when two bodies with different temperatures come into contact, both bodies will move toward a common temperature. When the bodies reach equilibrium, they share the same temperature.
For the purposes of illustration, we imagine two bodies with different temperatures that come into contact. They are allowed to exchange heat, but are not allowed to exchange particles.
Microstates
The two systems that are in thermal contact, each have a number of states that they can be in, given by {% \Omega(E_1) %} and {% \Omega(E_2) %}. The total number of states for the entire system is the multiplication of these two numbers
{% \Omega = \Omega(E_1) \Omega(E_2) %}
The two systems can exchange thermal energy, therefore, {% E_1 %} and {% E_2 %} are not constant, however,
they do not exchange energy with other systems, so the sum is a constant.
{% E = E_1 + E_2 %}
Probability of a Given Microstate
The argument here follows that present in blundell.
The system is in equilibrium when the number of states is maximized. We solve for the value of {% E_1 %} by solving for the maximum value of {% \Omega %}
{% \frac{d}{dE_1} (\Omega) = 0 %}
{% \Omega_2(E_2) \frac{d\Omega_1(E_1)}{E_1} + \Omega_1(E_1) \frac{d \Omega_2(E_2)}{dE_2} \frac{dE_2}{dE_1} = 0 %}
Because the sum of energies is constant, we must have
{% dE_1 = - dE_2 %}
which means
{% \frac{dE_2}{dE_1} = -1 %}
{% \frac{1}{\Omega_1} \frac{d \Omega_1}{dE_1} = \frac{1}{\Omega_2} \frac{d \Omega_2}{dE_2} %}
And finally we have
{% \frac{d ln \Omega_1}{dE_1} = \frac{d ln \Omega_2}{dE_2} %}
This formula defines a quantity which is the same for systems in thermal contact, which we then define to be the
temperature.
{% \frac{1}{k_B T} = \frac{d ln \Omega}{d E} %}