Why Do We Need to Outline a Proof for \beta=1/(Kb*T) in Stat Mech?

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we have to outline a proof to show that beta =1/(Kb*T) for a gas of fermions. we are supposed to put this system in thermal contact with a system obeying classical statistics, so that the two systems have the same beta, invoke the zeroth law to sat that they have the same temperature, and then show that beta=1/(Kb*T) for the classical system. The problem I have with this proof is that how do we know that just because we put the two system in contact they will have the same beta? Or if we assume that they have the same beta, why will they necessarily be in equilibrium? Also, does our proof apply to a gas of bosons as well? I don't see any reason why it shouldn't, but for some reason the question specifies for fermions only. thanks for your help.

Sachi
 
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How have you defined \beta? I need to know where you're coming from to offer specific advice.
 
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