# Definition of the number of microstates accessible to a system

by Sam_Goldberg
Tags: accessible, definition, microstates, number
 PF Gold P: 864 What is important to keep in mind is that when one talks about a number of accessible states, one is always speaking about a specific ensemble of states. In the most straightfoward way of thinking about the situation you describe the number of accessible states instantaneously increases because you instantly change the ensemble you are talking about. What happens immediately after the partition is physically removed is not part of the subject of equilibrium statistical mechanics because the system moves into a non-equilibrium state and does not sample from the equilibrium distribution of states. You could analyze the situation in different ways and come to different conclusions. You could consider the ensemble of states to include the entire container from the beginning, for instance. In that case, when the gas is confined to half the container by the partition, it is in a non-equilibrium state of this ensemble because it is kinetically trapped by the barrier from reaching certain states. This would not be a particularly useful way of analyzing the system, but my point is that in statistical mechanics, one has to keep in mind the ensemble one is talking about. As far as the inverse of the temperature goes, I think you're making a mistake. For each particle the number of positions it can be in doubles. If there are N particles, then the number of accessible states increases by a factor of $$2^N$$. Thus $$\Omega_{new} = 2^N\Omega_{old}$$ $$\rightarrow S_{new} = k\ln(\Omega_{new}) = k\ln(\Omega_{old})+k\ln{2^N} = S_{old} + k\ln(2^N)$$ $$\rightarrow \frac{1}{T_{new}} = \frac{dS_{new}}{dE}=\frac{dS_{old}}{dE}=\frac{1}{T_{old}}$$ If there was a similar change that did bring about a temperature change by passing through non-equilibrium states, then whether there is a gradual change in temperature or not depends on what definition of temperature you use. If you use a definition from the kinetic theory of gases that relies only on the velocities of the particles, then it will probably change gradually. If you use a statistical mechanical definition that depends on a derivative of the entropy, then it doesn't make sense to talk about temperature outside of equilibrium in the first place.