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## Main Question or Discussion Point

I have a question that has puzzled me during the last couple of days. Suppose that there is a system (e.g. a small box filled with gas) that is connected to a heat bath (much larger than the system) at a constant temperature T. The studied system and the heat bath are thought to be isolated from any other systems. The edges of the box could be interpreted as the interface to the heat bath.

Now we apply an amount of external work (W) to the system. The system is strongly thermally connected with the heat bath. Therefore as time passes, the work W will be transferred to the heat bath in the form of heat and the system will end up in a state similar to the initial state. From the Helmholtz free energy point of view we have dF = dW - SdT = dW, because the system is isothermal. Thus in the total process the free energy of the system has changed by an amount ΔF = W = ∫dW > 0.

On the other hand the system can be viewed as a canonical ensemble. Now we notice that the initial and final states of the system are similar and they are both described by the same partition function, [itex]Z_{init}=Z_{final}[/itex]. This occurs, since the additional energy W gets absorbed by the heath bath and the temperature T is constant. The free energy change in a canonical ensemble is obtained by

[itex]\Delta F = -k_BT \ln \frac{Z_{final}}{Z_{init}}[/itex].

Because of the similarity of the partition functions, the free energy change would now be zero. This is clearly in contradiction with the previous reasoning, could someone think of a good explanation for this? For instance, sometimes the canonical ensemble is derived by interpreting the heat bath and the system to form a microcanonical ensemble. This could cause problems with the present system, since external energy W is added and the total energy of the bath and the system would not be constant.

Thanks in advance!

Now we apply an amount of external work (W) to the system. The system is strongly thermally connected with the heat bath. Therefore as time passes, the work W will be transferred to the heat bath in the form of heat and the system will end up in a state similar to the initial state. From the Helmholtz free energy point of view we have dF = dW - SdT = dW, because the system is isothermal. Thus in the total process the free energy of the system has changed by an amount ΔF = W = ∫dW > 0.

On the other hand the system can be viewed as a canonical ensemble. Now we notice that the initial and final states of the system are similar and they are both described by the same partition function, [itex]Z_{init}=Z_{final}[/itex]. This occurs, since the additional energy W gets absorbed by the heath bath and the temperature T is constant. The free energy change in a canonical ensemble is obtained by

[itex]\Delta F = -k_BT \ln \frac{Z_{final}}{Z_{init}}[/itex].

Because of the similarity of the partition functions, the free energy change would now be zero. This is clearly in contradiction with the previous reasoning, could someone think of a good explanation for this? For instance, sometimes the canonical ensemble is derived by interpreting the heat bath and the system to form a microcanonical ensemble. This could cause problems with the present system, since external energy W is added and the total energy of the bath and the system would not be constant.

Thanks in advance!