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In canonical ensemble we treat the remainder systems as heat reservoir. Ensemble is collection of systems in all possible states of a system under study. Actually they are like mental photo copies of the system in different state. In canonical ensemble we take that energy is moving from remainder systems to the system and vice versa. But I am confused that how energy will transfer from one state of a system to other state of that same system? Thank you .

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2 Answers 2

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It is important to avoid mixing between different approaches. In a pure ensemble picture, there is a set of clones (the ensemble) of the system of interest subject to some individual and global constraints.

In the microcanonical ensemble, each member of the ensemble has the same energy, volume, and number of particles.

In the canonical ensemble, only the volume and number of particles of each member are fixed. The energy of each system can vary, although within the global constraint that the sum of individual system energies is constant. Such a theoretical construction allows using a max-entropy principle to determine the probability distribution function of the ensemble.

The exact physical mechanism allowing the transfer of energy from a system to another is not important, because the ensemble is not a physical object and what really matters is the theoretical mechanism allowing to introduce the concept of probability of individual microstates.

Notice that the ensemble method is different from the older approach based on the idea of a small system in thermal contact with a real heat reservoir. Both methods allow getting the same conclusions, after taking the proper thermodynamic limits. However, the ensemble method is considered conceptually more powerful.

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  • $\begingroup$ In canonical ensemble we've system and reservoir in equilibrium at temperature T but we still say that the system can change its energy. Why do we say that? If system changes its energy, it's temperature will also change? Can you please explain? $\endgroup$
    – Kashmiri
    May 11, 2022 at 12:43
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It is not actually the same system. The (sub-)system of interest is a part of a much bigger system. In many cases it is convenient to think of it as a collection of "identical" systems, i.e., the systems of similar size, composition, etc. - but they are not actually the same system.

It might be also useful to remind that even for microcanonical ensemble we assume that all the states of the system with the same energy are equally probable - even though an actual trajectory in the phase space may never pass through all the states of the same energy. When this reasoning is taken to the canonical ensemble, we assume that the other similar systems can be found in any of their states with the required energy.

Finally, there is obviously the energy flow between the system and its surroundings, but this energy flow is usually negligeable compared to the energy of the system. Indeed, the energy flow is proportional to the surface of the system, which scales as a square of the system size, whereas the energy of the system scales as its volume, i.e.,a s the cube of its size.

These points are usually covered in good texts on statistical physics, although easily overlooked in the first reading, when one focuses on equations.

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  • $\begingroup$ In canonical ensemble we've system and reservoir in equilibrium at temperature T but we still say that the system can change its energy. Why do we say that? If system changes its energy, it's temperature will also change? Can you please explain? $\endgroup$
    – Kashmiri
    May 11, 2022 at 12:43

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