Assumption of thermal equilibrium for ensembles

In summary, when deriving probabilities and distributions for a canonical ensemble, it is assumed that the system is in thermal equilibrium with the heat bath. This is typically done by assuming a constant temperature for the heat bath and considering the combined system of the heat bath and the system as an isolated system, using the max-entropy principle. This approach is outlined in H B Callen's book on thermodynamics.
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Take for example a canonical ensemble, to use the derived distributions and probabilities, the considered system has to be in thermal equilibrium with the heat bath.

Where in the derivation of the probabilities and distributions do we usually assume this? There is a point that I can identify assuming a constant temperature for the heat bath yes, but I can't find where we assume that the temperature of the system is equal to the temperature of the heat bath.
 
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You are deducing the results for a system in thermal interaction with an heat bath by considering the heat bath + the system as an isolated system and using the max-entropy principle for the combined system. See H B Callen's book on thermodynamics.
 

FAQ: Assumption of thermal equilibrium for ensembles

What is the assumption of thermal equilibrium for ensembles?

The assumption of thermal equilibrium for ensembles is that all particles within the system are in a state of equilibrium, meaning they have the same average energy and temperature. This allows for the use of statistical mechanics to describe the behavior of the system.

Why is the assumption of thermal equilibrium important?

The assumption of thermal equilibrium is important because it simplifies the mathematical calculations and analysis of a system. It allows for the use of statistical mechanics, which is a powerful tool for understanding the behavior of large systems.

What happens if the assumption of thermal equilibrium is not met?

If the assumption of thermal equilibrium is not met, the system is said to be in a non-equilibrium state. This means that the particles within the system have different temperatures and energies, and the system's behavior cannot be accurately described using statistical mechanics.

How is the assumption of thermal equilibrium applied in different ensembles?

The assumption of thermal equilibrium is applied differently depending on the ensemble being used. For example, in the microcanonical ensemble, the assumption is that the system is in equilibrium with its own energy, while in the canonical ensemble, the assumption is that the system is in equilibrium with its temperature.

What are the limitations of the assumption of thermal equilibrium?

The assumption of thermal equilibrium is limited by the fact that it is an idealized assumption and may not accurately reflect the behavior of real systems. Additionally, it may not be applicable in systems that are far from equilibrium, such as in highly non-linear or chaotic systems.

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