# Partition function in Statistical Physics

• Hymne
In summary, the conversation is about the different partition functions in statistical physics, including the microcanonical, canonical, and grand canonical partition functions. The speaker is struggling to understand when to use each one and what conditions are necessary for their use. They also mention the relationships between temperature, volume, and number of particles for each type of system. Overall, they are seeking clarification on the subject.
Hymne
Hi! I am for the moment reading a course in statistical physics where the author has definied not less then three diffrent partitionfunctions.

W, Z an Z which are called the microcanonical partitionfunction, canonical partitionfunction (?) and the grand canonical partitionfunction.

I have how ever a hard time keeping track of under which conditions to use which and what we assume that the system satifies when we use one of them.

microcanonical <-> E,V,N <-> isolated system (e.g. the universe (?), a very good coolbox)
canonical <-> T,V,N <-> closed rigid system that can exchange energy (e.g. a bottle in the coolbox)
grand canonincal <-> T,V,µ <-> open rigid system = exchange of energy and particles (e.g. the upper half of the bottle)

$$1/T := \frac{\partial S}{\partial <E>}$$ and $$-\mu / T := \frac{\partial S}{\partial <N>}$$

By their definitions you can see that they indeed characterize respective equilibrium in case of equality for both systems in contact.

## What is the partition function in statistical physics?

The partition function is a mathematical concept used in statistical physics to describe the distribution of particles in a system. It is denoted by the letter Z and is defined as the sum of all possible states of a system, each weighted by the Boltzmann factor e^(-E/kT), where E is the energy of the state, k is the Boltzmann constant, and T is the temperature.

## What is the significance of the partition function?

The partition function is a crucial tool in statistical physics as it allows us to calculate the thermodynamic properties of a system, such as the internal energy, entropy, and free energy. It also provides a link between the microscopic behavior of particles and the macroscopic properties of a system.

## How is the partition function related to the probability of a state?

The partition function is related to the probability of a state by the Boltzmann distribution, which states that the probability of a state is proportional to the Boltzmann factor e^(-E/kT) of that state. This means that states with higher energies are less likely to occur, while states with lower energies are more probable.

## What are the different types of partition functions?

There are several types of partition functions used in different contexts, such as the canonical partition function for systems with a fixed number of particles, the grand canonical partition function for systems with a varying number of particles, and the microcanonical partition function for systems with fixed energy.

## How is the partition function used to calculate thermodynamic properties?

The partition function can be used to calculate thermodynamic properties through various mathematical manipulations. For example, the internal energy can be calculated by taking the derivative of the partition function with respect to temperature, while the entropy can be calculated using the partition function and the Boltzmann distribution. Other properties, such as the free energy and specific heat, can also be obtained using the partition function.

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