# Microstate (statistical mechanics)

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Microstate (statistical mechanics)

In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature and pressure.[1]

A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates. This distribution describes the probability of finding the system in a certain microstate. In the thermodynamic limit, the microstates visited by a macroscopic system during its fluctuations all have the same macroscopic properties.

## Microscopic definitions of thermodynamic concepts

Statistical mechanics links the empirical thermodynamic properties of a system to the statistical distribution of an ensemble of microstates. All macroscopic thermodynamic properties of a system may be calculated from the partition function that sums the energy of all its microstates.

At any moment a system is distributed across an ensemble of N microstates, each denoted by i, and having a probability of occupation pi, and an energy Ei. These microstates form a discrete set as defined by quantum statistical mechanics, and Ei is an energy level of the system.

### Internal energy

The internal energy is the mean of the system's energy

$U = \langle E \rangle = \sum_{i=1}^N p_i \,E_i\ .$

This is a microscopic statement of the first law of thermodynamics.

### Entropy

The absolute entropy exclusively depends on the probabilities of the microstates and is defined as

$S = -k_B\,\sum_i p_i \ln \,p_i,$

where kB is Boltzmann's constant.

Entropy is formulated by the second law of thermodynamics. The third law of thermodynamics is consistent with this definition, since zero entropy means that the macrostate of the system reduces to a single microstate.

### Heat and work

Heat is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in energy levels of the system.

Work is the energy transfer associated to the effect of an ordered, macroscopic action on the system. If this action acts very slowly then the Adiabatic theorem implies that this will not cause a jump in the energy level of the system. The internal energy of the system can only change due to a change of the energies of the system's energy levels.

The microscopic definitions of heat and work are the following:

$\delta W = \sum_{i=1}^N p_i\,dE_i$
$\delta Q = \sum_{i=1}^N E_i\,dp_i$

so that

$~dU = \delta W + \delta Q.$

The two above definitions of heat and work are among the few expressions of statistical mechanics where the sum corresponding to the quantum case cannot be converted into an integral in the classical limit of a microstate continuum. The reason is that classical microstates are usually not defined in relation to a precise associated quantum microstate, which means that when work changes the energy associated to the energy levels of the system, the energy of classical microstates doesn't follow this change.

## References

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