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# Microstate (statistical mechanics)

In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits 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. In statistical mechanics, a macrostate is characterized by a probability distribution on a certain ensemble of microstates.

This distribution describes the probability of finding the system in a certain microstate as it is subject to thermal fluctuations.

Let us now turn to the case of large systems: even if those systems are theoretically able to fluctuate between very different microstates, observing such a fluctuation becomes less and less likely as the size of the system increases. This makes up for the thermodynamic limit. In this limit, the microstates visited by a system during its fluctuations all have the same bulk (or macroscopic) properties.

## Microscopic definitions of thermodynamic concepts

The definitions of this section link the thermodynamic properties of a system to its distribution on its ensemble (or set) of microstates. Note that all definitions and expressions of this section are valid even far away from thermodynamic equilibrium.

In this article we will consider a system which is distributed on an ensemble of N microstates. $p_i$ is the probability associated to the microstate i, and $E_i$ is its energy. Here microstates form a discrete set, which means we are working in quantum statistical mechanics, and $E_i$ is an energy level of the system.

### Internal energy

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

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

This definition is the traduction of the first law of thermodynamics.

### Entropy

The absolute entropy exclusively depends on the probabilities of the microstates. Its definition is the following:

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

where $k_B$ is Boltzmann's constant

Entropy evaluates according to the second law of thermodynamics. The third law of thermodynamics is consistent with this definition, since an absolute entropy of 0 means that the macrostate of the system reduces to a single microstate.

### Heat and work

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.

On the other hand heat is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in energy levels of the system.

The microscopic definitions of heat and work are the following:

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

So that

$~dU = delta W + delta Q$

Examples:

Warning: 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.