Microstate (statistical mechanics)

In contrast, the macrostate of a system refers to its macroscopic properties such as its 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 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 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 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 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

energy level of a microscopic component of a system through the direct effect of work, but it is possible to change the energy of the system's energy levels.

On the other hand energy levels for the microscopic components of the system.

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

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

Examples:

Warning: the two above definitions of heat and work are among the few expressions of 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.

Microstate (statistical mechanics)

Microscopic definitions of thermodynamic concepts

Internal energy

Entropy

Heat and work

See also