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# 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, pressure, volume and density.[1] Treatments on statistical mechanics, [2] [3] define a macrostate as follows. A particular set of values of energy, number of particles and volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate.

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 p_i, and an energy E_i. If the microstates are quantum-mechanical in nature, then these microstates form a discrete set as defined by quantum statistical mechanics, and E_i is an energy level of the system.

### Internal energy

The internal energy of the macrostate is the mean over all microstates of the system's energy

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

This is a microscopic statement of the notion of energy associated with the first law of thermodynamics.

### Entropy

For the more general case of the canonical ensemble, the absolute entropy depends exclusively on the probabilities of the microstates and is defined as

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

where k_B is Boltzmann's constant. For the microcanonical ensemble, consisting of only those microstates with energy equal to the energy of the macrostate, this simplifies to

S = k_B\,\ln W,

where W is the number of microstates. This form for entropy appears on Ludwig Boltzmann's gravestone in Vienna.

The second law of thermodynamics describes how the entropy of an isolated system changes in time. 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 and work can be distinguished if we take the underlying quantum nature of the system into account.

For a closed system (no transfer of matter), heat in statistical mechanics is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the quantum energy levels of the system, without change in the values of the energy levels themselves.[2]

Work is the energy transfer associated with an ordered, macroscopic action on the system. If this action acts very slowly, then the adiabatic theorem of quantum mechanics implies that this will not cause jumps between energy levels of the system. In this case, the internal energy of the system only changes due to a change of the system's energy levels.[2]

The microscopic, quantum 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 thermodynamic quantities defined in the quantum case find no analogous definition in the classical limit. The reason is that classical microstates are not defined in relation to a precise associated quantum microstate, which means that when work changes the total energy available for distribution among the classical microstates of the system, the energy levels (so to speak) of the microstates do not follow this change.