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Instatistical mechanics, amicrostate is a specific configuration of asystem that describes the precise positions and momenta of all the individual particles or components that make up the system. Each microstate has a certain probability of occurring during the course of the system'sthermal fluctuations.

In contrast, themacrostate of a system refers to its macroscopic properties, such as itstemperature,pressure,volume anddensity.^[1] Treatments onstatistical mechanics^[2][3] define a macrostate as follows: a particular set of values of energy, the number of particles, and the 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 aprobability distribution of possible states across a certainstatistical ensemble of all microstates. This distribution describes theprobability of finding the system in a certain microstate. In thethermodynamic limit, the microstates visited by a macroscopic system during its fluctuations all have the same macroscopic properties.

In aquantum system, the microstate is simply the value of thewave function.^[4]

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 thepartition function that sums${\displaystyle \text{exp}(-E_i/k_{\text{B}}T)}$ of all its microstates.

At any moment a system is distributed across an ensemble of${\displaystyle \mathrm{\Omega }}$ microstates, each labeled by${\displaystyle i}$, and having a probability of occupation${\displaystyle p_i}$, and an energy${\displaystyle E_i}$. If the microstates are quantum-mechanical in nature, then these microstates form a discrete set as defined byquantum statistical mechanics, and${\displaystyle E_i}$ is anenergy level of the system.

The internal energy of the macrostate is themean over all microstates of the system's energy$${\displaystyle U:=⟨E⟩=\sum _{i=1}^{\mathrm{\Omega }}{p}_i{E}_i}$$

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

For the more general case of thecanonical ensemble, the absoluteentropy depends exclusively on the probabilities of the microstates and is defined as$${\displaystyle S:=-k_{\text{B}}\sum _{i=1}^{\mathrm{\Omega }}{p}_i\ln (p_i)}$$where${\displaystyle k_{\text{B}}}$ is theBoltzmann constant. For themicrocanonical ensemble, consisting of only those microstates with energy equal to the energy of the macrostate, this simplifies to$${\displaystyle S=k_B\ln \mathrm{\Omega }}$$with the number${\displaystyle \mathrm{\Omega }}$ of microstates (often in combination with auniform or equal probability distribution for the microstates, which in the discrete case has${\displaystyle p_i:=1/\mathrm{\Omega }}$ for each i). This form for entropy appears onLudwig Boltzmann's gravestone in Vienna.

Thesecond law of thermodynamics describes how the entropy of an isolated system changes in time. Thethird 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 can be distinguished if we take the underlying quantum nature of the system into account and consider theLeibniz product rule for the differential of the internal energy:

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 changes in occupation numbers of the quantum energy levels of the system (i.e. with changes${\displaystyle d{p}_i}$ of the occupation probabilities of the states), 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 theadiabatic 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 changes${\displaystyle d{E}_i}$ of the system's energy levels.^[2]

The microscopic, quantum definitions of heat and work are the following:

so that$${\displaystyle dU=\delta W+\delta Q.}$$

The two above definitions of heat and work are among the few expressions ofstatistical 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.

The description of a classical system ofFdegrees of freedom may be stated in terms of a 2F dimensionalphase space, whose coordinate axes consist of theFgeneralized coordinatesq_i of the system, and itsF generalized momentap_i. The microstate of such a system will be specified by a single point in the phase space. But for a system with a huge number of degrees of freedom its exact microstate usually is not important. So the phase space can be divided into cells of the sizeh₀ = Δq_iΔp_i, each treated as a microstate. Now the microstates are discrete and countable^[5] and the internal energyU has no longer an exact value but is betweenU andU+δU, with$\delta U\ll U$.

The number of microstates Ω that a closed system can occupy is proportional to its phase space volume:$${\displaystyle \mathrm{\Omega }(U)=\frac{1}{h_0^{\mathcal{F}}}\int {\mathbf{1}}_{\delta U}(H(x)-U)\prod _{i=1}^{\mathcal{F}}d{q}_{i}d{p}_i}$$where${\mathbf{1}}_{\delta U}(H(x)-U)$ is anIndicator function. It is 1 if the Hamilton functionH(x) at the pointx = (q,p) in phase space is betweenU andU+ δU and 0 if not. The constant$1/h_0^{\mathcal{F}}$ makes Ω(U) dimensionless. For an ideal gas is${\displaystyle \mathrm{\Omega }(U)\propto \mathcal{F}{U}^{\frac{\mathcal{F}}{2}-1}\delta U}$.^[6]

In this description, the particles are distinguishable. If the position and momentum of two particles are exchanged, the new state will be represented by a different point in phase space. In this case a single point will represent a microstate. If a subset ofM particles are indistinguishable from each other, then theM! possible permutations or possible exchanges of these particles will be counted as part of a single microstate. The set of possible microstates are also reflected in the constraints upon the thermodynamic system.

For example, in the case of a simple gas ofN particles with total energyU contained in a cube of volumeV, in which a sample of the gas cannot be distinguished from any other sample by experimental means, a microstate will consist of the above-mentionedN! points in phase space, and the set of microstates will be constrained to have all position coordinates to lie inside the box, and the momenta to lie on a hyperspherical surface in momentum coordinates of radiusU. If on the other hand, the system consists of a mixture of two different gases, samples of which can be distinguished from each other, sayA andB, then the number of microstates is increased, since two points in which anA andB particle are exchanged in phase space are no longer part of the same microstate. Two particles that are identical may nevertheless be distinguishable based on, for example, their location. (Seeconfigurational entropy.) If the box contains identical particles, and is at equilibrium, and a partition is inserted, dividing the volume in half, particles in one box are now distinguishable from those in the second box. In phase space, theN/2 particles in each box are now restricted to a volumeV/2, and their energy restricted toU/2, and the number of points describing a single microstate will change: the phase space description is not the same.

This has implications in both theGibbs paradox andcorrect Boltzmann counting. With regard to Boltzmann counting, it is the multiplicity of points in phase space which effectively reduces the number of microstates and renders the entropy extensive. With regard to Gibbs paradox, the important result is that the increase in the number of microstates (and thus the increase in entropy) resulting from the insertion of the partition is exactly matched by the decrease in the number of microstates (and thus the decrease in entropy) resulting from the reduction in volume available to each particle, yielding a net entropy change of zero.

• Quantum statistical mechanics
• Degrees of freedom (physics and chemistry)
• Ergodic hypothesis
• Phase space

• Some illustrations of microstates vs. macrostates

• Statistical mechanics

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