Since the probabilities must add up to 1, the probability P is the inverse of the number of microstates W within the range of energy, The range of energy is then reduced in width until it is infinitesimally narrow, still centered at E. In the limit of this process, the microcanonical ensemble is obtained.
[1] Because of its connection with the elementary assumptions of equilibrium statistical mechanics (particularly the postulate of a priori equal probabilities), the microcanonical ensemble is an important conceptual building block in the theory.
[3][4] On the other hand, most nontrivial systems are mathematically cumbersome to describe in the microcanonical ensemble, and there are also ambiguities regarding the definitions of entropy and temperature.
Generally, fluctuations are negligible if a system is macroscopically large, or if it is manufactured with precisely known energy and thereafter maintained in near isolation from its environment.
More generally, the correspondence between these ensemble-based definitions and their thermodynamic counterparts is not perfect, particularly for finite systems.
[10] Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size.
This smoothing effect is usually negligible in macroscopic systems, which are sufficiently large that the free energy can approximate nonanalytic behavior exceedingly well.
However, the technical difference in ensembles may be important in the theoretical analysis of small systems.
[1] Early work in statistical mechanics by Ludwig Boltzmann led to his eponymous entropy equation for a system of a given total energy, S = k log W, where W is the number of distinct states accessible by the system at that energy.
Boltzmann did not elaborate too deeply on what exactly constitutes the set of distinct states of a system, besides the special case of an ideal gas.
This topic was investigated to completion by Josiah Willard Gibbs who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the microcanonical ensemble described in this article.
[1] Gibbs investigated carefully the analogies between the microcanonical ensemble and thermodynamics, especially how they break down in the case of systems of few degrees of freedom.
In many realistic cases a system is thermostatted to a heat bath so that the energy is not precisely known.
In quantum mechanics, diagonalization provides a discrete set of microstates with specific energies.
To construct the microcanonical ensemble, it is necessary in both types of mechanics to first specify a range of energy.
(a function of H, peaking at E with width ω) will be used to represent the range of energy in which to include states.
An example of this function would be[1] or, more smoothly, A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by
For very small energy width, the ensemble does not exist at all for most values of E, since no states fall within the range.
When the ensemble does exist, it typically only contains one (or two) states, since in a complex system the energy levels are only ever equal by accident (see random matrix theory for more discussion on this point).
Moreover, the state-volume function also increases only in discrete increments, and so its derivative is only ever infinite or zero, making it difficult to define the density of states.
This problem can be solved by not taking the energy range completely to zero and smoothing the state-volume function, however this makes the definition of the ensemble more complicated, since it becomes then necessary to specify the energy range in addition to other variables (together, an NVEω ensemble).
In classical mechanics, an ensemble is represented by a joint probability density function ρ(p1, ... pn, q1, ... qn) defined over the system's phase space.
The probability density function for the microcanonical ensemble is: where Again, the value of W is determined by demanding that ρ is a normalized probability density function: This integral is taken over the entire phase space.
Based on the above definition, the microcanonical ensemble can be visualized as an infinitesimally thin shell in phase space, centered on a constant-energy surface.
For an ideal gas, the energy is independent of the particle positions, which therefore contribute a factor of
The temperature is given by which agrees with analogous result from the kinetic theory of gases.
Calculating the pressure gives the ideal gas law: Finally, the chemical potential
is The microcanonical phase volume can also be calculated explicitly for an ideal gas in a uniform gravitational field.
, confined in a thermally isolated container that is infinitely long in the z-direction and has constant cross-sectional area
(averaged over all heights) is The analogues of these equations in the canonical ensemble are the barometric formula and the Maxwell–Boltzmann distribution, respectively.