In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it cannot exchange energy or particles with its environment, so that (by conservation of energy) the energy of the system does not change with time.
The primary macroscopic variables of the microcanonical ensemble are the total number of particles in the system (symbol: ), the system's volume (symbol: ), as well as the total energy in the system (symbol: ). Each of these is assumed to be constant in the ensemble. For this reason, the microcanonical ensemble is sometimes called the ensemble.
In simple terms, the microcanonical ensemble is defined by assigning an equal probability to every microstate whose energy falls within a range centered at . All other microstates are given a probability of zero. Since the probabilities must add up to 1, the probability is the reciprocal of the number of microstates within the range of energy,
<math display="block">P = 1/W,</math>
The range of energy is then reduced in width until it is infinitesimally narrow, still centered at . In the limit of this process, the microcanonical ensemble is obtained. It is sometimes considered to be the fundamental distribution of equilibrium statistical mechanics. It is also useful in some numerical applications, such as molecular dynamics. 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. For these reasons, other ensembles are often preferred for theoretical calculations.
The applicability of the microcanonical ensemble to real-world systems depends on the importance of energy fluctuations, which may result from interactions between the system and its environment as well as uncontrolled factors in preparing the system. 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. In such cases the microcanonical ensemble is applicable. Otherwise, different ensembles are more appropriate – such as the canonical ensemble (fluctuating energy) or the grand canonical ensemble (fluctuating energy and particle number).
Properties
Thermodynamic quantities
The fundamental thermodynamic potential of the microcanonical ensemble is entropy. There are at least three possible definitions, each given in terms of the phase volume function . In classical mechanics this is the volume of the region of phase space where the energy is less than . In quantum mechanics is roughly the number of energy eigenstates with energy less than ; however this must be smoothed so that we can take its derivative (see the Precise expressions section for details on how this is done). The definitions of microcanonical entropy are:
In the microcanonical ensemble, the temperature is a derived quantity rather than an external control parameter. It is defined as the derivative of the chosen entropy with respect to energy. For example, one can define the "temperatures" and as follows:
<math display="block">\begin{align}
\frac{1}{T_v} &= \frac{dS_v}{dE}, &
\frac{1}{T_s} &= \frac{dS_s}{dE} = \frac{dS_\text{B{dE}.
\end{align}</math>
Like entropy, there are multiple ways to understand temperature in the microcanonical ensemble. More generally, the correspondence between these ensemble-based definitions and their thermodynamic counterparts is not perfect, particularly for finite systems.
The microcanonical pressure and chemical potential are given by:
<math display="block"> \frac{p}{T}=\frac{\partial S}{\partial V}; \qquad \frac{\mu}{T}=-\frac{\partial S}{\partial N}</math>
Phase transitions
Under their strict definition, phase transitions correspond to nonanalytic behavior in the thermodynamic potential or its derivatives. Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size. This contrasts with the canonical and grand canonical ensembles, for which phase transitions can occur only in the thermodynamic limit – i.e., in systems with infinitely many degrees of freedom. Roughly speaking, the reservoirs defining the canonical or grand canonical ensembles introduce fluctuations that "smooth out" any nonanalytic behavior in the free energy of finite systems. 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.
The preferred solution to these problems is avoid use of the microcanonical ensemble. In many realistic cases a system is thermostatted to a heat bath so that the energy is not precisely known. Then, a more accurate description is the canonical ensemble or grand canonical ensemble, both of which have complete correspondence to thermodynamics.
Precise expressions for the ensemble
The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration – quantum or classical – since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, diagonalization provides a discrete set of microstates with specific energies. The classical mechanical case involves instead an integral over canonical phase space, and the size of microstates in phase space can be chosen somewhat arbitrarily.
To construct the microcanonical ensemble, it is necessary in both types of mechanics to first specify a range of energy. In the expressions below the function <math>f{\left(\tfrac{H - E}{\omega}\right)}</math> (a function of , peaking at with width ) will be used to represent the range of energy in which to include states. An example of this function would be
- is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.
Again, the value of is determined by demanding that is a normalized probability density function:
<math display="block">W = \int \cdots \int \frac{1}{h^n C} f{\left(\tfrac{H-E}{\omega}\right)} \, dp_1 \cdots dq_n </math>
This integral is taken over the entire phase space. The state volume function (used to calculate entropy) is defined by
<math display="block">v(E) = \int \cdots \int_{H < E} \frac{1}{h^n C} \, dp_1 \cdots dq_n .</math>
As the energy width is taken to zero, the value of decreases in proportion to as .
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. Although the microcanonical ensemble is confined to this surface, it is not necessarily uniformly distributed over that surface: if the gradient of energy in phase space varies, then the microcanonical ensemble is "thicker" (more concentrated) in some parts of the surface than others. This feature is an unavoidable consequence of requiring that the microcanonical ensemble is a steady-state ensemble.
Examples
Ideal gas
The fundamental quantity in the microcanonical ensemble is <math>W(E, V, N)</math>, which is equal to the phase space volume compatible with given <math>(E, V, N)</math>. From <math>W</math>, all thermodynamic quantities can be calculated. For an ideal gas, the energy is independent of the particle positions, which therefore contribute a factor of <math>V^N</math> to <math>W</math>. The momenta, by contrast, are constrained to a <math>3N</math>-dimensional (hyper-)spherical shell of radius <math>\sqrt{2mE}</math>; their contribution is equal to the surface volume of this shell. The resulting expression for <math>W</math> is:
<math display="block">
W = \frac{V^N}{N!} \frac{2\pi^{3N/2{\Gamma(3N/2)}\left(2mE\right)^{(3N-1)/2}
</math>
where <math> \Gamma(\cdot) </math> is the gamma function, and the factor <math>N!</math> has been included to account for the indistinguishability of particles (see Gibbs paradox). In the large <math>N</math> limit, the Boltzmann entropy <math>S = k_{\mathrm{B \log W</math> is
<math display="block">
S = k_\text{B} N \log \left[ \frac VN \left(\frac{4\pi m}{3}\frac EN\right)^{3/2}\right] + \frac{5}{2} k_\text{B} N + O\left( \log N \right)
</math>
This is also known as the Sackur–Tetrode equation.
The temperature is given by
<math display="block">
\frac{1}{T} \equiv \frac{\partial S}{\partial E} = \frac{3}{2} \frac{N k_\text{B{E}
</math>
which agrees with analogous result from the kinetic theory of gases. Calculating the pressure gives the ideal gas law:
<math display="block">
\frac{p}{T} \equiv \frac{\partial S}{\partial V} = \frac{N k_\text{B{V} \quad \rightarrow \quad pV = N k_\text{B} T
</math>
Finally, the chemical potential <math>\mu</math> is
<math display="block">
\mu \equiv -T \frac{\partial S}{\partial N} = -k_\text{B} T \log \left[\frac{V}{N} \, \left(\frac{4 \pi m E}{3N} \right)^{3/2} \right]
</math>
Ideal gas in a uniform gravitational field
The microcanonical phase volume can also be calculated explicitly for an ideal gas in a uniform gravitational field.
The results are stated below for a 3-dimensional ideal gas of <math>N</math> particles, each with mass <math>m</math>, confined in a thermally isolated container that is infinitely long in the z-direction and has constant cross-sectional area <math>A</math>. The gravitational field is assumed to act in the minus z direction with strength <math>g</math>. The phase volume <math>W(E, N)</math> is
<math display="block">
W(E, N) = \frac{(2 \pi)^{3N/2} A^N m^{N/2{g^N \, \Gamma{\left(\frac{5N}{2}\right) E^{\frac{5N}{2}-1}
</math>
where <math>E</math> is the total energy, kinetic plus gravitational.
The gas density <math>\rho(z)</math> as a function of height <math>z</math> can be obtained by integrating over the phase volume coordinates. The result is:
<math display="block">
\rho(z) = \left(\frac{5N}{2} - 1 \right) \frac{mg}{E}\left(1 - \frac{mgz}{E} \right)^{\frac{5N}{2}-2}
</math>
Similarly, the distribution of the velocity magnitude <math>\left|\mathbf{v}\right|</math> (averaged over all heights) is
<math display="block">
f(|\mathbf{v}|)
= \frac{\Gamma{\left(\frac{5N}{2}\right){\Gamma{\left(\frac{3}{2}\right)} \, \Gamma{\left(\frac{5N}{2} - \frac{3}{2}\right)
\cdot \frac{m^{3/2} {\left|\mathbf{v}\right|}^2}{2^{1/2} E^{3/2
\cdot \left(1 - \frac{m {\left|\mathbf{v}\right|}^2}{2 E} \right)^
</math>
The analogues of these equations in the canonical ensemble are the barometric formula and the Maxwell–Boltzmann distribution, respectively. In the limit <math>N \to \infty</math>, the microcanonical and canonical expressions coincide; however, they differ for finite <math>N</math>. In particular, in the microcanonical ensemble, the positions and velocities are not statistically independent. As a result, the kinetic temperature, defined as the average kinetic energy in a given volume <math>A \, dz</math>, is nonuniform throughout the container:
<math display="block">
T_{\text{kinetic = \frac{3 E}{5 N - 2} \left(1 - \frac{mgz}{E} \right)
</math>
By contrast, the temperature is uniform in the canonical ensemble, for any <math>N</math>.
See also
- Isolated system
- Ergodic hypothesis
- Loschmidt's paradox
- Canonical ensemble
- Grand canonical ensemble