Microcanonical ensemble

The microcanonical ensemble in statistical physics describes a system with a fixed total energy in thermodynamic equilibrium. It differs, for example, so from the canonical ensemble in which a thermal contact with the environment is, the fluctuating total energy allowed at a fixed temperature.

The only information about a quantum system is that the energy equal to or out of the interval with which the conditions of the extrinsic parameters such as volume and number of particles, have to be compatible. This corresponds to a system within a totally closed box (no energy or particle exchange with the environment, no external fields). The potential outside the box is infinite, so that the Hamiltonian has only discrete energy eigenvalues ​​and the states are countable.

• 5.1 Ideal Gas
• 5.2 Uncoupled Oscillators

Quantum mechanically

In equilibrium, the density operator of the system does not change. According to the von Neumann equation interchanged in equilibrium the density operator with the Hamiltonian ( commutator is equal to zero ). Therefore, common eigenstates can choose, ie the energy eigenstates of are also eigenstates of.

It restricts a Hilbert space of a subspace that is spanned by the state vectors of the eigenvalue (own space ). Be a complete orthonormal system ( VONS ), namely the eigenstates of the Hamiltonian, the subspace spanned by the basis vectors for the. The energy is generally degenerate (hence is a condition not unique by specifying determined, but a further quantum number is necessary); the degree of degeneracy corresponds to the dimension of the subspace.

The Hamiltonian can not be distinguished ( degeneracy ) between the basis states; since no base state preference, so each base state assigned a priori the same probability: After the maximum entropy method, the system can be described by the state, which maximizes the entropy. The entropy is exactly at a maximum when each basis vector has the same probability.

Therefore, the density operator of the microcanonical ensemble gives to

With the microcanonical partition function (and often referred to )

The trace of an operator is defined as follows: for any of VONS

That each energy eigenstate with energy has the same probability, is the basic assumption of the equilibrium statistics. From it all equilibrium properties can be derived for closed or open systems (eg, canonical or grand canonical ensemble).

Classic

Analogously, the classical microcanonical equilibrium state for N particles ( phase space density)

With the classical microcanonical partition function (total number of accessible microstates, the same total energy have )

With

Where for N identical particles of the factor the multiple counting of indistinguishable particles prevents

And for different types of particles with particle numbers and the factor.

The microcanonical partition function can be defined as the surface energy surface and ( with the -dimensional gradient ) conceived in the phase space and is the derivative of the volume of the energy shell:

Performs one -dimensional coordinates on the energy shell, and a coordinate that is perpendicular to it, and decomposes the Hamiltonian with, as can be the partition function written as a surface integral:

The gradient of the Hamiltonian is perpendicular to the velocity in phase space, ie, so that the velocity is always tangential to the energy shell. However, both are identical in magnitude. Thus, the sum of the surface elements is divided by the velocity in phase space, so that areas with high velocity less to the integral contribute (see also: ergodic hypothesis ).

Entropy

The entropy can be expressed by the partition function:

This can be derived from the definition of the entropy, the partition function is the same.

Thermodynamics

The entropy (hereinafter referred to ) of the total energy, depending on the volume and number of particles (because the Hamiltonian is dependent on and ). The total energy in thermodynamics usually called internal energy. Now the derivatives of the entropy are investigated.

Temperature

If two closed systems in weak thermal contact, then per unit time only little energy is exchanged so that the subsystems remain approximately in equilibrium and the entropy of the total system can be written additively. The total energy remains constant. If the energy of a system, so must be in the same degree that of the other decrease. By exchange of energy balance of the entire system is achieved, where the total entropy takes a maximum:

In equilibrium, the derivatives of the entropy with respect to the energy of the subsystems are equal. We define here about the temperature:

Thus, the temperatures of the two subsystems are equal in equilibrium.

Pressure

The pressure is defined by

So the isentropic ( = const ) energy change per unit volume. The entropy is differentiated according to the volume:

Thus we obtain

Chemical potential

The chemical potential is defined by

So the isentropic change of energy per particle. The entropy is differentiated according to the number of particles:

It follows

In general it can be stated: If the Hamiltonian of an external parameter dependent (eg volume or number of particles ), the derivative of the entropy is equal to a constant energy by:

Thermodynamic potential

In summary, the derivatives with respect to energy, volume and number of particles can be represented:

The total differential of entropy is:

The entropy can dissolve after the energy. The energy of the thermodynamic potential of the Mikrokanonik. With it, the above derivations compactly written as a gradient of the potential:

The total differential of the energy is thus:

This is the fundamental equation of thermodynamics.

Examples

Ideal Gas

An example of a prepared mikrokanonisch system can be calculated with the conventional methods, is the ideal gas; Derivation under Sackur - Tetrode equation.

Uncoupled oscillators

Another example is a system of non-interacting similar harmonic oscillators. Their total Hamiltonian is

Where the occupation number operator of the th oscillator. For a given total energy

Is now to be calculated the partition function. This is equal to the degeneracy of the energy E or equal to the number of ways to distribute indistinguishable quanta of energy on multiple besetzbare oscillators or distribute the number of ways indistinguishable quanta of energy to simply besetzbare oscillators ( the combinatorial problem indistinguishable balls be distributed to multiple besetzbare pots to arrange equivalent indistinguishable to the task indistinguishable balls and inner walls in a row):

From this it can calculate the entropy:

For large and one can develop the logarithm of the faculty with the Stirling formula to first order and also the one negligible compared to the huge number:

Rearranging and using supplies the entropy, which is extensive because of the pre-factor ( and are in fact intensive):

Then the temperature can be calculated:

For ( the minimum total energy ) is the temperature and increases strictly monotonically with the energy. For large, the temperature goes against asymptotically.

Finally, one can still solve for the energy. The energy increases monotonically with temperature:

For the total energy and increases monotonically with temperature. For large energy goes against asymptotically.

The chemical potential is:

For or is the chemical potential and falls monotonically with the energy and with temperature. For large or the chemical potential is negative and goes asymptotically to or.