Partition function (statistical mechanics)
The partition function is an essential tool of statistical physics. Because of the English term partition function, the partition function is also called the partition function, which is not to be confused with the partition function from the combinatorics. From a partition function ( the function, not the value ), all thermodynamic quantities can be derived. If the number of particles N are large enough, you can view the system as a continuous and formulate the partition function as a state integrals.
Microcanonical partition function
The microcanonical partition function is used to describe a closed system with constant internal energy ( ), volume ( ) and particle number () without exchange with the environment in thermodynamic equilibrium. The corresponding ensemble is called microcanonical ensemble. It should already be noted at this point that there are two different definitions for the microcanonical partition function: When a definition is summed over all states with energy less and the other definition is only about the energies in the energy shell around summed.
Countable states
First, such systems are considered which can be located in one of a finite or countable number of microstates (systems with uncountable / continuous states are discussed further below ).
For such systems, is (in the first definition), the microcanonical partition represented by the number of those micro-states of a closed system, for a given energy and volume of particles (and possibly other parameters ), the total energy is less than or equal to:
In the second common definition of the microcanonical partition function of this is given by the energy of which is the number of states in the interval:
( Ie in the state of maximum entropy), the system is in equilibrium, then the probability of finding a particular microstate:
Continuous states
In classical mechanics systems are often considered the micro- state can change continuously. One example is the ideal gas. The room (also called phase space ) of an ideal gas consisting of particles having dimensions: dimensions for the location coordinates and the momenta. Each point in the space corresponds to a phase state of the system with energy, the Hamiltonian of the system of particles and volume. As considered in the Mikrokanonik closed systems have a constant energy, give the allowed states in the space a hypersurface on which the system can move. The partition function of such a gas is the area enclosed by this hypersurface volume, which can be written as integral condition:
Where the Heaviside function. Thus, the density of states is determined by:
Of course, the following applies:
The probability of the gas encountered by a particular state around, is:
And the Dirac δ function.
Often you can also find different definition of microcanonical partition function. Summed or integrated over the energy shell is then up to the hypersurface of the system in space. The shell has thereby the width. The discrete version is ( as described above ):
For continuous systems, the partition function is then:
For the values of and approach each other, since almost all states in the shell are edge.
Canonical partition
In the canonical ensemble not the energy of the system is given, but the temperature. This ensemble is also called the Gibbs ensemble ( see also Canonical state). The partition function is
With the Boltzmann constant
Is the probability of a micro- state
The canonical partition function is
It is the Hamiltonian. The Gibbs factor comes from the indistinguishability of the particles. When wegließe this factor would have N distinct states instead and in comparison to many micro-states, which would result in the Gibbs paradox: two electrodes separated by a partition wall of the same ideal gas quantities have the same temperature and the same pressure. When pulling out the partition is observed without the factor erroneously a gain in entropy.
Grand canonical partition function
In the large canonical ensemble, the chemical potential is given instead of the number of particles. Is the probability of a particular micro- state
The partition function is
In integral notation the partition function and the partition function is
You can get the grand canonical partition function of the canonical partition function and the fugacity:
Calculation of the thermodynamic potentials
Here is
- The entropy
- The free energy and
- The grand potential.