Primon gas

The Primonengas is an example model that connects individual concepts from quantum physics, the physics of heat and number theory. It consists of hypothetical particles called Primonen, the so called, because their energy is determined by primes.

Survey

The idea of ​​Primonengases goes back to Bernard Julia.

Primonen are bosons and do not interact with each other, for example, they do not collide with each other.

Quantum -theoretical description

One Primon

The eigenstates of the individual particles have energies that are proportional to the logarithms of primes:

With

In this " numbering " of the eigenstates with a subset of the natural numbers are not eigenstates " omitted "; it is merely a convenient name.

Many-body system

An eigenstate of a system of any number of Primonen, since they are bosons, are described as follows: in the state to be prime particles are ( Fock space ).

This is analogous to a natural number prime factorization, wherein the prime factor in the nth power occurs. Since every natural number has a unique prime factorization ( Fundamental Theorem of Arithmetic ), corresponding to each natural number a state of Primonengases and vice versa. The number thereby contains all the information on the occupation numbers of the single-particle ( but it is not the total number of Primonen ). It is therefore reasonable to name the state by this number.

With

Is the energy of Vielteilchenzustandes

Examples

• The state does not Primonen and has the total energy 0
• The state contains eight particles in state 2 (the lowest single-particle state ) and has the power.
• The state consists of three particles in the state 2, two particles in the state 3, and a particle in the state 5, the total energy.

Thermodynamic Description

The partition function is equal to the Riemann zeta function:

Here is the Boltzmann constant and temperature in degrees Kelvin. The divergence of the zeta function at corresponds to the divergence of the partition function at the Hagedorn temperature.

Fermions

One can alternatively also consider fermionic Primonen.

It can be occupied only once each single particle. This also leads to an interesting number-theoretic statement: the numbers must be then that is square-free.