Primon gas

The primon gas is an example model that combines individual concepts from quantum physics , the physics of heat and number theory . It consists of hypothetical particles, the primons , which are named because their energy is determined by prime numbers .

Overview

The idea of ​​primon gas goes back to Bernard Julia .

Primons are bosons and do not interact with one another, for example they do not collide with one another.

Quantum theoretical description

Single primon

The eigen-states of the individual particles have energies that are proportional to the logarithms of the prime numbers: ${\displaystyle \log p}$

${\displaystyle H|p\rangle =E_{p}|p\rangle }$

With

${\displaystyle E_{p}=E_{0}\log p.}$

In this “numbering” of the eigenstates with a subset of the natural numbers, no eigenstates are “left out”; it is just a practical naming convention.

Many-body system

An eigenstate of a system made up of any number of primons can, since it is a matter of bosons, be described as follows: In the state of the prime number there are particles ( Fock space ). ${\displaystyle p}$ ${\displaystyle k_{p}}$

This is analogous to the prime factorization of a natural number , in which the prime factor occurs to the power of -th. Since every natural number has a unique prime factorization ( fundamental theorem of arithmetic ), every natural number corresponds to a state of the primon gas and vice versa. The number contains the entire information about the occupation numbers of the single-particle states (but it is not the total number of primons). It therefore makes sense to name the condition by this number . ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle k_{p}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle |n\rangle =|k_{2},k_{3},k_{5},k_{7},k_{11}\ldots ,k_{p}\ldots \rangle }$

With

${\displaystyle n=2^{k_{2}}\cdot 3^{k_{3}}\cdot 5^{k_{5}}\cdot 7^{k_{7}}\cdot 11^{k_{11}}\ldots p^{k_{p}}\ldots }$

The energy of the many-particle state is

${\displaystyle E(n)=\sum _{p}k_{p}E_{p}=E_{0}\cdot \sum _{p}k_{p}\log p=E_{0}\log n}$

Examples

• The state does not contain any primons and has a total energy of 0. ${\displaystyle |1\rangle }$
• The state contains eight particles in state 2 (the lowest single-particle state) and has the energy . ${\displaystyle |256\rangle }$ ${\displaystyle \log(256)E_{0}}$
• The state contains three particles in state 2, two particles in state 3 and one particle in state 5. The total energy is . ${\displaystyle |360\rangle }$ ${\displaystyle \log(360)E_{0}}$

Thermodynamic description

The canonical partition function is equal to the Riemann zeta function : ${\displaystyle Z}$

${\displaystyle Z(T):=\sum _{n=1}^{\infty }\exp \left({\frac {-E(n)}{k_{\mathrm {B} }T}}\right)=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{0}\log n}{k_{\mathrm {B} }T}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)}$

It is , the Boltzmann's constant and the temperature in Kelvin. The divergence of the zeta function at corresponds to the divergence of the sum of states at the Hagedorn temperature . ${\displaystyle s=E_{0}/k_{\mathrm {B} }T}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$ ${\displaystyle s=1}$ ${\displaystyle T=E_{0}/k_{\mathrm {B} }}$

Fermions

Alternatively, one can also consider fermionic primons.

Each single-particle state can only be occupied once. This also leads to an interesting number-theoretic statement: the numbers must then be square-free . ${\displaystyle n}$

Individual evidence

1. ↑ Bernard L. Julia: Statistical theory of numbers. In: JM Luck, P. Moussa, M. Waldschmidt (Eds.): Number Theory and Physics. Proceedings of the Winter School , Les Houches, France, March 7-16, 1989 '(Springer Proceedings in Physics, Vol. 47) Springer, Berlin 1990, ISBN 0387521291 , pp. 276-293.

Web links

• John Baez : This Week's Finds in Mathematical Physics, Week 199 . December 8, 2003