# How chaos theory relates two seemingly different areas of physics

### A new study at TU Wien has revealed how chaos theory connects quantum theory and thermodynamics, two seemingly separate areas of physics.

A single particle does not have a temperature, it only has a certain energy or speed. Only when many particles with random velocity distributions are present can a well-defined temperature be derived.

The relationship between thermodynamics and quantum physics has been the subject of growing interest in recent years. researchers in Technical University of Vienna have used computer simulations to investigate this relationship and have found that chaos plays an important role. The simulations indicate that the laws of thermodynamics can only be derived from quantum physics when chaos is present.

#### Boltzmann: Anything is possible, but it may be unlikely

Air molecules flying randomly in a room can assume an unimaginable number of different states: different locations and different speeds are allowed for each individual particle. But not all of these states are equally likely. “Physically, it would be possible for all the energy in this space to be transferred to a single particle, which would then move at extremely high speeds while all other particles remain motionless,” says Professor Iva Brezinova from the Institute for Theoretical Physics at TU Vienna. “But this is so unlikely that it will practically never be observed.”

The probabilities of the different allowed states can be calculated according to a formula that the Austrian physicist Ludwig Boltzmann established according to the rules of classical physics. And from this probability distribution, the temperature can also be read: it is only determined for a large number of particles.

#### The entire world as a single quantum state.

However, this causes problems when it comes to quantum physics. When a large number of quantum particles are in play at the same time, the equations of quantum theory become so complicated that even the best supercomputers in the world have no chance of solving them.

In quantum physics, individual particles cannot be considered independent of each other, as is the case with classical billiard balls. Each billiard ball has its own individual trajectory and its own individual location at all times. Quantum particles, on the other hand, have no individuality: they can only be described together, in a single large quantum wave function.

“In quantum physics, the entire system is described by a single large quantum state of many particles,” says Professor Joachim Burgdörfer (TU Wien). “How a random distribution, and therefore a temperature, should arise from this, remained an enigma for a long time.”

#### Chaos theory as a mediator

A team from TU Wien has now been able to show that chaos plays a key role. To do this, the team ran a computer simulation of a quantum system consisting of a large number of particles: many indistinguishable particles (the “heat bath”) and one of a different type of particle, the “sample particle.” which acts as a thermometer. Each individual grand system quantum wave function has a specific energy, but not a well-defined temperature, like a single classical particle. But if you now select the sample particle from the single quantum state and measure its velocity, you can surprisingly find a velocity distribution that corresponds to a temperature that conforms to the well-established laws of thermodynamics.

“Whether it fits or not depends on chaos, that’s what our calculations clearly showed,” says Iva Brezinova. “We can specifically change the interactions between the particles in the computer and thus create a completely chaotic system, or one that shows no chaos at all, or something in between.” And in doing so, one finds that the presence of chaos determines whether a quantum state of the sample particle shows a Boltzmann temperature distribution or not.

“Without making any assumptions about random distributions or thermodynamic rules, thermodynamic behavior emerges from quantum theory on its own, if the combined system of sample particles and heat bath behaves quantum chaotically. And how well this behavior conforms to the well-known Boltzmann formulas is determined by the force of chaos”, explains Joachim Burgdörfer.

This is one of the first cases where the interplay between three major theories has been rigorously demonstrated using many-particle computer simulations: quantum theory, thermodynamics, and chaos theory.

Reference: “Canonical density matrices of eigenstates of mixed systems” by Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer and Iva Březinová, November 29, 2022, *entropy*.

DOI: 10.3390/e24121740