How chaos theory connects two seemingly different areas of physics

Particle thermometer

One of the particles acts as a “thermometer,” and the entire system is simulated on a computer. Credit: TU Wien

A new study at TU Wien reveals how chaos theory links 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. A well-defined temperature can only be derived when many particles with random velocity distributions are present.

The relationship between thermodynamics and quantum physics has been the subject of increasing interest in recent years. Researchers at TU Wien used computer simulations to investigate this relationship, and found that chaos plays an important role. 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 also be improbable

Air particles flying randomly in a room can assume an unimaginable number of different states: different positions and different speeds are allowed for each individual particle. But not all of these countries are equally likely. says Professor Eva Brezinova of the Institute for Theoretical Physics at TU Wien. “But this is so unlikely that it will not be noticed in practice.”

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

The whole world as a single quantum state

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

In quantum physics, individual particles cannot be considered independently of each other, as is the case with classic billiard balls. Each billiard ball has its own individual path and individual location at each point in time. On the other hand, quantum particles are not individual – they can only be described together, in one large quantum wave function.

“In quantum physics, the entire system is described by one large multiparticle quantum state,” says Professor Joachim Burgdorfer (TU Wien). “How the random distribution and thus temperature should arise from this has long been a mystery.”

Chaos theory as a mediator

A team at TU Wien has now been able to show that chaos plays a major role. To do this, the team ran computer simulations of a quantum system made up of a large number of particles — many indistinguishable (“thermal bath”) and one of a different type of particle, the “sample particle” whose thermometer operates. Each individual quantum wavefunction of a large system has a specific energy, but not a well-defined temperature – just like an individual classical particle. But if you now choose a sample particle from the single quantum state and measure its velocity, you can surprisingly find a velocity distribution corresponding to a temperature that fits well-established laws of thermodynamics.

“It depends on whether it is messy or inappropriate – this is clearly shown by our calculations,” says Iva Brezinova. “We can specifically change the interactions between particles on the computer and thus create either a completely chaotic system, or a system that shows no chaos at all — or anything in between.” In doing so, one finds that the presence of chaos determines whether or not the quantum state of a sample particle exhibits a Boltzmann temperature distribution.

“Without making any assumptions about random distributions or thermodynamic rules, thermodynamic behavior arises from quantum theory alone — if the combined system of sample particles and the thermal bath behaves quantum chaotically. Joachim Burgdorfer explains how such behavior fits the well-known Boltzmann equations and is determined by chaos power.

This is one of the first cases in which the interaction of three important theories has been rigorously demonstrated by computer simulations of many particles: quantum theory, thermodynamics, and chaos theory.

Reference: “Canonical Density Matrices from Eigenstates of Mixed Systems” By Mehdi Korebaz, Stefan Donsa, Fabian Lackner, Joachim Burgdorfer, Eva Bezinova, November 29, 2022, Available here. entropy.
DOI: 10.3390/e24121740

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