How chaos theory connects two seemingly different fields of physics

How chaos theory connects two seemingly different fields of physics

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

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

A single particle does not possess 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 a significant role. The simulations indicate that the laws of thermodynamics can only be derived from quantum physics when chaos is present.

Boltzmann: Everything is possible, but it can be improbable

Air molecules flying randomly around 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. “From a physical point of view, 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 the other particles stand still,” says Professor Iva Brezinova from the Institute of Theoretical Physics at TU Wien. “But this is so unlikely that it will virtually never be noticed.”

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 then be read and: it is only determined 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 at 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 independently 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 every moment. 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 with many particles,” says Professor Joachim Burgdörfer (TU Wien). “How a random distribution should arise, and therefore a temperature from it, remained a puzzle for a long time.”

Chaos theory as a mediator

A team from TU Wien has now succeeded in demonstrating that chaos plays a key role. To do this, the team performed a computer simulation of a quantum system that consists of a large number of particles – many undifferentiated particles (the “heat bath”) and one of a different type of particle, the “particle of sample” that acts as a thermometer. Each individual quantum wave function of the large system has a specific energy, but not a well-defined temperature – just like a single classical particle. But if you now pick the sample particle from the single quantum state and measure its velocity, you can surprisingly find a velocity distribution corresponding to a temperature that fits the well-established laws of thermodynamics.

“Whether it fits or not depends on the chaos – that’s what our calculations clearly showed,” says Iva Brezinova. “We can specifically change the interactions between particles on the computer and thus create either a completely chaotic system or one that doesn’t look chaotic at all — or something in between.” And in doing so, it is found that the presence of chaos determines whether or not a quantum state of the sample particle exhibits a Boltzmann temperature distribution.

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

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

Reference: “Canonical density matrices from eigenstates of mixed systems” by Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer and Iva Březinová, 29 Nov 2022, Entropy.
DOI: 10.3390/e24121740

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