How chaos theory mediates between quantum theory and thermodynamics

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

One particle has no temperature. It has a certain energy or a certain speed – but that can’t be translated into a temperature. Only when dealing with random velocity distributions of many particles does a well-defined temperature emerge.

How can the laws of thermodynamics arise from the laws of quantum physics? This is a topic that has attracted increased attention in recent years. At TU Wien (Vienna), this question has now been pursued by computer simulations, which have shown that chaos plays a crucial role: only chaos follows the known rules of thermodynamics from quantum physics.

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, from 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 a 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. Quantum particles, on the other hand, have no individuality—they can only be described together in one large quantum wave function.

“In quantum physics, the entire system is described by a single 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 the large system has a certain energy, but not a well-defined temperature – just like a single 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 the mess or inappropriateness – this is clearly shown by our calculations,” says Eva Brizinova. “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 all by itself—if the combined system of sample particles and the thermal bath behaves quantum chaotically. The suitability of this behavior is determined by the well-known Boltzmann formulas of the chaos force. Joachim Burgdorfer explains.

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.

Publication of the research in the journal entropy.

more information:
Mehdi Korebaz et al., Fundamental Density Matrices from Eigenstates for Mixed Systems, entropy (2022). DOI: 10.3390/e24121740

Provided by the Vienna University of Technology

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