Thermodynamics of diffusion processes and quantum motion

A. Gamba

Gerhard Grössing, Siegfried Fussy, J Mesa Pascasio +1 more

By modelling quantum systems as emerging from a (classical) sub-quantum thermodynamics, the quantum mechanical "decay of the wave packet" is shown to simply result from sub-quantum diffusion with a specific diffusion coefficient varying in time due to a particle's changing thermal environment. It is thereby proven that free quantum motion strictly equals ballistic diffusion. The exact quantum mechanical trajectory distributions and the velocity field of the Gaussian wave packet are thus derived solely from classical physics. Moreover, also quantum motion in a linear (e.g., gravitational) potential is shown to equal said ballistic diffusion. Quantitative statements on the trajectories' characteristic behaviours are obtained which provide a detailed "micro-causal" explanation in full accordance with momentum conservation.

Sebastian Deffner, Steve Campbell

Peng Wang, Maimaitiniyazi Maimaitiabudula

In order to solve the current lack of rigorous theoretical models in the machine learning process, in this paper the iterative motion process of machine learning is modeled by using quantum dynamic method based on the principles of first-principles thinking. This approach treats the iterative evolution of algorithms as a physical motion process, defines a generalized objective function in the parameter space of machine learning algorithms, and regards the iterative process of machine learning as the process of seeking the optimal value of this generalized objective function. In physical terms, this process corresponds to the system reaching its ground energy state. Since the dynamic equation of a quantum system is the Schrödinger equation, we can obtain the quantum dynamic equation that describes the iterative process of machine learning by treating the generalized objective function as the potential energy term in the Schrödinger equation. Therefore, machine learning is the process of seeking the ground energy state of the quantum system constrained by a generalized objective function. The quantum dynamic equation for machine learning transforms the iterative process into a time-dependent partial differential equation for precise mathematical representation, enabling the use of physical and mathematical theories to study the iterative process of machine learning. This provides theoretical support for implementing the iterative process of machine learning by using quantum computers. In order to further explain the iterative process of machine learning on classical computers by using quantum dynamic equation, the Wick rotation is used to transform the quantum dynamic equation into a thermodynamic equation, demonstrating the convergence of the time evolution process in machine learning. The system will be transformed into the ground energy state as time approaches infinity. Taylor expansion is used to approximate the generalized objective function, which has no analytical expression in the parameter space. Under the zero-order Taylor approximation of the generalized objective function, the quantum dynamic equation and thermodynamic equation for machine learning degrade into the free-particle equation and diffusion equation, respectively. This result indicates that the most basic dynamic processes during the iteration of machine learning on quantum computers and classical computers are wave packet dispersion and wave packet diffusion, respectively, thereby explaining, from a dynamic perspective, the basic principles of diffusion models that have been successfully utilized in the generative neural networks in recent years. Diffusion models indirectly realize the thermal diffusion process in the parameter space by adding Gaussian noise to and removing Gaussian noise from the image, thereby optimizing the generalized objective function in the parameter space. The diffusion process is the dynamic process in the zero-order approximation of the generalized objective function. Meanwhile, we also use the thermodynamic equation of machine learning to derive the Softmax function and Sigmoid function, which are commonly used in artificial intelligence. These results show that the quantum dynamic method is an effective theoretical approach to studying the iterative process of machine learning, which provides a rigorous mathematical and physical model for studying the iterative process of machine learning on both quantum computers and classical computers.

L. F. Lemmens, F. Brosens, J. T. Devreese

For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteracting system. For motion in one dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved one-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.

Felix C. Binder, Luis Alfonso Correa, Christian Gogolin +2 more