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arXiv:2307.10795 · Chaos An Interdisciplinary Journal of Nonlinear Science

Dissipative systems fractionally coupled to a bath

Audrique Vertessen, Robin C. Verstraten, C. Morais Smith

Published Jun 1, 2024• 3 citations
arXivHTMLPDF

Abstract

Quantum diffusion is a major topic in condensed-matter physics, and the Caldeira-Leggett model has been one of the most successful approaches to study this phenomenon. Here, we generalize this model by coupling the bath to the system through a Liouville fractional derivative. The Liouville fractional Langevin equation is then derived in the classical regime, without imposing a non-Ohmic macroscopic spectral function for the bath. By investigating the short- and long-time behavior of the mean squared displacement, we show that this model is able to describe a large variety of anomalous diffusion. Indeed, we find ballistic, sub-ballistic, and super-ballistic behavior for short times, whereas for long times, we find saturation and sub- and super-diffusion.

DOI
10.1063/5.0204304

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Quantum diffusion is a major topic in condensed-matter physics, and the Caldeira-Leggett model has been one of the most successful approaches to study this phenomenon. Here, we generalize this model by coupling the bath to the system through a Liouville fractional derivative. The Liouville fractional Langevin equation is then derived in the classical regime, without imposing a non-Ohmic macroscopic spectral function for the bath. By investigating the short- and long-time behavior of the mean squared displacement, we show that this model is able to describe a large variety of anomalous diffusion. Indeed, we find ballistic, sub-ballistic, and super-ballistic behavior for short times, whereas for long times, we find saturation and sub- and super-diffusion.
We study a classical system-bath model in which a system particle is linearly coupled to a bath of harmonic oscillators. In a system subject to white and correlated Gaussian noises, we derive an expression for the joint probability density, and the mean squared values of the system-bath particles are calculated. For white noise, the mean squared values of the system-bath particles exhibit an anomalous time dependence, which is different from that of normal diffusion. In particular, for a correlated Gaussian noise, the mean squared displacement and mean squared velocity of the bath particle show superspreading growths of t5t^5t5 and t3t^3t3 in t≪τt{\ll}τt≪τ, respectively. When τ=0τ=0τ=0, the mean squared velocity of a quantum particle under random noise is proportional to ttt, while the mean squared velocity of a bath particle in the presence of correlated Gaussian noise increases in proportion to t2t^2t2. This anomalous transport phenomenon results from the mixed derivative structure of the master equation, which couples with the transport coordinates in the diffusion dynamics of the relative coordinates. This result shows that the removal of dissipation in the Caldeira-Leggett framework leads to fundamentally different transport mechanisms characterized by non-diffusive quantum diffusion.
Fluctuation and dissipation are byproducts of coupling to the ``environment.'' The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g., for a neutral particle in quantum electrodynamics. We argue that the absence of linear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalous diffusion, unfolding as a power-law distribution in space, reminiscent of L\'evy flights. We suggest condensed matter analogs where similar effects may arise.
The standard Brownian diffusion process governs the scaling of a random walk, where a tracer in a two dimensional spacetime is governed by the variance of the displacement of a particle distribution σ(t) ∝ t , γ = 1. In other words, if we denote by P (x, t) the relative probability of particles at location x ∈ R at time t, the classical diffusion equation ∂P/∂t = 12∂ p/∂x can be solved by means of the Fourier transform which inverts to a normal probability distribution with the mean zero and standard deviation t. In the last decade, it has been realized that the understanding of complex dynamical systems has required the implementation of the subdiffusion (slower spreading rate, i.e. γ 1) anomalous random processes. Anomalous superdiffusion can be modeled using infinite variance particle jumps that lead to space-fractional derivatives in the governing diffusion equation. Anomalous subdiffusion can be modeled using IID infinite mean waiting times between particle jumps, leading to a fractional time derivative in the governing diffusion equation [1]. The aim of this short communication is to prove that the anomalous subdiffusion can be realized in quantum physical phenomena without implementing any kind of time-fractional derivative.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
The standard system-plus-reservoir approach used in the study of dissipative systems can be meaningfully generalized to a dissipative coupling involving the momentum, instead of the coordinate: the corresponding equation of motion differs from the Langevin equation, so this is called anomalous dissipation. It occurs for systems where such coupling can indeed be derived from the physical analysis of the degrees of freedom that can be treated as a dissipation bath. Starting from the influence functional corresponding to anomalous dissipation, it is shown how to derive the effective classical potential that gives the quantum thermal averages for the dissipative system in terms of classical-like calculations; the generalization to many degrees of freedom is given. The formalism is applied to a single particle in a double well and to the discrete phi(4) model. At variance with the standard case, the fluctuations of the coordinate are enhanced by anomalous dissipative coupling.
Abstract This thesis has two parts, in the first, the Caldeira-Leggett model is introduced; its derivation and general consequences are explored following a paper by Caldeira and Leggett[1]. An operator-formalism shortcut through some of the more mathematically cumbersome parts of the derivation of the model is also developed. The correlation of the force resulting from reservoir-interaction is examined in the high- and low-temperature limits, and the Langevin equation is shown to emerge in the classical limit.Abstract The second part introduces decoherence through a thought experiment that demonstrates the destructive effect of random phase shifts on interference terms, and then follows another paper by Caldeira and Leggett[2] in applying their model further to study the phenomenon of dissipative decoherence. The time-evolution of the interference terms in a superposition of Gaussian wave packets in a harmonic oscillator potential is studied when interacting with a heat bath, and they are shown to vanish at a rate much faster than the relaxation of the system.