Quantum Turbulence Arises from Stochastic Forces Linked to Dissipation

Wael Itani, from American University of Beirut, and colleagues have uncovered a fundamental connection between classical turbulence and quantum mechanics, challenging conventional understanding of viscous forces. Viscosity cannot solely originate from Hamiltonian quantum mechanics, requiring an open quantum treatment. The work reveals a stochastic Navier-Stokes equation derived from quantum state diffusion, linking dissipation and stochastic forcing through Lindblad operators. This approach recovers the well-established Landau-Lifshitz framework and, through the topology of wavefunction zeros, provides a new mechanism for the Migdal area law governing circulation statistics, validated even at quantum scales.

Madelung transforms and Born-Markov approximations resolve inconsistencies in quantum fluid

The Madelung transform, a mathematical technique initially developed to connect quantum mechanics with classical hydrodynamics, proved central to this work, enabling translation of quantum concepts into the language of classical fluid dynamics. This transform recasts the time-dependent Schrödinger equation into a fluid-dynamical form, expressing the quantum wave function in terms of a fluid density and velocity potential. However, applying this transform to the fundamental Schrödinger equation alone yielded forces acting in a way inconsistent with observed viscosity. Specifically, the Madelung transform, when applied naively, produces only gradient forces, whereas viscous forces in classical fluids are inherently solenoidal, possessing zero divergence. This discrepancy highlights a fundamental incompatibility between a purely Hamiltonian quantum description and the observed behaviour of viscous fluids. To overcome this, the Born-Markov approximation was employed, simplifying a quantum system’s interaction with its environment by focusing on immediate impacts and ignoring more distant effects, much like approximating a bouncing ball without accounting for air resistance. The Born-Markov approximation assumes that the environment has a short correlation time and a weak coupling to the system, allowing for a reduction of the environmental degrees of freedom. This simplification is crucial for making the problem tractable and for connecting the quantum description to a classical, dissipative picture.

A direct approach was chosen to address inconsistencies arising when applying the Madelung transform to the Schrödinger equation. The resulting calculations involved Lindblad operators with k 2 scattering rates, unravelled using quantum state diffusion into a stochastic nonlinear Schrödinger equation. The Lindblad operators describe the irreversible interaction between the quantum system and its environment, leading to decoherence and dissipation. The k 2 scattering rates indicate that the probability of scattering events is proportional to the square of the wavevector, a characteristic feature of certain quantum interactions. Quantum state diffusion (QSD) is a stochastic method used to simulate the time evolution of a quantum system by representing the wavefunction as a collection of trajectories. This allows for the calculation of ensemble averages and the derivation of effective classical equations of motion. This simplification allows a focus on immediate effects rather than distant ones, streamlining the complex interactions within the quantum system. Consequently, the model provides a more accurate representation of quantum fluid behaviour, bridging the gap between the unitary evolution of the Schrödinger equation and the dissipative behaviour of classical fluids. The stochastic nonlinear Schrödinger equation obtained through QSD incorporates both deterministic and random terms, reflecting the interplay between quantum coherence and environmental noise.

Open quantum systems explain turbulent flow viscosity and the Migdal area law

Circulation statistics now exhibit the Migdal area law even when the de Broglie length exceeds the Kolmogorov scale, a threshold previously inaccessible to numerical verification. The de Broglie length, representing the quantum mechanical wavelength of a particle, and the Kolmogorov scale, the smallest length scale in turbulent flow, define a regime where quantum effects are typically negligible. The fact that the Migdal area law holds even when the de Broglie length is larger than the Kolmogorov scale demonstrates the significance of quantum effects in influencing the statistical properties of turbulence. This advancement, achieved through an open quantum treatment, allows for analysis at quantum scales where traditional methods relying on loop-functional saddle points fail. Loop-functional methods, commonly used in turbulence research, become inaccurate when quantum effects become dominant. Itani and colleagues demonstrate that viscosity, a property defining fluid resistance, cannot originate solely from Hamiltonian quantum mechanics; instead, it necessitates accounting for a system’s interaction with its environment. This finding underscores the importance of considering the system as an open quantum system, constantly exchanging energy and information with its surroundings.

Their approach links dissipation and stochastic forcing via Lindblad operators, recovering the Landau-Lifshitz framework and offering a new explanation for the observed area law governing swirling flows. The Landau-Lifshitz framework is a classical theory describing the dynamics of magnetic fluids, and its recovery within this quantum context suggests a deep connection between these seemingly disparate phenomena. These operators generate both viscous damping and random forcing with k 2 scattering rates, achieved by reducing complex quantum interactions to simpler, one-particle dynamics and employing quantum state diffusion. The resulting stochastic Navier-Stokes equation satisfies the fluctuation-dissipation relation, linking noise and dissipation. This relation is a fundamental principle in statistical physics, stating that the strength of the fluctuations is proportional to the strength of the dissipation. Furthermore, analysis of the wavefunction’s zeros revealed the Migdal area law for circulation statistics, even when the de Broglie length exceeds the Kolmogorov scale, a previously unverified regime. The Migdal area law states that the area enclosed by a loop of fluid circulation is proportional to the square root of the time elapsed. While the model accurately predicts circulation at the ensemble level, identifying this behaviour on single fluid trajectories remains an open challenge. Understanding the behaviour of individual fluid particles is crucial for developing a complete understanding of turbulence.

Quantum origins of viscosity explored through fluid particle ensembles

Turbulence, a persistent challenge in fluid dynamics, has long resisted complete theoretical description, and understanding its origins remains a key goal for physicists and engineers alike. The inherent complexity of turbulent flows, characterised by chaotic and multiscale interactions, makes it difficult to develop accurate theoretical models. Itani and colleagues’ work offers a compelling, if incomplete, bridge between the quantum and classical realms, suggesting viscosity, a fluid’s resistance to flow, requires more than just traditional Hamiltonian mechanics. Deriving classical fluid behaviour from quantum mechanics requires acknowledging the system’s interaction with its surroundings; a purely Hamiltonian description is insufficient. This highlights the limitations of purely reductionist approaches and the need for a more holistic understanding of complex systems.

By employing the Madelung transform, scientists successfully generated a stochastic Navier-Stokes equation, the cornerstone of fluid dynamics, from a spinless Schrödinger equation, but only through an ‘open quantum treatment’. This approach not only recovers established physics but also offers a new explanation for the Migdal area law governing swirling flows, validated even at quantum scales. The ability to derive classical equations of motion from quantum mechanics is a significant achievement, providing a deeper understanding of the fundamental principles governing fluid behaviour. Currently, confirmation exists only when considering large ensembles of fluid particles, leaving open the question of whether individual trajectories also adhere to these quantum-derived rules. Investigating the behaviour of individual fluid particles is a crucial next step in validating the model and gaining a more complete understanding of turbulence. The research highlights the importance of open quantum systems in understanding complex fluid behaviour, suggesting that the environment plays a crucial role in shaping the properties of fluids.

By linking quantum mechanics to fluid dynamics, researchers demonstrated that viscosity, a fluid’s resistance to flow, requires considering a system’s interaction with its surroundings, rather than solely relying on traditional mechanics. This work successfully derived a stochastic Navier-Stokes equation from a spinless Schrödinger equation, offering a new explanation for the behaviour of swirling flows and validating the Migdal area law at quantum scales. The findings currently apply to the average behaviour of many fluid particles, and further investigation into individual particle behaviour is needed to fully confirm the model. This research emphasises the importance of ‘open quantum systems’ in understanding complex fluid behaviour.