Fastest Change in Physics Limited by Planck Time

Scientists have long sought to understand the minimum time required for a system to reach local thermal equilibrium. Marvin Qi from the Leinweber Institute for Theoretical Physics & James Franck Institute, University of Chicago, and Alexey Milekhin from the Department of Physics and Astronomy, University of Kentucky, alongside Luca Delacr etaz from the Leinweber Institute for Theoretical Physics & James Franck Institute, University of Chicago, demonstrate a rigorous lower bound on this ‘equilibration time’, conjecturing it is fundamentally limited by the Planckian time. Their research establishes this bound by analysing the emergence of hydrodynamic behaviour in conserved densities, revealing a dimensionless coefficient dependent only on dimensionality and the type of behaviour, irrespective of the underlying thermalisation mechanism. This universally applicable result, achieved through careful consideration of real-time thermal correlators, offers significant insight into the foundations of statistical mechanics and applies to a broad range of physical systems, even those lacking a quasiparticle description or exhibiting inelastic scattering.

Within a cryostat chilled to near absolute zero, delicate measurements track how quickly order arises from chaos. This pursuit reveals a fundamental limit to how rapidly any physical system can reach stability. The universal timescale, linked to the very fabric of spacetime, governs the emergence of predictable behaviour in everything from fluids to quantum materials.

Scientists have long recognised the importance of the Planckian timescale, ħ/T, in quantum statistical physics — recent attention focuses on a compelling conjecture: that this timescale fundamentally limits how quickly quantum many-body systems reach local equilibrium. With a local equilibration time τeq greater than or equal to the Planckian time, and scientists have now moved beyond theoretical motivation to formally establish this bound. Defining τeq as the moment when a hydrodynamic description accurately captures the behaviour of conserved densities within a system.

By examining the analytic properties of real-time thermal correlators. They have derived a rigorous lower limit on the onset of hydrodynamic behaviour within a ‘regulated’ thermal two-point function — establishing a sharply defined value for τeq has remained a challenge until now. This effort adopts a universal definition, linking τeq to the emergence of hydrodynamic behaviour, and a slow active characteristic of conserved densities present in diverse systems ranging from spin chains to metals and quantum field theories.

Once interactions increase, a hydrodynamic description emerges more rapidly, prompting the question of whether arbitrarily strong interactions could lead to arbitrarily fast equilibration. Investigations reveal this is not the case, as hydrodynamic fluctuations introduce limitations at earlier times, bounding τeq from below. The precise quantification of this bound required a novel approach.

By analysing the behaviour of a specific ‘two-sided’ correlation function, a measure of density fluctuations, its rate of change is constrained — this constraint, in turn, directly limits the onset time of hydrodynamic behaviour. Establishing a lower bound of τeq ≥ αħ/T is therefore possible, and unlike previous attempts, the dimensionless coefficient α depends solely on the system’s dimensionality and the type of hydrodynamic or diffusive behaviour observed. Remaining independent of the specific thermalization mechanism or microscopic details.

Here, the bound is not merely a consequence of the Heisenberg uncertainty principle. In turn, this posits a relationship between energy and time fluctuations. Instead, it arises from the inherent properties of how information propagates within the quantum many-body system — this bound applies universally, encompassing systems with or without quasiparticle descriptions and even those exhibiting inelastic scattering. Meanwhile, this finding has implications for understanding transport and thermalization in strongly interacting systems, and potentially offering new avenues for exploring materials beyond the reach of traditional weak-coupling methods.

Defining hydrodynamic emergence via regulated thermal correlators

A detailed examination of thermal two-point functions underpinned this effort, allowing researchers to establish a rigorous lower bound on the onset of hydrodynamic behaviour in conserved densities. Initially, real-time thermal correlators were analysed for their analytic properties, providing a foundation for defining a time scale at which a hydrodynamic description emerges.

Such an approach moved beyond observing hydrodynamic behaviour and instead sought to define a fundamental limit on its appearance. Calculations focused on a ‘regulated’ thermal two-point function, a specific mathematical construct designed to isolate and quantify the emergence of hydrodynamic behaviour. Through analysing this function, scientists determined a dimensionless coefficient dependent solely on dimensionality and the type of hydrodynamic or diffusive behaviour. Remaining independent of the thermalisation mechanism.

In turn, this ensured the bound’s universality, applying to a broad range of local systems, even those lacking a quasiparticle description or exhibiting inelastic scattering. Meanwhile, the project did not limit itself to diffusion alone, but instead considered superdiffusion and subdiffusion by generalizing the density correlator to account for varying late-time behaviours.

Spatially resolved correlators were employed when examining sound modes, adjusting calculations to follow the sound front’s propagation. At the same time, the effort also investigated the possibility of bounding exponential decay rates of correlators, though it found no universal upper limit without additional constraints. Scientists proposed an alternative Planckian conjecture concerning the slowest non-hydrodynamic decay rate, sometimes termed the ‘Liouvillian gap’. It might be bounded by T/ħ.

At the core of the methodology lay The assessment of the two-sided density correlator, F(t) = Tr √ρn(t)√ρn, to rigorously bound the onset of hydrodynamic behaviour by the Planckian time, τeq ≥α ħ T. This approach, while employing a specific correlator, aimed to provide insights applicable to conventional real-time observables and potentially detectable through techniques like time-resolved near-field spectroscopy.

A dimensionally dependent lower bound on local thermalisation timescales

Scientists establish a lower bound of d 2 πħT on the time scale for the emergence of hydrodynamic behaviour in d spatial dimensions. This bound, derived from analysing the analytic properties of real-time thermal correlators, signifies a fundamental limit on how quickly systems reach local thermal equilibrium. Specifically, The project formalises a conjecture that the local equilibration time is bounded below by the Planckian time. Defining a time scale at which a hydrodynamic description becomes valid for conserved densities.

Calculations reveal that this dimensionless coefficient depends solely on dimensionality and the type of hydrodynamic behaviour. Remaining independent of the thermalisation mechanism or microscopic details. Determining the precise relationship between scattering time and the onset of hydrodynamic behaviour proved central to this effort. Once the analytic structure of correlators was examined, the rate of change of the normalised two-point function, F nn (t), is bounded by πTħ.

This inequality, while not directly applicable at arbitrarily early times, establishes a limit on the onset of hydrodynamic behaviour, implying that for any time t > τ eq (ε), d 2 t ≤ πTħ + O(ε). The assessment extended beyond simple diffusion, encompassing other hydrodynamic behaviours like sound modes and both sub- and superdiffusion, all subject to similar bounds.

At the heart of the method lies the application of complex analysis, specifically leveraging the Schwarz-Pick theorem to bound the function f(z) = 1 − FAB(z) / C, where FAB(z) represents the correlator of two Hermitian operators A and B, and C is a constant. When considering the conditions for establishing property 3, the symmetric Green’s function, GS AB (t), must dominate the anti-symmetric one, GA AB (t), at late times, with GA AB (t) / GS AB (t) = O(βħt).

Also, the real part of FAB(z) must remain positive for Re z ≥ t 0. Indicating a specific regime where hydrodynamics governs the system’s behaviour. These conditions, if violated, would imply an infinite equilibration time, rendering the Planckian bound trivial. Inside the half-strip defined by t > T 0 and |τ| ≤ β/2, |f(z)| ≤ 1, utilising the Phragmén-Lindelöf principle.

This bound, combined with the established inequalities, in the end leads to the Planckian bound on the emergence of diffusion. By assuming that Re FAB(z) > 0 for Re z ≥ t 0, The team were able to prove the bound, then relaxing this assumption to derive a weaker, yet more general, Planckian limit.

Establishing a Planckian limit to the speed of thermal equilibration

Scientists have long sought to define the minimum speed at which systems reach thermal equilibrium, a fundamental limit dictated by the principles of quantum mechanics and relativity. Recent work establishes a definitive lower bound on this ‘equilibration time’, linking it directly to the Planckian time, the smallest unit of time with physical meaning.

Beyond simply confirming a theoretical conjecture, this project provides a universal benchmark applicable to a surprisingly broad range of physical systems, from the behaviour of electrons in metals to the dynamics of black holes. For years, pinning down this timescale proved difficult because existing methods relied heavily on specific models or assumptions about the underlying physics.

This new approach bypasses those limitations by focusing on the mathematical properties of how systems respond to disturbances, rather than the details of the disturbance itself. Instead of modelling the complex interactions within a material, researchers examined the general characteristics of thermal fluctuations, revealing a surprisingly consistent lower limit.

The implications extend far beyond fundamental physics. Offering a new lens through which to understand phenomena like high-temperature superconductivity and the behaviour of strongly interacting quantum materials. These materials are now seen as potential building blocks for future technologies. Applying this theoretical bound to real-world systems remains a challenge.

Through measurement of the relevant thermal correlations with sufficient precision is difficult, particularly in complex materials where other effects can obscure the signal — unlike previous attempts to define this limit, this effort does not offer a pathway to immediately improve materials or devices. The connection between this minimum equilibration time and the emergence of macroscopic properties like viscosity or electrical conductivity requires further investigation. The focus will likely shift towards exploring how deviations from this bound might signal new physics, or identifying systems where this limit is approached most closely. Potentially unlocking new avenues for materials design and quantum technologies.