Planckian dissipation from classical hydrodynamics

This paper demonstrates that the requirement for a quantum system to remain describable by classical hydrodynamics at low temperatures necessitates a finite classical region within the light cone, which in turn forces the effective relaxation rate to be at least Planckian, thereby deriving Planckian scaling of transport coefficients as a consequence of hydrodynamic self-consistency rather than microscopic quantum constraints.

The Gist

The Big Question: Why Do Quantum Things Relax So Fast?

Imagine you drop a pebble into a pond. The ripples spread out, but eventually, the water settles down. In the world of quantum physics (the world of atoms and subatomic particles), scientists have noticed something strange: many materials "settle down" or relax at a speed that depends only on the temperature and a tiny number called Planck's constant.

It's as if the universe has a universal speed limit for how fast things can calm down, and that limit is set by the temperature. This is called the Planckian bound. For years, physicists have asked: Why is there this limit? Is it a fundamental law of the quantum world, or is it something else?

The Paper's New Idea: The "Blur" Effect

This paper proposes a different way to look at the problem. Instead of asking what the quantum rules force the system to do, the authors ask: What does it take for a quantum system to still look like a "classical" system?

Think of Classical Hydrodynamics (the math we use to describe water flowing or heat spreading) as a high-definition movie. It's crisp, clear, and follows simple rules.
Think of Quantum Mechanics as that same movie, but viewed through a pair of glasses that slightly blur the image.

The paper argues that the "quantum blur" happens on a specific timescale (the Planckian time). If the movie is moving slowly, the blur doesn't matter; the water still looks like water. But if the movie is moving too fast, the blur smears everything out, and the simple rules of classical hydrodynamics break down.

The Experiment: Three Types of "Flow"

To test this, the authors imagined three different ways a substance could flow or spread, like three different types of traffic:

1. Diffusion (Instant Spread): Imagine a crowd of people instantly appearing everywhere. This is the standard way we usually think heat spreads. It has no speed limit.
2. Telegraph (The Light Cone): Imagine a crowd running, but they can't run faster than a specific speed (like the speed of light). There is a sharp "front" where the crowd hasn't reached yet.
3. Diffusive-Telegraph (The Smoothed Front): A mix of the two, where the front is a bit fuzzy but still has a speed limit.

They tracked how "correlations" (how much one part of the system knows about another part) decayed over time in these scenarios.

The Discovery: Two Zones Inside the Cone

When they applied the "quantum blur" to these scenarios, they found the space inside the "light cone" (the area where information can travel) splits into two distinct zones:

• The Classical Zone (The Center): Near the center of the flow (where things are moving slowly), the "blur" is too weak to matter. The system behaves exactly like a classical fluid. The math works perfectly.
• The Quantum Zone (The Edge): As you get closer to the edge of the light cone (where things are changing very rapidly), the "blur" takes over. The simple classical rules stop working. The system starts behaving in a strictly quantum way, decaying at the "Planckian rate."

The Analogy: Imagine walking through a foggy forest.

• In the middle of the forest, the fog is thin. You can see the trees clearly (Classical Zone).
• As you walk toward the edge where the wind is blowing the fog in fast, the fog gets so thick you can't see the trees at all; you only see a white wall (Quantum Zone).

The "Price" of Being Classical

Here is the main punchline of the paper:

If you want a system to remain describable by simple, classical hydrodynamics (the clear view) all the way down to very low temperatures, you have to pay a price.

That price is that the system's relaxation rate (how fast it settles) cannot be arbitrarily slow. It must be at least as fast as the "Planckian rate."

If the system tried to relax slower than this rate, the "quantum blur" would become so dominant that the classical description would break down immediately. The system would be forced to become "quantum" everywhere, even in the center.

So, the Planckian bound isn't a mysterious rule forcing quantum systems to be fast. Instead, it is the minimum speed required for a system to stay "classical" enough for us to use our standard hydrodynamic equations.

Summary

• The Problem: Why do quantum systems relax at a speed set only by temperature?
• The Mechanism: Quantum mechanics acts like a "blur" on fast-changing details.
• The Result: If a system changes too slowly, the blur ruins the classical picture. To keep the classical picture valid, the system must change fast enough to stay ahead of the blur.
• The Conclusion: The "Planckian bound" is the speed limit a system must obey just to remain describable by classical physics. It's not a constraint from the quantum world; it's the cost of staying classical.

Technical Summary

Technical Summary: Planckian Dissipation from Classical Hydrodynamics

Problem Statement
A central puzzle in quantum many-body physics is the observation that strongly correlated materials (e.g., cuprates, heavy fermions) often exhibit relaxation timescales determined solely by temperature ($T$) and Planck's constant ($\hbar$), defined as $\tau_{Pl} = \hbar / k_B T$. This "Planckian bound" appears in various contexts, including shear viscosity bounds ($\eta/s \ge \hbar/4\pi k_B$), chaos bounds (Lyapunov exponents $\lambda_L \le 2\pi k_B T/\hbar$), and local equilibration times. While these bounds are ubiquitous, the microscopic mechanism enforcing them remains elusive. Previous work suggested that the quantum fluctuation-dissipation theorem (FDT), when expressed in the time domain, acts as a "blurring" operation on correlation functions at the scale $\Omega^{-1} = \hbar \beta / \pi$. This paper investigates what the self-consistency of a classical hydrodynamic description imposes on a quantum system, specifically asking: up to what temperatures can a hydrodynamic solution be consistently viewed as classical?

Methodology
The authors analyze the interplay between the quantum FDT and generic phenomenological hydrodynamic equations within a causal light cone of velocity $v$. They assume a conserved density $n(x,t)$ and its symmetrized correlator $C(x,t)$ follows a large-deviation ansatz inside the light cone ($|x| < vt$):
$$C(x, t) \simeq \frac{1}{\sqrt{t}} e^{-\lambda t \Phi(x/vt)}$$
where $\Phi(a)$ describes the spatial profile, vanishing at $a=0$ (time axis) and growing toward the light cone ($a=1$). The decay rate $\lambda$ is related to the diffusion constant $D$ and velocity $v$ by $\lambda = v^2/4D$.

The core of the methodology involves applying the time-domain quantum FDT, which relates the regulated correlator $F(x,t)$ and integrated response $\Psi(x,t)$ to $C(x,t)$ via convolution with kernels of width $\Omega^{-1} = \hbar \beta / \pi$. This convolution acts as a "blurring" filter that washes out temporal details faster than the Planckian time.

The authors track the behavior of these functions along rays $x = avt$ inside the light cone for three specific phenomenological equations:

1. Diffusion Equation: Infinite propagation speed (requires an external $v$).
2. Telegraph Equation: Finite relaxation time $\tau$, finite speed $v$, and a sharp light cone.
3. Diffusive-Telegraph Equation: A Jeffreys-type refinement smoothing the crossover.

For each case, they perform a saddle-point analysis of the convolution integral to determine the late-time decay rate of $F(x,t)$ and compare it to the hydrodynamic decay of $C(x,t)$.

Key Results
The analysis reveals a bifurcation of the light cone interior into two distinct regimes based on the ratio of the hydrodynamic decay rate $\lambda(a)$ to the Planckian frequency $\Omega$:

1. Classical Region: Near the time axis ($a < a_c$), the hydrodynamic decay is slow ($\lambda(a) \ll \Omega$). The blurring kernel is too narrow to alter the decay profile. Consequently, $F \simeq C$ and the fluctuation-dissipation ratio $X \simeq 1$. The system behaves classically, and the hydrodynamic description remains self-consistent.
2. Quantum Region: Near the light cone ($a > a_c$), the hydrodynamic decay is rapid ($\lambda(a) \gg \Omega$). The blurring dominates, and the regulated correlator $F$ acquires a decay rate set by $\Omega$ rather than the hydrodynamic rate. The functions $F$ and $C$ diverge, signaling a breakdown of the classical fluctuation-dissipation relation.

The boundary between these regimes is a critical ray $x = a_c vt$. The position $a_c$ is determined by the condition where the saddle point of the integral coincides with the integration boundary.

Derivation of the Planckian Bound
The authors argue that for a system to remain describable by classical hydrodynamics in a finite region of spacetime as the temperature approaches zero ($\beta \to \infty$, $\Omega \to 0$), the critical ray $a_c$ must remain finite.

• If $a_c \to 0$, the classical region vanishes entirely, and the system is quantum everywhere.
• To maintain a finite classical wedge ($a_c > 0$) as $\Omega \to 0$, the hydrodynamic rate $\lambda$ must scale with $\Omega$.

This requirement leads directly to the inequality:
$$\lambda = \frac{v^2}{4D} \lesssim \Omega = \frac{\pi}{\hbar \beta}$$
Rearranging this yields a lower bound on diffusivity:
$$D \gtrsim \frac{v^2 \hbar \beta}{4\pi}$$
This scaling recovers known bounds, such as the Kovtun–Son–Starinets bound on shear viscosity ($\eta/s \ge \hbar/4\pi k_B$) when identifying $D$ with momentum diffusivity and $v$ with the speed of light.

Significance and Claims
The paper claims that Planckian scaling of the diffusion constant is not necessarily a direct quantum constraint on microscopic dynamics, but rather the "price" a system pays to remain describable by classical hydrodynamics down to low temperatures.

• Mechanism: The Planckian bound emerges from the self-consistency requirement that the classical hydrodynamic ansatz must not be "blurred out" by the quantum FDT in a finite region of spacetime.
• Generality: The derivation is generic and independent of the specific microscopic model, relying only on the existence of a light cone and diffusive behavior.
• Reframing: The authors reframe Planckian dissipation as the condition for the validity of a classical hydrodynamic description. If the dissipation rate were slower than the Planckian scale, the classical description would become inconsistent with the quantum FDT, necessitating a switch to a genuinely quantum hydrodynamic theory.

The work provides a unified perspective linking transport bounds, equilibration times, and chaos bounds to the fundamental structure of the quantum fluctuation-dissipation theorem in the time domain.