Using the wavelet transform to separate scales in the Schr"odinger equation and subsequently derive the Boltzmann equation

arXiv:2511.21601 · quant-ph, cond-mat.other · Submitted 2025-11-26 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Using the wavelet transform to separate scales in the Schr"odinger equation and subsequently derive the Boltzmann equation".

Mira: The paper presents a novel derivation of classical mechanics from quantum mechanics by deriving the Boltzmann equation directly from the Schrödinger equation through formal mathematical manipulation.

Kai: First, who's behind it and why it matters.

Title and authors: Mira: So, focusing on this paper, "Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation," what we're really looking at is their method for connecting quantum mechanics and classical kinetic theory through scale separation. They show how different limits of quantum mechanics map onto distinct parts of kinetic theory one.

Kai: I agree with that, Mira; it’s wild to see how they take something as abstract as a wave function and systematically peel back its layers to get kinetic theory without just making some shaky assumptions one. They are aiming for a single mathematical derivation from first principles one.

Lev: If the core of the paper is this single manipulation, then for someone working on simulations, it means we don't have to worry about patching together different theoretical models; it’s one unified mathematical structure one.

Kai: They start by finding an equation for that smooth envelope A and then use that to derive the probability density rho = A*A, which is a key intermediate step in their process one.

Mira: After that, they introduce wave packets into the theory to account for external potentials, which leads them to a smooth function A(x, p) dependent on both coordinate and momentum one.

Lev: I'm interested in how the Liouville equation emerges from that process; if it’s a natural consequence of their procedure, it suggests we're following the physically expected path for these kinds of systems one.

Kai: That leads them to the one-dimensional Liouville equation, and then they generalize this to many-particle systems using a Slater determinant for fermions and expanding the kinetic energy operator one.

Mira: The generalization is where it gets heavy; they get an equation of evolution for A in arbitrary degrees of freedom, which is then shown to be equivalent to the Liouville equation for the probability density one.

Lev: Dealing with that many-particle structure is what separates this from simpler models; if they can handle that complexity, it means we're looking at something much closer to realistic physical systems one.

Kai: Finally, they use these results to derive the non-collision part of the Boltzmann equation by averaging Liouville’s equation when interactions are excluded, which is Equation thirty-two one.

Mira: And then they get the full collision integral term by considering interactions derived from quantum transitions and transition rates one.

Lev: So, this summary shows a clear progression: from wave packet decomposition to a many-particle evolution equation, and finally assembling the full Boltzmann equation from those pieces one.

Kai: This whole process is quite systematic; they start with the wave function structure, move to potentials using wave packets, generalize to many particles using fermions, and then assemble the kinetic theory terms one.

Mira: And that assembly step is what really earns them credit because they show how all these pieces fit together mathematically without needing extra assumptions between them one.

Lev: It’s a solid framework because it shows the path from a fundamental quantum equation to a macroscopic transport equation, which is exactly what we need for hardware modeling one.

The paper's summary: Kai: Moving on to the actual summary of "Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation," they show how this method directly connects quantum mechanics and classical kinetic theory through scale separation.

Mira: Basically, they manage to derive the Boltzmann equation as a single piece of formal mathematical manipulation from first principles without resorting to non-rigorous reasoning one. They resolve contradictions around time-reversibility by treating the different limits of quantum mechanics as separate phenomena one.

Lev: That means if we follow their structure, we bypass having to justify disparate parts separately; it’s one unified mathematical structure one.

Kai: The derivation involves using a specific form for the wave function and combining equations for time and space derivatives to get an equation for the smooth envelope A, leading to Equation four one.

Mira: Then they derive the probability density as rho = A*A through multiplication by its complex conjugate and adding equations, which is a core intermediate step one.

Lev: If we look at how they introduce external potentials, that's where things get interesting because it forces them to move beyond just the free particle case one.

Kai: They introduce wave packets to incorporate these potentials, which leads them to an expression for A(x, p) that is a smooth function of both coordinate and momentum one.

Mira: This step results in the one-dimensional Liouville equation, which they then recast into the Hamiltonian form involving rho one.

Lev: I'm interested in how they handle the complexity when moving to many particles; if it's a natural consequence of their procedure, it suggests we're following the physically expected path for these kinds of systems one.

Kai: They generalize this to fermionic particles using a Slater determinant and then expand the kinetic energy operator to get that equation for A in arbitrary degrees of freedom one.

Mira: The final result is an equation of evolution for A, which they prove is equivalent to the Liouville equation for the probability density one.

Lev: That means if they can handle that many-particle structure, it implies we're looking at something much closer to realistic physical systems one.

Kai: They then combine everything with previous work to get the non-collision part of the Boltzmann equation by averaging Liouville’s equation when interactions are excluded, which is Equation thirty-two one.

Mira: And they get the full collision integral term by considering interactions derived from quantum transitions and transition rates one.

Lev: So, this summary shows a clear progression: from wave packet decomposition to a many-particle evolution equation, and finally assembling the full Boltzmann equation from those pieces one.

Mira: And that assembly step is what really earns them credit because they show how all these pieces fit together mathematically without needing extra assumptions between them one.

Lev: It’s a solid framework because it shows the path from a fundamental quantum equation to a macroscopic transport equation, which is exactly what we need for hardware modeling one.

The paper's improvements: Kai: Now let's look at the suggested refinements in "Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation." The authors point out how their method is compatible with existing methods used for deriving transition rates between quantum states due to interactions one.

Mira: They suggest this compatibility means their approach works well with established methods, which is good because it implies we don't have to reinvent the wheel for calculating collision integrals one.

Lev: That compatibility is crucial because it suggests that if we can successfully implement their framework, we don't have to reinvent the wheel for calculating collision integrals one.

Kai: They also highlight how they manage the potential term by considering a specific scale x —it needs to be larger than the fast oscillations but smaller than the characteristic variation of A one.

Mira: That constraint on x is important because it defines the precise boundary where their method is valid, and if we choose that scale incorrectly, the derivation breaks down one.

Lev: From a research standpoint, this gives us a clear guideline for setting up our computational boundaries so we know exactly where our semi-classical approximation is safe to use one.

Kai: They also suggest using this formal separation of the potential into "sharp" (collision integral) and "smooth" parts, which they say is essential for avoiding infrared divergences in perturbation theory one.

Mira: That idea of separating the potential into these two parts seems very useful; it gives us a way to tackle those mathematical instabilities in other quantum transport models one.

Lev: If we can apply this separation idea to other problems, it means we can systematically eliminate those divergences, which is a major win for theoretical modeling one.

Kai: So the improvements focus on using this formal structure to make the theory more stable and applicable in broader contexts than just one specific derivation one.

Mira: It sounds like the authors are essentially providing a toolkit that helps researchers manage mathematical instabilities across different quantum transport models one.

Lev: That systematic approach to handling those divergences is a major win for theoretical modeling one.

Conclusion: Kai: To wrap up our discussion on "Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation," we’ve seen how this work connects quantum mechanics and classical mechanics through scale separation.

Mira: It really shows how they managed to derive the Boltzmann equation as a single piece of formal mathematical manipulation from first principles without resorting to non-rigorous reasoning one.

Lev: From my side, this means we have a clearer pathway for understanding how noise transitions from quantum behavior into the classical kinetic description, which is important when we think about running things on real hardware one.

Mira: Exactly; by showing how different limits of quantum mechanics map onto distinct parts of kinetic theory, they resolve that old contradiction around time-reversibility by treating those limits as separate phenomena one.

Lev: For error correction, this suggests we have a potentially cleaner way to analyze the underlying noise mechanisms that affect qubit coherence on real devices one.

Kai: That's a big deal because it gives us a concrete way to look at how quantum systems behave when they start looking like classical ones in transport applications one.

Mira: And I think it opens up new avenues for studying how quantum effects fade away systematically as we move toward those classical descriptions, which is really useful for condensed matter theory one.

Lev: For simulation purposes, this provides a rigorous way to approach the dynamics of these systems before we try to build complex models one.

Kai: I think this whole process confirms that the relationship between quantum and classical mechanics isn't just an approximation but a structured limit of one into the other one one.

Mira: We’ve seen how they managed to derive the Boltzmann equation as a single piece of formal mathematical manipulation from first principles using wavelet transforms in this paper, "Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation."

Lev: This means we have a clearer pathway for understanding how noise transitions from quantum behavior into the classical kinetic description.

Kai: That’s right; it really shows how we can connect quantum and classical mechanics through scale separation.

Centro Atomico Constituyentes, CNEA

quant-ph, cond-mat.other

Submitted: 2025-11-26

Updated: 2026-10-07

Comments: 16 pages, 1 figure

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 62/100

The gist: The paper presents a novel derivation of classical mechanics from quantum mechanics by deriving the Boltzmann equation directly from the Schrödinger equation through formal mathematical manipulation.

Key concepts

Wavelet Transform: This mathematical tool is used to analyze the Schrödinger equation by decomposing the wave function into components representing different spatial scales or resolutions. This allows researchers to isolate specific features of the quantum state, such as smooth envelopes versus fast oscillations, which is crucial for the subsequent derivation.
Wave Packet Decomposition: The method involves writing a wave function as a product of a smooth envelope and a fast-oscillating part. By manipulating equations related to time and space derivatives, this decomposition leads to an equation governing the evolution of the smooth envelope, which is key to finding classical descriptions.
Boltzmann Equation: This is the equation that describes how particle distribution functions evolve over time in kinetic theory. The paper derives it by showing it emerges as a specific limit—the non-collision part—of quantum mechanics when interparticle interactions are averaged out, providing a bridge to classical statistical mechanics.

Terminology

Boltzmann Equation — 55× in this paper · explained in 2 episodes
Wavelet Transform — 8× in this paper · explained in 2 episodes
Generalization — 4× in this paper · explained in 13 episodes
Core — 3× in this paper · explained in 2 episodes
Lindblad master equation — 1× in this paper · explained in 2 episodes

Summary

The paper presents a novel derivation of classical mechanics from quantum mechanics by deriving the Boltzmann equation directly from the Schrödinger equation through formal mathematical manipulation. This approach aims to bridge the gap between quantum and classical descriptions by showing how different limits of quantum mechanics correspond to distinct parts of kinetic theory, resolving apparent contradictions regarding time-reversibility.

The gist

The Boltzmann equation is derived from the Schrödinger equation as a single piece of formal mathematical manipulation, without any non-rigorous plausible reasoning used to glue together its different parts.

Derivation via Wave Packet Decomposition

The method begins by seeking the spatial part of the wave function in a specific form: Ψ(x) = X p Ap(x)ϕ p(x), where Ap(x) is a smooth envelope and ϕ p(x) is the fast oscillating part. The derivation proceeds by combining equations for time and space derivatives to obtain an equation for the smooth envelope A, leading to the relation A˙ = −p/m A′ (Equation 4). By multiplying this by its complex conjugate and adding it to a similar equation for A∗, the probability density ρ is derived as ρ = A∗A.

Incorporating External Potentials and Wave Packets

The theory introduces wave packets to incorporate external potentials. This involves substituting the wave function form into the Schrödinger equation and cutting out a piece of the wave function in a coordinate region near x0 and a momentum region near p0. Through approximations, this leads to an expression for A(x, p) that is a smooth function of two variables, coordinate and momentum. This step results in the one-dimensional Liouville equation: ρ˙ = −p/m∂ρ/∂x + ∂U/∂x ∂ρ/∂p (Equation 15), which can be recast as the Hamiltonian form: ρ˙ + ∂H/∂p∂ρ/∂x − ∂H/∂x∂ρ/∂p = 0 (Equation 16).

Generalization to Many-Particle Systems

The derivation is extended to the many-particle case, specifically for fermionic particles using a Slater determinant for the fast oscillating part. The kinetic energy operator is applied, and after careful expansion of the resulting summands, a key result is obtained: ∂A/∂x1 p1 + ∂A/∂x2 p2 + … = ψ p(x) (Equation 26). This generalization leads to an equation of evolution for the amplitude A in arbitrary many degrees of freedom: A˙ + Xn i=1 ∂H/∂pi ∂Ak/∂xi − ∂H/∂xi ∂Ak/∂pi = 0 (Equation 31), which is then shown to be equivalent to the Liouville equation for the probability density.

Derivation of the Boltzmann Equation

The final step involves combining the results from Sections II and III with previous work. By averaging Liouville’s equation when interparticle interactions are excluded, averaging Liouville’s equation indeed yields the non-collision part of the Boltzmann equation: ˙f(x, p) + ∂H/∂p∂f/∂x − ∂H/∂x∂f/∂p = 0 (Equation 32). Furthermore, by considering the interaction term derived from quantum transitions (the collision integral), which is related to transition rates, the full Boltzmann equation is obtained: ˙f(x, p) + ∂H/∂p∂f/∂x − ∂H/∂x∂f/∂p = St(f) (Equation 48). This derivation resolves the paradox between classical time-reversibility and the irreversibility of the Boltzmann equation by identifying them as two different limits of quantum mechanics—the long-wavelength (non-collision) and short-wavelength (collision) limits, respectively.

Applications to Physical Kinetics

The derived current density formula, ⃗j(x) = e Z ∫d3p(2πħ)/3⃗p m f(x) (Equation 42), demonstrates that the particles behave as classical point particles under the given assumptions. The derivation for other physical quantities like charge, energy, and similar densities is analogous and simpler. This systematic derivation justifies the use of classical concepts for quasiparticles in solid-state physics by showing their emergence from quantum mechanics under specific limiting conditions. The theory also provides a formal separation of the potential into sharp (collision integral) and smooth (force term) parts, which is crucial for eliminating infrared divergences in perturbation theory.

How it works

The derivation starts with the Schrödinger equation and decomposes the wave function into a fast-oscillating part and a smooth envelope to derive an equation for the latter.
Wave packets are introduced by localizing the wave function in space and momentum regions, leading to an expression for A(x, p) as a smooth function of two variables.

Improvements for AI systems

Based on the provided scientific paper, here are the specific improvements that could be made to AI systems:

Improving Real-Time Kinetic Simulations (Molecular Dynamics/Transport):
Developing Time-Reversible, Non-Coherent Kinetic Models:
Creating Robust Collision Integral Algorithms for High-Frequency Fields:
Enhancing Predictive Power for Quasiparticles in Condensed Matter:

Here is a detailed breakdown of what the improved AI system can do in each area:

AIs can perform highly accurate, real-time simulations of particle transport (e.g., gas dynamics, fluid flow) by directly implementing the derived Liouville equation (Eq. 31) and its resulting macroscopic form (Eq. 48).
The AI can model systems where the non-collisional component is governed by classical, time-reversible dynamics, allowing for deterministic trajectory predictions under external fields (low-frequency limits).
The AI can develop sophisticated algorithms to calculate the collision integral term in the Boltzmann equation (Eq. 48) by treating interparticle interactions as sharp or short-ranged, utilizing the transition rate matrix formalism derived from quantum mechanics. This allows for accurate kinetic modeling even when dealing with systems where classical assumptions are locally valid but quantum transitions occur during collisions.
The AI can accurately predict transport properties of quasiparticles (like electrons in solids or phonons) by using the derived formulas for current density (Eq. 42) and other related observables, effectively bridging the gap between microscopic quantum descriptions and macroscopic classical descriptions used in areas like field-effect transistor modeling.

Abstract

Here we continue our investigation aimed at the derivation of the Boltzmann equation from the first principles of quantum mechanics. The key difficulty that we overcome in this manuscript is the incompatibility between traditional methods employed in the derivation of classical mechanics from quantum mechanics and methods used to describe transitions between quantum states due to interaction. Here we use a novel technique that is similar to the wavelet transform method that originally appeared in the context of signal processing. With this method, we can simultaneously work with slowly varying potentials that give rise to classical-like wave-function behaviour, motion along a prescribed trajectory, and sharp potentials that lead to behaviour that is best understood in terms of quantum transitions. Working with smooth potentials in this framework, we derive the Liouville equation from the Schrödinger equation. When using this method for sharp potentials, we naturally obtain the description of system dynamics in terms of a transition-rate matrix. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the transition-rate matrix, we can deduce the collision integral. We combine the results to derive the Boltzmann equation, the main formula of physical kinetics.