A Schrödinger-like equation for the Thermodynamics of a particle in a box

This paper proposes a Schrödinger-like wave equation within a canonical Hamiltonian framework to describe the thermodynamics of a particle in an expanding or contracting 1D box, successfully linking mechanical behavior to entropy production, reproducing the universal quantum of heat conductance, and confirming consistency with quantum mechanics in quasi-static regimes while revealing adiabaticity breakdown under far-from-equilibrium conditions.

The Gist

Imagine you have a tiny, invisible billiard ball bouncing back and forth inside a hallway. This is the classic "particle in a box" problem physicists love to solve. Usually, we look at this in two separate ways:

1. Mechanics: How fast is the ball moving? How hard does it hit the walls?
2. Thermodynamics: How much heat is generated? How does the "disorder" (entropy) of the system change?

For over a century, these two worlds have been kept in separate rooms. This paper tries to knock down the wall between them. The author, A. Faigon, proposes a new way to look at this bouncing ball that treats heat and entropy as if they were forces and motion.

Here is the breakdown of the paper's big ideas, using everyday analogies.

1. The "Evolutive" Hallway (The Main Idea)

Usually, if you stretch a hallway while a ball is bouncing inside, the ball speeds up. In standard physics, we calculate the new speed. In this paper, the author says: "Let's treat the stretching of the hallway not just as a mechanical change, but as a flow of heat."

He invents a new set of variables (a new language) to describe the ball:

• The Old Way: We track position ($x$) and momentum ($p$).
• The New Way: We track a "Thermodynamic Momentum" ($f$) and a "Thermodynamic Angle" ($g$).

The Analogy: Imagine the ball is a runner on a track.

• In the old view, we measure how fast the runner is going.
• In this new view, we measure how much "sweat" (entropy) the runner produces as the track gets longer.
• The author shows that the "sweat" produced is mathematically identical to the "work" done by the runner. He creates a Hamiltonian (a master equation for energy) where the "energy" is actually the heat flowing in or out.

2. The "Bounce" as a Clock

The paper focuses on a specific moment: every time the ball hits the wall.

• The Metaphor: Think of the ball hitting the wall as a clock ticking.
• If the hallway is expanding, the ball has to run a bit further to hit the next wall.
• The author realized that the frequency of these bounces is directly linked to Temperature.
• The change in the distance between bounces is directly linked to Entropy (disorder).

By watching how the "clock" (the bounces) speeds up or slows down as the box expands, you can calculate exactly how much heat is being created or destroyed.

3. The "Quantum" Heat Pipe

The paper also looks at what happens when the box size stays the same, but the temperature changes (like putting a cold box into a hot room).

• The Result: The author calculated how fast heat flows through the walls.
• The Surprise: The math came out to match a famous, mysterious number in physics called the "Universal Quantum of Heat Conductance."
• Why it matters: This is like building a model of a car engine and finding that your math perfectly predicts the speed of light. It suggests that this new "thermodynamic mechanics" isn't just a guess; it's deeply connected to the fundamental laws of the universe.

4. The "Ghost" Wave (The Schrödinger-Like Equation)

This is the most sci-fi part. The author takes his new "Thermodynamic Hamiltonian" and turns it into a wave equation (like the famous Schrödinger equation used in quantum mechanics).

• The Concept: Instead of a wave describing where a particle might be, this new wave describes how the entropy (disorder) of the system evolves.
• The Test: He tested this "Entropy Wave" against a known solution for a ball in an expanding box.
  • Slow Expansion: The "Entropy Wave" matched the old "Mechanical Wave" perfectly.
  • Fast Expansion: When the box expands too quickly (far from equilibrium), the old mechanical rules break down, but the new "Entropy Wave" still works, predicting that the system gets "confused" and jumps to a new state.

The Metaphor: Imagine a dancer (the particle).

• If the music slows down gradually, the dancer's steps (mechanics) and their breathing (thermodynamics) stay in sync.
• If the music suddenly speeds up, the dancer stumbles. The old rules say "the dancer falls." The new rule says "the dancer's breathing pattern changes into a new rhythm," and the new equation predicts exactly what that new rhythm looks like.

5. Why Should You Care?

This paper is a bridge.

• Historical Context: Great physicists like Boltzmann and Helmholtz tried to link mechanics and heat a long time ago, but they mostly gave up in favor of statistics (counting probabilities).
• The New Hope: This paper suggests we can go back to the "mechanical" roots but include heat and entropy as fundamental parts of the motion, not just side effects.
• Future Use: This could help us understand how tiny machines (like nanobots) work when they are moving fast and getting hot, or how energy flows in systems that are far from being stable.

Summary in One Sentence

The author has rewritten the laws of motion for a bouncing ball so that heat and disorder are treated as forces and movement, creating a new "wave equation" that predicts how systems behave when they are pushed to their limits.

Technical Summary

1. Problem Statement

The paper addresses the challenge of unifying classical mechanics and thermodynamics, specifically for a particle in a one-dimensional (1D) box undergoing expansion or contraction. While the mechanical behavior of such a system is well-understood (via Newtonian or Hamiltonian mechanics) and its quantum mechanical treatment exists (e.g., Doescher's 1969 solution), there is a gap in formulating a canonical Hamiltonian framework that directly incorporates thermodynamic evolution (specifically entropy production) for systems far from equilibrium.

The author seeks to:

• Revisit the "particle in a box" problem using action-angle-like variables with direct thermodynamic interpretations.
• Derive a "Schrödinger-like" wave equation that describes the system's evolution beyond the quasi-static (adiabatic) limit.
• Bridge the gap between mechanical work/energy and thermodynamic heat/entropy within a single formalism.

2. Methodology

The author develops a novel "Evolutive Hamiltonian" ($H_{ev}$) framework by mapping mechanical variables to thermodynamic counterparts.

A. Variable Transformation

The core of the methodology involves redefining the phase space variables:

• Mechanical Momentum ($p$) & Coordinate ($q$): Mapped to a new momentum-like variable $f \equiv p \cdot l$ (where $l$ is the box length) and a phase coordinate $g \equiv q/l$.
• Thermodynamic Interpretation:
  • The product $f \cdot \dot{g}$ is identified with thermal energy ($kT$).
  • The logarithmic change in $f$ ($d \ln f$) is identified with entropy change ($dS/k$).
  • The kinetic term in the new Hamiltonian corresponds to reversible heat ($dQ_{rev}$).

B. Construction of the Evolutive Hamiltonian

The author constructs a differential form analogous to the mechanical Hamiltonian:
$$dH_{ev} = \dot{g}df - \dot{f}dg$$

• Kinetic Term ($dK_{ev}$): Identified as $dQ_{rev}(f)$.
• Potential Term ($dV_{ev}$): Identified as $-dQ_{rev}(g)$, representing the work done against the changing volume.
• This formulation allows the system to be treated as a conservative Hamiltonian system where the "evolution" is driven by the potential term, leading to entropy production ($\dot{S} \neq 0$) when the box size changes non-adiabatically.

C. Derivation of the Schrödinger-like Equation

To handle far-from-equilibrium conditions where the trajectory $g(t)$ is no longer well-defined (i.e., when changes are too rapid for a classical path), the author promotes the variables to operators:

• The momentum variable $f$ is replaced by the operator $\hat{f} = -i\hbar \frac{\partial}{\partial g}$.
• This yields a wave equation of the form $\hat{H}_{ev}\Psi = E_{ev}\Psi$, where the wavefunction $\Psi(g)$ encodes the probability distribution of the system's state and its entropy evolution.

3. Key Contributions

1. Evolutive Hamiltonian Formalism: The paper introduces a Hamiltonian scheme where the "momentum" is the adiabatic invariant ($p \cdot l$) and its variation drives entropy production. This provides a rigorous mechanical basis for the First Law of Thermodynamics ($TdS = dE + PdV$) within a canonical framework.
2. Universal Quantum of Heat Conductance: By analyzing heat transfer at constant volume using the derived Lagrangian, the author calculates the thermal conductance $\sigma$. In the quantum limit, this matches the universal quantum of heat conductance ($G_Q = \frac{\pi^2 k_B^2 T}{3h}$), validating the framework against known quantum thermal limits.
3. Schrödinger-like Equation for Thermodynamics: The derivation of a wave equation (Eq. 41 and 43) that describes the thermodynamic state of the system. The solutions to this equation contain information about entropy evolution, effectively treating thermodynamic evolution as a wave phenomenon in the "phase" space of the box.
4. Reconstruction of Quantum Mechanics: The paper demonstrates that the standard quantum mechanical wavefunction $\Psi(x,t)$ for a particle in an expanding box (as derived by Doescher) can be reconstructed from the author's thermodynamic wavefunction $\Psi(g)$ by mapping the phase coordinate $g$ back to physical position $x$.

4. Results

• Quasi-Static Regime: In the limit of slow changes (quasi-static), the solutions of the new Schrödinger-like equation show exact agreement with classical thermodynamic results and the WKB approximation. The wavefunction amplitude scales as $f^{-1/2}$, consistent with classical probability density.
• Far-from-Equilibrium Regime: As the expansion rate increases (non-adiabatic), the classical trajectory assumption breaks down. The wave equation predicts a breakdown of adiabaticity, where the system transitions to different quantum states.
• Comparison with Doescher's Solution:
  • For slow expansion ($\alpha = 50$), the author's solution matches Doescher's quantum solution perfectly.
  • For moderate expansion ($\alpha = 5$), deviations appear as the assumption of constant adiabatic invariant ($f=$ constant) fails.
  • For rapid expansion ($\alpha = 2.4$), the model predicts a transition to a higher quantum state, illustrating the system's inability to remain in the ground state under rapid non-equilibrium conditions.
• Entropy Production: The formalism explicitly shows that entropy production ($\dot{S}$) is proportional to the rate of change of the box length squared ($\dot{l}^2$), confirming that entropy is generated whenever the process is not perfectly reversible.

5. Significance

• Unification of Mechanics and Thermodynamics: The paper provides a rare, explicit canonical framework that treats thermodynamic variables (entropy, heat) as dynamic variables within a Hamiltonian/Lagrangian structure, similar to position and momentum.
• New Tool for Non-Equilibrium Physics: The derived "Schrödinger-like" equation offers a new mathematical tool to study systems far from equilibrium, potentially applicable to nanoscale devices, quantum thermodynamics, and heat engines where standard statistical mechanics might be difficult to apply directly.
• Validation of Quantum Limits: The successful derivation of the universal quantum of heat conductance from a purely mechanical/thermodynamic Hamiltonian model strengthens the link between microscopic mechanics and macroscopic thermal transport.
• Conceptual Shift: It reframes the "particle in a box" not just as a mechanical or quantum problem, but as a fundamental thermodynamic system where the "wavefunction" represents the evolution of entropy and state probability.

In conclusion, A. Faigon's work successfully bridges the gap between classical mechanics, thermodynamics, and quantum mechanics by proposing an "Evolutive Hamiltonian." This framework not only reproduces known results in the quasi-static limit but also provides a predictive wave-equation model for non-equilibrium thermodynamic processes.