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Quantum states of macrosystems and entropy

This paper critiques the traditional Boltzmann definition of entropy as the logarithm of quantum states, proposing instead that entropy arises from subquantum processes and is mathematically expressed as the ratio of the logarithm of a macroscopic system's maximum state realizations to the frequency of its quantum state occurrences over a given observation period.

Original authors: Maria Polski, Vladimir Skrebnev

Published 2026-06-02✓ Author reviewed ⓘ

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Maria Polski, Vladimir Skrebnev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: What is Entropy?

Imagine you are trying to understand a giant, bustling city (a macrosystem). In standard physics, we usually think of "entropy" (a measure of disorder or chaos) as simply counting how many different ways the city's buildings could be arranged. The more ways they can be arranged, the higher the entropy.

The authors of this paper argue that this standard way of thinking is wrong. They claim that entropy isn't just a static count of "possible" arrangements. Instead, they say entropy is the result of invisible, super-fast movements happening underneath the surface of reality.

The Problem with the Old View

The paper starts by critiquing the famous formula $S = \ln W$ .

The Old View: Imagine a gas in a box. Physicists say the gas has a total energy $E$ . They assume all the possible energy levels for the gas are squeezed into a tiny, narrow band right around that energy $E$ . They count how many "quantum states" (specific energy levels) fit in that band and call that number $W$ . Then, they say Entropy is just the log of that number.
The Authors' Critique: The authors say this is like assuming a movie is just a single, frozen frame. They argue that energy levels in a real system aren't stuck in a tiny, narrow band that changes depending on the temperature. They say it's physically impossible for the "rules" of the system (the energy spectrum) to shift just because the energy changes.
The Metaphor: Imagine a piano. The keys (energy levels) are fixed. You can't say the keys move closer together just because you are playing a louder song (higher energy). The authors argue that the standard formula assumes the piano keys are magically rearranging themselves to fit the song, which isn't real.

The New View: The "Subquantum" Dance

So, if it's not about counting static states, what is it? The authors propose a new explanation involving subquantum processes.

The Analogy: The Invisible Dancers
Imagine a macroscopic system (like a cup of coffee) is a stage.

The Visible Show: We see the coffee sitting there, calm and still.
The Invisible Reality: Underneath, there are "subquantum processes" (invisible dancers) moving so fast that we can't see them. These dancers are constantly jumping between different energy states.
The Visits: Every time a dancer jumps to a specific energy level, it's a "visit." Over a period of time, the system "visits" many different states.
The Count: The authors argue that entropy is actually a ratio of two things:
- The Top: How many different ways the system could arrange these visits to reach a stable, balanced state (equilibrium).
- The Bottom: How many times the system actually did visit those states during the observation time.

The "Visit Configuration"
Think of a deck of cards.

You have a total energy (the sum of the card values).
There are many different ways to shuffle the cards to get that same total sum.
The authors say the "subquantum processes" are the shuffling.
The system naturally shuffles itself into the arrangement that has the most possible permutations (the most ways to be shuffled). This is the state of "thermodynamic equilibrium."

The Connection to Boltzmann

The paper pays homage to Ludwig Boltzmann, a 19th-century physicist who first tried to link entropy to probability.

Boltzmann called the different ways to arrange gas molecules "Komplexions" (complexions).
He realized that the state with the most ways to be arranged is the one the gas naturally settles into.
The authors agree with Boltzmann's math but disagree with the modern quantum interpretation. They say Boltzmann was right about the "counting of arrangements," but modern physicists have mistakenly applied this to static "quantum states" rather than the dynamic "visits" caused by subquantum processes.

The Conclusion: What is Entropy Really?

The authors conclude that entropy is not a static number of "possible states."

The Final Metaphor:
Imagine a busy highway.

Old View: Entropy is just counting how many different lanes exist on the highway.
Authors' View: Entropy is a measure of the traffic flow. It is the ratio of the number of ways the cars could be distributed to keep traffic moving smoothly, divided by the actual number of times cars passed a specific point.

They argue that entropy is the result of these invisible, rapid "subquantum" jumps. The system naturally evolves toward the state where these jumps can happen in the maximum number of ways. This is why things naturally move toward "disorder" (equilibrium)—because that is the state with the most "dance moves" available to the invisible subquantum processes.

In short: The paper claims we have been looking at entropy as a photo (a static count of states), when we should be looking at it as a video (a record of rapid, invisible transitions). Entropy is the measure of how many ways those transitions can happen to keep the system balanced.

Technical Summary: Quantum States of Macrosystems and Entropy

Problem Statement
The paper addresses the fundamental physical interpretation of entropy in contemporary physics. Specifically, it critiques the standard formulation $S = \ln W$ (where $S$ is entropy and $W$ is the number of quantum mechanical states), arguing that this expression relies on a physically untenable assumption. The authors contend that the prevailing view—that a macrosystem's energy levels concentrate in an "extremely narrow interval" around the total energy $E$ with equal probability—is inconsistent with the nature of energy spectra, which are independent of the system's specific energy value $E$ . Consequently, the authors posit that defining entropy as the logarithm of a number of states that "do not exist" in this specific manner is physically incorrect.

Methodology
The authors employ a theoretical analysis combining statistical mechanics, thermodynamics, and a proposed framework of "subquantum processes." Their methodology proceeds as follows:

Critique of Standard Derivation: They analyze the canonical distribution and the assumption of energy concentration, demonstrating logical inconsistencies regarding the independence of energy spectra from the system's total energy.
Thermodynamic Derivation: They derive entropy starting from the probability density $\rho$ and the canonical distribution, utilizing the relationship between Helmholtz free energy ( $F$ ), internal energy ( $E$ ), and temperature ( $T$ ) to express entropy as a dimensionless quantity derived from $\ln \rho$ .
Subquantum Process Framework: The authors introduce the concept that quantum transitions (emission/absorption of energy) are driven by underlying "subquantum processes." They model the system's evolution as a series of "visits" to specific energy states over an observation period.
Combinatorial Analysis: Drawing on Boltzmann's concept of Komplexions (permutations), they calculate the number of ways ( $P$ ) a specific configuration of visits can be realized. They apply the law of conservation of energy to these visit configurations, identifying the configuration with the maximum number of permutations ( $P_{max}$ ) as the state of thermodynamic equilibrium.

Key Contributions

Rejection of $S = \ln W$ : The paper argues that the standard interpretation of entropy as the logarithm of the number of quantum states is a misinterpretation of Boltzmann's work and is physically flawed.
Subquantum Origin of Entropy: The authors propose that entropy is the product of "subquantum processes." They assert that while quantum mechanics describes the probabilities of states, it does not describe the transitions themselves; these transitions are caused by subquantum regularities that manifest as random processes in statistical physics.
New Entropy Formulations: The paper derives two primary expressions for entropy:
- Formula (22): $S = \frac{\ln P_{max}}{N}$ , where $P_{max}$ is the maximum number of ways to realize the canonical distribution configuration, and $N$ is the total number of acts (visits) of occurrence of all states during the observation time.
- Formula (28): A generalized form relating entropy to the ratio of the logarithm of the number of realizations to the number of occurrences.
Historical Contextualization: The authors re-examine Boltzmann's 1877 work, suggesting that his concept of Komplexions (permutations of molecules) was a precursor to the concept of subquantum visit configurations, and that Boltzmann's use of probability theory was a "premonition" of quantum theory, even though he operated within classical mechanics.

Results
The analysis yields the conclusion that entropy is not a static count of quantum states but a dynamic ratio. Specifically, entropy is expressed as the ratio of the logarithm of the maximum number of realizations (permutations) of a macroscopic system's states with a given total energy to the number of occurrences of its quantum states over a specific observation period.
The authors demonstrate that the irreversible evolution of a macrosystem toward thermodynamic equilibrium is driven by subquantum processes that maximize the number of permutations ( $P$ ), thereby maximizing entropy. They show that the canonical distribution emerges naturally as the state where the number of visit permutations is maximized, consistent with the conservation of total energy.

Significance
The paper claims its primary significance lies in clarifying the "true physical nature" of entropy. By shifting the focus from a static count of non-existent concentrated states to a dynamic ratio of subquantum visit realizations, the authors aim to resolve the conceptual confusion surrounding the formula $S = \ln W$ . They suggest that their work, alongside their previous papers [2, 3], provides a more accurate understanding of entropy as a characteristic of phenomena driven by subquantum regularities, rather than a mere mathematical abstraction of quantum state counting. The authors express hope that this perspective will aid in better understanding the physical meaning of entropy in the context of the world around us.