Non-Statistical Analytic Entropy-Multiplicative Entropy Formula Answers Planck’s Question

1. The Concept of Entropy: From Statistical Description to Analytic Dynamics

Traditionally, entropy is understood as a measure of "disorder" or "randomness" in a system. However, "disorder" is inherently abstract and difficult to quantify directly—it relies on statistical counting of microstates (e.g., Boltzmann’s S = k ln W), rather than dynamic tracking of actual energy flows. Although Planck sought an analytic expression for entropy at the birth of quantum theory, entropy has remained largely confined within probabilistic and statistical frameworks, failing to emerge as a real-time evolving dynamical variable.

This statistical formulation leads to a fundamental mismatch in resolution: quantum processes occur on timescales as fine as the Planck time, governed precisely by the Schrödinger equation; spacetime geometry in general relativity evolves continuously and deterministically. Yet, thermodynamic entropy—and its irreversibility—has no place in the core equations of either theory. As a result, the two pillars of modern physics—quantum field theory and general relativity—fail to naturally incorporate the laws of thermodynamics, especially the second law and its arrow of time.

The deeper issue lies in the fact that statistical entropy is essentially a "retrospective summary"—a macroscopic description that cannot capture the path of entropy change during individual quantum transitions or local geometric changes. It cannot answer: "How does entropy change during a single energy exchange event? " Consequently, it fails to embed thermodynamic dynamics into quantum evolution or spacetime geometry.

The multiplicative analytic entropy proposed here aims to overcome this limitation. By defining entropy as the product of energy values across all Space Elementary Quanta (SEQs), S = ∏ mᵢ, this form abandons statistical assumptions and instead tracks every discrete energy redistribution event, enabling high-resolution dynamic modeling of entropy. Each energy transfer satisfying mᵢ > mⱼ + ℎ strictly increases entropy, embedding the arrow of time directly into the dynamical rules. This not only aligns entropy’s resolution with quantum processes but also opens a new mathematical pathway to integrate thermodynamic irreversibility into both the Schrödinger equation and Einstein’s field equations.

This formulation originates from the degree of energy uniformity: the greater the entropy, the more uniform the energy distribution becomes. Energy flows explicitly from higher-energy quanta to adjacent lower-energy ones, in direct correspondence with classical thermodynamic descriptions of heat conduction. Under this principle, we introduce multiplicative entropy—while the total energy of a system remains conserved, the cumulative product of individual energy levels increases monotonically as the distribution evolves toward uniformity

3.1. Definition of Multiplicative Entropy-Energy Product Entropy

I define multiplicative entropy as the product of the energies carried by each unit in a closed system. This definition of entropy is computable, and can also serve as an entropy coordinate for simulating the evolution of physical systems. When logarithmized, this multiplicative form reduces to the traditional additive entropy expression used in statistical mechanics.

Under the constraint of the second law of thermodynamics, energy always flows from units with higher energy to those with lower energy in a closed system. The total energy of all units remains constant (energy conservation), but the product of their energies keeps increasing — that is, the more uniform the energy distribution becomes, the larger the value of the multiplicative entropy.

When all units carry exactly the same amount of energy, the system reaches its maximum possible entropy — a state known as heat death.

This definition provides a much clearer and more intuitive description of entropy increase than the classical statistical formulation.

3.2. Quantum Thermodynamics Perspective

Speaking of quantum thermodynamics, if we assume that the quantum is the fundamental building block of space, then under the framework of a quantized space network, all states become analytic rather than statistical, specifically at the Planck time scale.

Let’s revisit the rule defined by the second law of thermodynamics: energy only flows from high-energy regions to low-energy ones. In essence, the process described by the second law — entropy increase — is a process of energy homogenization.

3.3. Comparison Between Multiplicative Entropy and Classical Statistical Entropy

Dimension	Multiplicative Entropy	Traditional Statistical Entropy
Process Explicitness	Yes, every step has clear changes	No, only describes macroscopic end states
Preserves Microscopic Details	Yes, path-dependent	No, only cares about probability distributions
Has Temporal Directionality	Yes, defined by local energy flow	No, needs extra assumption for time arrow
Suitable for Simulations	Yes, good for numerical modeling	No, mainly used for theoretical analysis
Is an Analytical Function Form	Yes, not based on statistical average	No, relies on ensemble average

Why Is Multiplicative Entropy More Suitable for Describing Energy Homogenization?

Take for example a system composed of N basic quanta. According to the second law, energy transfers from high-energy quanta to low-energy ones, gradually making the overall energy distribution more uniform.

Assume each quantum carries energy mᵢ. At any given transformation state, the total energy of the system is:

∑ mᵢ = constant; This satisfies energy conservation in a closed system.

How do we characterize the entropy change during this evolution? My proposed multiplicative entropy offers a more intuitive and precise analytical tool:

The entropy S of the system at a certain transformation state is defined as:

S = ∏ mᵢ

Under the constraint of energy conservation and directional energy transfer (from high to low), the more uniform the energy distribution becomes, the larger the product becomes.

When all quanta carry equal energy, the system reaches its maximum entropy — heat death occurs.

After applying the logarithm, this formula transforms into the classical additive entropy form. However, I argue that the multiplicative form more accurately captures the spontaneous and irreversible nature of entropy increase, offering greater clarity and visual intuition.

3.4. Computable Definition of Multiplicative Entropy: Time–Entropy Mapping

In this entropy definition, the entropy value of a closed system at a given moment (i.e., during a specific state transition) is calculated as the product of the energy norms of all space elementary quanta (SEQ) involved in the spatial transformation at that moment.

Analysis Formula of Entropy

In this definition of entropy, the entropy value of a closed system at a given moment (i.e., during a specific state transition) is calculated as the multiplicative product of the energy norms of all SEQ involved in that transition(that moment’s space transformation).

Entropy value of closed system S=∏mᵢ, i∈N

(1)

Energy of closed system=constant=∑mᵢ, i∈N

(2)

Sₘₐₓ≤mᵢⁿ , When all mᵢ are equal or differ only by Planck's constant h

(Where mᵢ refers to the energy carried by the ith SEQ during a single transformation of the closed system, where each energy state mᵢ is an integer multiple of Planck’s constant h, mᵢ=nᵢh, nᵢ∈N)

3.5. Energy Transfer Rules and Triggering Conditions:

Energy exchange occurs between adjacent SEQ (i,j) if and only if the following thermodynamic gradient exists: mᵢ＞mⱼ+h, Energy transfer occurs only in discrete quanta of Planck's constant h, mᵢ→ mᵢ−h; mⱼ→mⱼ+h (Planck's constant:h)

Numerical Example: System States and Entropy Evolution

Table 1. Simplified Entropy Increase Demonstration.

System State	SEQ Energy Distribution Etotal=∑mᵢ=12	Entropy S=∏mᵢ	Remarks
Initial non-equilibrium state	[3, 1, 5, 3]	45	-
Intermediate state	PathA [3, 1, 4, 4]; PathB [3, 2, 4, 3]；	A48; B72；	-
Final state	PathA [3, 2, 3, 4]; PathB [3, 3, 3, 3]；	A72; B81；	Due to adjacent energy transfer with minimal quanta h, this system cannot reach maximum entropy in case A

Note:The above analysis demonstrates that different entropy-increasing pathways exhibit distinct sequences of entropy variation.

Table 1. Simplified Entropy Increase Demonstration.

System State	SEQ Energy Distribution Etotal=∑mᵢ=12	Entropy S=∏mᵢ	Remarks
Initial non-equilibrium state	[3, 1, 5, 3]	45	-
Intermediate state	PathA [3, 1, 4, 4]; PathB [3, 2, 4, 3]；	A48; B72；	-
Final state	PathA [3, 2, 3, 4]; PathB [3, 3, 3, 3]；	A72; B81；	Due to adjacent energy transfer with minimal quanta h, this system cannot reach maximum entropy in case A

Note:The above analysis demonstrates that different entropy-increasing pathways exhibit distinct sequences of entropy variation.

3.6. Logarithmic Relation:

After logarithmic transformation, lnS aligns with the conventional Boltzmann entropy form, while the multiplicative formulation naturally suits discrete systems.

3.7. Proof of Spontaneous Entropy Increase

Spontaneity Theorem of entropy increase (Second Law of Thermodynamics):

For every possible energy transfer process, the total entropy change satisfies ΔS≥0.

Proof Outline: Let the pre-transfer states be mᵢ=a, mⱼ=b (a＞b+h);

The post-transfer entropy ratio is:

Sₜ₁/Sₜ₂=(a−h)(b+h)/ab=1+h(a−b−h)/ab＞1 (Planck's constant h)

This demonstrates how entropy increases along various paths, reflecting the irreversibility and path dependence of thermodynamic processes in real systems.

Moreover, each system state can be assigned a unique entropy value — an "entropy coordinate" , which provides a powerful tool for computer simulations at the quantum level.

3.8. Analysis of the Maximum Entropy Principle

In the SEQ model, the maximum entropy principle is manifested through the driving tendency of entropy increase: energy not only flows from a higher-energy SEQ to an adjacent lower-energy SEQ, but it also follows the path with the largest energy difference.

For example, consider an SEQ i, which is adjacent to two other SEQ j and k. The energies carried by nodes i, j, and k are A, B, and C respectively, where A > B > C. In this case, there are two possible energy transfer paths from SEQ i according to the principle of entropy increase: path i→j or path i→k. We will analyze the entropy change for each path separately.

Under the constraint of energy conservation (i.e., A + B + C = constant), the entropy of the local system composed of these three SEQ before energy transfer is:S = A×B×C

Path i→j: transferring energy to node j (which has relatively higher energy):

S′= (A − 1)(B + 1)C = (A − 1)(BC + C)

Path i → k: transferring energy to node k (which has lower energy):

S″= (A − 1)B(C + 1) = (A − 1)(BC + B)

Here, "1" represents one unit of Planck constant h. Since B > C ⇒ BC + B > BC + C ⇒ S″> S′, the entropy increases more along the i→k path—that is, the path with the larger energy difference leads to a greater increase in entropy.

This deduction can be easily generalized to cases where the number of adjacent nodes is greater than two, so a detailed proof is omitted here.

However, it should be emphasized that the path with the maximum energy difference is not necessarily unique. Therefore, for states that have not yet occurred, the future still retains sufficient degrees of freedom — the evolution is not entirely deterministic.

From the perspective of this model, the apparent randomness in the wave function described by the Schrödinger equation arises from the non-uniqueness of the maximum entropy path—multiple microscopic energy redistribution trajectories, giving rise to probabilistic outcomes without requiring fundamental indeterminism.

3.9. Dimensional Structure in the Model: Dimensionless Energy and Entropy

In this model, all physical quantities are reconstructed based on the quantization of Planck’s constant ℎ. Energy is defined as discrete units carried by Space Elementary Quanta (SEQs), with each SEQ carrying an energy value that is an integer multiple of ℎ, i.e., mᵢ = nᵢℎ, where nᵢ ∈ ℕ. Since ℎ is normalized as a natural unit, energy becomes a dimensionless counting variable—representing the distribution of "energy quanta" across the system.

Accordingly, entropy is defined as the product of all SEQ energy values: S = ∏ mᵢ. Since each mᵢ is already dimensionless (normalized relative to ℎ), their cumulative product remains dimensionless. This definition frees entropy from reliance on statistical ensembles or probability weights, making it a directly computable, path-dependent dynamical variable capable of precisely characterizing evolution through every energy redistribution event.

Given the model’s prior assumption that each Space Elementary Quantum (SEQ) possesses a ground-state energy quantized in integer multiples of ℎ ,[3] the energy value of every mᵢ is at least 1; consequently, in this framework, the multiplicative entropy of any physical system is strictly greater than 1.

4. Novelty of the Concept

After extensive searches using general search engines and academic databases, I find that the analytical entropy expression based on energy products I propose here offers a fundamentally new alternative outside the traditional statistical framework. This approach transforms entropy from a statistical concept into an analytic one, and to my knowledge, no similar formulation has been previously proposed.

Based on this idea, I introduce the concept of Analytic Quantum Thermodynamics, which allows us to better understand the essence of thermodynamic processes — essentially, they are processes of energy homogenization. And to describe such processes, the multiplicative entropy based on multiplication is a new tool.

Distinction from Existing “Multiplicative Entropy” Concepts

Although the term “multiplicative entropy” or “multiplicative form of entropy” does appear in some literature, these are still variations of traditional statistical entropy, not departing from the following frameworks:

4.1. Statistical Approach Based on Number of States

• For example, Boltzmann entropy
• Or Shannon entropy

These so-called “multiplicative forms” often involve multiplying entropies of subsystems or using multiplicative techniques in mathematical derivations.

But fundamentally, they remain logarithmic functions over probability distributions.

4.2. No Microscopic Path Recording Function

• They describe only macroscopic final-state differences, without retaining intermediate processes.
• They do not capture how each individual energy transfer affects entropy.

4.3. Unique Features of This Multiplicative Entropy Theory

This model proposes a new way of constructing physical quantities, assigning them clear physical meaning and evolutionary rules:

• It does not start from probability theory, but from the dynamical process of energy redistribution.
• It naturally captures the trend toward homogenization through the product of energy values.
• It provides a computable entropy coordinate, ideal for simulation, prediction, and visualization of system evolution.
• It remains analytically valid at the Planck scale, suggesting natural extension into quantum thermodynamics.
• This multiplicative entropy framework is grounded in a 3+1-dimensional realist foundation—it operates entirely within the physical spacetime we observe, without introducing extra dimensions beyond the three spatial and one temporal dimensions, or invoking abstract mathematical constructs detached from empirical reality.

This study presents an updated extension of a portion of the author’s prior work [3]. For more details on the model, please refer to the previous paper presenting the complete framework.

5. Summary

As early as 1901, Planck attempted to find an analytic representation for entropy [4]. Subsequently, scholars such as Evaldo M.F. Curado and Fernando D. Nobre [5,6]; and Tirnakli, U.[7] have made outstanding contributions in this direction. Building upon these pioneering works, this paper proposes a novel form of analytic entropy—multiplicative entropy—for readers' reference and further exploration.