Energy Product Entropy VS Statistical Entropy-An Analytic Quantum Thermodynamics Perspective

2.1. 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.

2.2. 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

2.3. 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).

3.1. Entropy value of closed system S=∏mᵢ, i∈N   ①

Energy of closed system=constant=∑mᵢ, i∈N   ②

3.2. 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)

3.3. 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.

3.4. Logarithmic Relation:

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

3.5. 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.

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 an excellent 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:

5.1. 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.

5.2. 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.

☑ Unique Features of This Multiplicative Entropy Theory

Feature	This Multiplicative Entropy	Existing “Multiplicative Entropy”
Can Track Evolutionary Paths	✅ Yes, with clear changes at each step	No, only describes macroscopic end states
Preserves Microscopic Details	✅ Yes, path-dependent	No, focuses only on probability distributions
Has Temporal Directionality	✅ Yes, defined by local energy flow	No, needs extra assumption for time direction
Suitable for Simulations	✅ Yes, ideal for numerical modeling	No, mainly used for theoretical analysis
Is an Analytical Function Form	✅ Yes, not based on statistical averaging	No, relies on ensemble averages

☑ Why This Is a “New Discovery”

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 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.