Quantum states of macrosystems and entropy

The paper examines and critiques the expression of entropy as the logarithm of the number of quantum states of a physical system. Boltzmann method of expressing entropy as the logarithm of the number of states of a gas with a given total energy is analyzed. We demonstrate that entropy is the product of subquantum processes and show that entropy is expressed as the ratio of the logarithm of the maximum number of realizations, over the observation period, of a macroscopic system’s states with a given total energy, to the number of occurrences of its quantum states over this time.

💡 Research Summary

The paper sets out to question the widely‑used identification of entropy with the logarithm of the number of quantum states, S = k ln W, and to propose an alternative formulation rooted in what the authors call “sub‑quantum processes.” The authors begin by reviewing the traditional statistical‑mechanical derivation of the Boltzmann entropy, emphasizing that it rests on the assumption that all quantum energy levels of a macroscopic system are confined to an extremely narrow interval around the total energy E. They argue that this assumption is physically untenable because the spectrum of a macroscopic system is generally broad and independent of the instantaneous value of E. Consequently, they claim that the conventional identification of W with the number of accessible quantum states is meaningless.

To replace the dubious W, the authors rewrite the canonical probability density ρ(E) = e^{–βE}/Z and show that –ln ρ(E) = (E + F)/kT, where F = –kT ln Z is the Helmholtz free energy. Using the thermodynamic identity E – F = TS, they obtain the familiar relation S = –k ln ρ(E). In this view, entropy is directly linked to the probability density of the total energy rather than to a count of microstates.

The core of the paper introduces the notion of sub‑quantum processes as the hidden dynamical events that cause a system to transition between energy eigenstates. The authors note that ordinary quantum mechanics provides only the amplitudes c_n(t) = c_n(0) e^{–iE_n t/ħ} and the associated probabilities |c_n|², but it does not describe the physical mechanism of the transitions themselves. They hypothesize that an underlying, much faster stochastic dynamics—termed sub‑quantum processes—randomly “visits” the various energy levels during an observation interval.

Within this framework they define N as the total number of visits to any energy level during the observation time, and ν_n as the number of visits to the specific level E_n. The number of distinct visitation configurations (i.e., the number of ways to distribute the N visits among the levels) is given by the multinomial coefficient

  P = N! / (ν_1! ν_2! … ν_l! …).

Applying Stirling’s approximation, they maximize ln P under the constraint ∑ ν_n = N and the normalization ∑ ν_n e^{–βE_n}= N. The maximization yields ν_n ∝ e^{–βE_n}, which reproduces the canonical Boltzmann distribution. Substituting the optimal ν_n back into ln P gives

  ln P_max = –ln ρ(E) = S/k,

so that the entropy is identified with the logarithm of the maximal number of visitation configurations per observation time.

The authors then revisit Boltzmann’s original 1877 work on the distribution of kinetic energy among gas molecules. They reinterpret Boltzmann’s “Komplexion” (the number of permutations of molecules among energy cells) as an early precursor of the modern microstate count W. By showing that Boltzmann’s combinatorial treatment leads to the same exponential occupation numbers ν_k ∝ e^{–βE_k}, they argue that Boltzmann’s method already contained the seeds of the sub‑quantum picture, albeit without an explicit quantum framework.

In the discussion, the authors generalize the entropy formula to S = ln P_N, where P_N is the number of possible configurations of N visits. They claim that irreversible evolution driven by sub‑quantum processes pushes the system toward the configuration that maximizes P, i.e., the state of maximum entropy. The paper concludes by lamenting that the historical “S = k ln W” formula appears to be a convenient but misleading transcription of Boltzmann’s combinatorial result, and that their sub‑quantum interpretation offers a more physically grounded understanding of entropy.

Overall, the manuscript provides a mathematically consistent derivation of the canonical distribution from a maximization of a combinatorial factor associated with hypothetical fast stochastic processes. However, the central hypothesis—existence of sub‑quantum processes that generate the visitation statistics—remains untested and lacks a concrete physical model. The definition of the observation time, the meaning of N and ν_n in real systems, and the connection between the abstract “visits” and measurable quantities are not fully clarified. Consequently, while the paper raises interesting conceptual questions about the foundations of statistical mechanics, it would benefit from a more detailed proposal for experimental verification or a microscopic theory that could substantiate the existence of the proposed sub‑quantum dynamics.