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arXiv:2602.06060

Quantum states of macrosystems and entropy

Maria V. Polski, Vladimir A. Skrebnev

Published Jan 25, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2602.06060

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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.
This paper critically considers the main interpretations of the wave function and offers an interpretation in which wave function is a consequence of subquantum processes taking place at the level of the organization of matter which underlies the phenomena described by quantum mechanics. We show that this interpretation provides a common ground to explain problematic issues of quantum theory (wave function collapse, electrons passing through two slits, behavior of electrons within the atom, quantum entanglement). The paper also shows that entropy is determined by the maximum number of macrosystem states which appear as the result of subquantum processes.
We show that the second law of thermodynamics is rooted in quantum mechanics, inasmuch as allowing the substitution of the Boltzmann constant k and temperature with respectively the Planck constant and quantum numbers. In particular, we will see that the entropy S becomes proportional to the natural logarithm of the average of the squared quantum numbers n2‾, where each quantum number is associated with a quantum state of the constituents of the thermodynamical system under consideration. It is important to stress that the present approach furnishes the corrected Boltzmann entropy expression. Thus, instead of S=klnΩ, with Ω in the former signifying the number of Boltzmann microstates, we land at S=klnn2‾ The results obtained are discussed.
Abstract The quantum thermodynamics of the ideal gas is treated by means of the particle entropies σ=ka. The non-dimensional entropy numbers a are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f=exp(α−a), the density of entropy levels D(a), the dimensionless chemical potential α=μ/kT, etc. – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A, the thermodynamic entropy S (equation of Sackur-Tetrode), the internal and the Helmholtz free energy E and F, respectively, etc. Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S=S max. Against this, the entropy S ′ of an arbitrary non-equilibrium state is always smaller, S′ < S max (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, dS/dt=0, regardless whether we consider an equilibrium state, S=S max, or a non-equilibrium state, S=S′. Consequently, S cannot irreversibly change from S′ to S max. However, any irreversible process is accompanied by a positive entropy production P=dS/dt>0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G =i[ P , t ], which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t. Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt, which are expressed by the uncertainty relation of the N-particle system, ΔPΔt≥(1/2)|〈 G 〉|=(γ/2)k. This P − t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S′ to S max. Now the irreversible steps are correctly described by the entropy production P=dS/dt > 0, and the thermal equilibrium by P=0, ∆P=0, and thus the lifetime ∆t=∞.
The paper analyzes the entropy of a system composed by non-interacting and indistinguishable particles whose quantum state numbers are modelled as independent and identically distributed classical random variables. The crucial observation is that, under this assumption, whichever is the number of particles that constitute the system, the occupancy numbers of system's quantum (micro)states are multinomially distributed. This observation leads to an entropy formula for the physical system, which is nothing else than the entropy formula of the multinomial distribution, for which we claim novelty, in the sense that it is proposed here for the first time that the entropy of the multinomial distribution is the entropy of the physical system. The entropy formula of the multinomial distribution unveils yet unexplored connections between information theory and statistical mechanics, among which we mention the connection between conditional entropy of the random microstate given the random occupancy numbers and the Boltzmann-Planck entropy log⁡(W)\log(W)log(W) and between these two and the Gibbs correction term log⁡(N!)\log(N!)log(N!), thermalization and communication-theoretic preparation of a thermal state, accessible information of the thermal state and physical entropy of the thermalized system. A noticeable specific result that descends from our approach is the exact quantum correction to the textbook Sackur-Tetrode formula for the entropy of an ideal gas at the thermal equilibrium in a container.
This chapter has two goals. The first is to develop the quantum version of multiplicity. That will show us how entropy is expressed in quantum theory. The second goal is to develop a general quantitative definition of temperature. Entropy plays a key role in that definition. The density of states When a physical system has reached thermal equilibrium, its macroscopic properties do not change with time. In quantum theory, the energy eigenstates of an isolated system provide predictions and estimates that are constant in time; therefore such states are appropriate for a quantum description of thermal equilibrium. To be sure, the information at hand will not enable us to select a single state as uniquely the correct state to use. We will be driven to consider many states and to form sums over them. This section develops a mathematical technique for working with such sums. When the system is both isolated and of finite size (as we shall specify here), the energy eigenstates form a discrete set, whose members we can arrange and label in order of increasing energy. (If any two distinct states happen to have the same energy, we just assign them consecutive labels.) Typically, the states will be densely spaced in energy, and so a sum over a range of states can often be approximated adequately by an integral with respect to energy, provided that we have constructed an appropriate density of states : a function that specifies the number of energy eigenstates per unit energy interval.