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arXiv:2610.18553

On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems

Cedric Igelspacher, Roderich Tumulka

quant-phSubmitted Oct 13, 2026• Announced Oct 14, 2026• 0 citations
v1 · Oct 13, 2026v2 · Oct 26, 2026
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Abstract

We consider a closed macroscopic quantum system in a pure state $ψ$ evolving unitarily with Hamiltonian $H$ in a finite-dimensional Hilbert space $\mathscr{H}$. Following von Neumann, we assume a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces $\mathscr{H}_ν$, where each $ν$ represents a macro state, and $\mathscr{H}_ν$ contains the pure states compatible with $ν$. We define the quantum Boltzmann entropy for a macro state $ν$ as $S_\mathrm{qB}(ν) = k_{\mathrm{B}} \log \dim \mathscr{H}_ν$, where $k_\mathrm{B}$ is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that $S_\mathrm{qB}$ predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of $H$ and $\mathscr{H}_ν$.

Comments
13 pages, 4 figures

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We consider a closed macroscopic quantum system in a pure state ψψψ evolving unitarily with Hamiltonian HHH in a finite-dimensional Hilbert space H\mathscr{H}H. Following von Neumann, we assume a decomposition of H\mathscr{H}H into mutually orthogonal subspaces Hν\mathscr{H}_νHν​, where each ννν represents a macro state, and Hν\mathscr{H}_νHν​ contains the pure states compatible with ννν. We define the quantum Boltzmann entropy for a macro state ννν as SqB(ν)=kBlog⁡dim⁡HνS_\mathrm{qB}(ν) = k_{\mathrm{B}} \log \dim \mathscr{H}_νSqB​(ν)=kB​logdimHν​, where kBk_\mathrm{B}kB​ is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqBS_\mathrm{qB}SqB​ predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of HHH and Hν\mathscr{H}_νHν​.
For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
Abstract We consider a macroscopic quantum system with unitarily evolving pure state ψt∈H\psi _t\in \mathcal {H}ψt​∈H ψ t ∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces Hν\mathcal {H}_\nu Hν​ H ν (macro spaces) of H\mathcal {H}H H . Let PνP_\nu Pν​ P ν denote the projection to Hν\mathcal {H}_\nu Hν​ H ν . We prove two facts about the evolution of the superposition weights ∥Pνψt∥2\Vert P_\nu \psi _t\Vert ^2∥Pν​ψt​∥2 ‖ P ν ψ t ‖ 2 : First, given any T>0T>0T>0 T > 0 , for most initial states ψ0\psi _0ψ0​ ψ 0 from any particular macro space Hμ\mathcal {H}_\mu Hμ​ H μ (possibly far from thermal equilibrium), the curve t↦∥Pνψt∥2t\mapsto \Vert P_\nu \psi _t\Vert ^2t↦∥Pν​ψt​∥2 t ↦ ‖ P ν ψ t ‖ 2 is approximately the same (i.e., nearly independent of ψ0\psi _0ψ0​ ψ 0 ) on the time interval [0, T]. And second, for most ψ0\psi _0ψ0​ ψ 0 from Hμ\mathcal {H}_\mu Hμ​ H μ and most t∈[0,∞)t\in [0,\infty )t∈[0,∞) t ∈ [ 0 , ∞ ) , ∥Pνψt∥2\Vert P_\nu \psi _t\Vert ^2∥Pν​ψt​∥2 ‖ P ν ψ t ‖ 2 is close to a value MμνM_{\mu \nu }Mμν​ M μ ν that is independent of both t and ψ0\psi _0ψ0​ ψ 0 . The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.
For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
A way to construct Boltzmann entropy, i.e., the entropy as a function of a microscopic pure state, for quantum field systems is proposed. Operators that shift the field in wave-vector space are used in the construction. By employing an assumption that some terms emerging due to the shift are negligible in the thermodynamic limit, it is shown that, for almost all states in the ensemble of pure states corresponding to a thermodynamic state, the value of the proposed Boltzmann entropy coincides with that of the thermodynamic entropy for the thermodynamic state. For general self-interacting fields, the Boltzmann entropy evolves with time under Hamiltonian dynamics, so that it is capable of characterizing the thermalization of isolated quantum field systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.