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Journal of Mathematical Physics

The Boltzmann’s formula for quantum unbounded lattice spin systems

Takahiro Kajisa

Published Apr 1, 2025• 0 citations
Publisher

Abstract

The Boltzmann’s formula, formally known as S = log W, provides a basic concept of a relation between the logarithmic number of micro states log W and the thermodynamic entropy S. In this paper, we prove the Boltzmann’s formula in quantum unbounded lattice systems that includes not only the quantum anharmonic oscillator but also the Bose–Hubbard type model with negative interactions. Our approach is based on a particle number cut-off method, mini-max theory, and convex analysis. In particular, the cut-off method, nurtured in my previous work on classical non-compact spin systems [T. Kajisa. Rev. Math. Phys. 36(08), 2450021 (2024)] is well harmonized with the non-commutative approximation method developed in [Stachura et al., J. Math. Phys. 61(11), 113303 (2020)]. Finally, we present how to relax the assumption of the negativity of interactions.

DOI
10.1063/5.0248081

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The Boltzmann’s formula, formally known as S = log W, provides a basic concept of a relation between the logarithmic number of micro states log W and the thermodynamic entropy S. In this paper, we prove the Boltzmann’s formula in quantum unbounded lattice systems that includes not only the quantum anharmonic oscillator but also the Bose–Hubbard type model with negative interactions. Our approach is based on a particle number cut-off method, mini-max theory, and convex analysis. In particular, the cut-off method, nurtured in my previous work on classical non-compact spin systems [T. Kajisa. Rev. Math. Phys. 36(08), 2450021 (2024)] is well harmonized with the non-commutative approximation method developed in [Stachura et al., J. Math. Phys. 61(11), 113303 (2020)]. Finally, we present how to relax the assumption of the negativity of interactions.
We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.
In this work, we compare three formulations of thermodynamic entropy for a non-interacting bosonic gas: (i) the grand-canonical Bose-Einstein entropy, (ii) the finite-N canonical entropy obtained from the exact partition function (Ziff, Uhlenbeck, and Kac construction), and (iii) the entropy of a multinomial distribution with Boltzmann categorical probabilities and temperature determined from Clausius' equation. It is well-known that the grand-canonical Bose-Einstein systematically overestimates entropy of canonical systems in regimes where particle-number fluctuations are significant. The exact canonical entropy correctly enforces the particle-number constraint, but recent experimental results suggest that it also overestimates particle-number fluctuations below the crossover temperature. The multinomial distribution is less common in thermodynamics. It addresses in a mathematically exact way the puzzle of the famous -log(N!) term introduced by Gibbs as a deus ex machina in discussions of thermodynamic entropy. One remarkable consequence is that the entropy of the multinomial distribution overcomes the issue of negative entropy at low temperature that affects the Gibbs and the Sackur-Tetrode entropies at low temperature. The analysis presented in the paper shows that the multinomial distribution, equipped with a categorical distribution calibrated in such a way that the resulting multinomial entropy fits Clausius' equation, provides accurate approximations to the canonical entropy in the classical regime, while it is smaller than the canonical entropy below the crossover temperature. One feature of the multinomial distribution is that it predicts lower peak variance of the number of particles in the ground state than the canonical distribution. This is in agreement with recent experimental results; hence, this paper identifies the thermodynamically calibrated multinomial distribution as a candidate alternative to the canonical distribution for thermodynamic bosonic entropy in finite systems.
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.
Abstract We employ a mathematical framework based on the Riemann-Hilbert approach developed by Bettelheim et al (2022 J. Phys. A: Math. Gen. 55 135001) to study logarithmic negativity of two intervals of free fermions in the case where the size of the intervals as well as the distance between them is macroscopic. We find that none of the eigenvalues of the density matrix become negative, but rather they develop a small imaginary value, leading to non-zero logarithmic negativity. As an example, we compute negativity at half-filling and for intervals of equal size we find a result of order <?CDATA (log⁡(N))−1(\log(N))^{-1}(log(N))−1?> ( log ( N ) ) − 1 , where N is the typical length scale in units of the lattice spacing. One may compute logarithmic negativity in further situations, but we find that the results are non-universal, depending non-smoothly on the Fermi level and the size of the intervals in units of the lattice spacing.
The field of quantum mechanics has revolutionised physics as a subject. Areas such as information theory, computer science and physical sensing have all been affected by the tremendous successes of various quantum protocols. In this thesis I present my contribution to the development of such non-classical protocols. In classical communication theory a message is always carried by physical particles that interact with a transmitter, after which they travel to a receiver. In this thesis I outline a quantum protocol which allows a receiver to obtain a message without receiving any physical object or particles that have interacted with the transmitter—that is, counterfactually. I build my protocol for counterfactual communication on the principles of interaction-free measurements, ensuring that information always propagates in the opposite direction to the protocol particles. The protocol shows how quantum mechanics breaks the previous premise of communication theory. From the perspective of local observers, it is a beautiful manifestation of the non-locality of interaction-free measurements. Furthermore, it is highly robust against experimental errors and external disturbances. The majority of this part of the thesis is based on my published article ‘Quantum counterfactual communication without a weak trace’ [Phys. Rev. A 94, 062303 (2016)]. Previous to my work, Salih et al. attempted to design a counterfactual communication protocol [Phys. Rev. Lett. 110, 170502 (2013)]. This protocol has been highly controversial. As counterfactual phenomena impose restrictions on the inter-measurement paths of quantum particles, and the physical reality of such paths lacks description in the Copenhagen interpretation of quantum mechanics, an extension of current quantum theory is required to facilitate a discussion. In this thesis I present an operational and interpretation-independent methodology, enabling the discussion of inter-measurement paths of quantum particles. I start by considering the interferometers of counterfactual protocols, making the basic assumption that any quantum evolution naturally involves uncontrolled weak interactions. I then show how the Fisher information of these weak interactions, available at the output of counterfactual experiments, can be used to discuss the pre-measurement past of the particles. Based on this analysis, the protocol developed by Salih et al. is found to strongly violate counterfactuality. However, my protocol is more flexible in that it allows particles to propagate in the opposite direction to the message. This leads to counterfactuality being satisfied—even in the presence of large experimental errors. These results are observed both analytically and numerically. This part of the thesis is based on my article ‘Evaluation of counterfactuality in counterfactual communication protocols’ [Phys. Rev. A 96, 062316 (2017)]. The numerical methods are inspired by another of my publications: ‘Protocol for fermionic positive-operator-valued measures’ [Phys. Rev. A 96, 052305 (2017)]. Finally, as the Fisher information measure is found to be useful in evaluating counterfactual protocols, I extend my work by investigating the quantum Fisher information in experiments with general discrete quantum circuits. I prove that the quantum Fisher information of a two-level interaction in a quantum circuit can be expressed by a simple formula. Under certain phase-relations, the formula provides a straightforward connection between the abstract concept of the inter-measurement wavefunction and the quantum Fisher information at the output. With regard to how the information obtained from a certain volume of space influences our perception of classical objects, I argue that the quantum Fisher information measure is highly useful in describing quantum objects. If this measure is applied to observers with a limited set of the experimental measurement outcomes, a quantum object can appear to follow non-classical discontinuous paths. This supports the remarkable conclusion that our perception of the past of a quantum object is subjectively dependent on the measurement we conduct on it.
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.