The paper analyzes the probability distribution of the occupancy numbers and the entropy of a system at the equilibrium composed by an arbitrary number of non-interacting bosons. The probability distribution is obtained through two approaches: one involves tracing out the environment from a bosonic eigenstate of the combined environment and system of interest (the empirical approach), while the other involves tracing out the environment from the mixed state of the combined environment and system of interest (the Bayesian approach). In the thermodynamic limit, the two coincide and are equal to the multinomial distribution. Furthermore, the paper proposes to identify the physical entropy of the bosonic system with the Shannon entropy of the occupancy numbers, fixing certain contradictions that arise in the classical analysis of thermodynamic entropy. Finally, by leveraging an information-theoretic inequality between the entropy of the multinomial distribution and the entropy of the multivariate hypergeometric distribution, Bayesianism of information theory and empiricism of statistical mechanics are integrated into a common "infomechanical" framework.
We study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function
Ψ from a micro-canonical energy shell of a quantum system
S weakly coupled to a large but finite quantum system
B, the reduced density matrix
ρ^ΨS=trB∣Ψ⟩⟨Ψ∣ is approximately equal to the canonical density matrix
ρ^can=Zcan−1exp(−βH^S). Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix
ρ^gc=Zgc−1exp(−β(H^S−μ1N^1S−…−μrN^rS)) with
N^iS the number operator for molecules of type
i in the system
S. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems
S defined by a spatial region which particles may enter or leave. It includes statements about how
ρ^gc arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace
Hgmc⊂HS⊗HB (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical
Ψ in
Hgmc, as well as statements about the distribution of the (conditional) wave function
ψS of
S, which turns out to be a so-called GAP or Scrooge measure. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.
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 analytical probability distribution of finite systems obeying Bose-Einstein statistics in one, two, and three dimensions are investigated by using a canonical ensemble approach. Starting from the canonical partition function of the system, a unified approach is provided to study the probability distribution of a condensate for various confinements and in different dimensions. Based on the probability distribution function, it is straightforward to obtain the mean ground state occupation number and particle number fluctuations of the condensate. It is found that the particle number fluctuations in a trapped Bose gas are strongly dependent on the type of confining potential and on the dimensionality of the system.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (
N→∞) limit the density matrix emerges for small subsystems from almost all pure states of large quantum many-body systems. This notion of canonical is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. We show that quantum chaos plays a crucial role in the emergence of typicality for large but finite quantum systems. We demonstrate that the degree of typicality can be quantitatively controlled and tuned by the degree of quantum chaoticity present in the many-body system.