Grand-canonical Typicality Achieves Approximate Density Matrices for Weakly Coupled Systems

Understanding how macroscopic systems settle into thermodynamic equilibrium remains a central challenge in physics. Cedric Igelspacher, Roderich Tumulka, and Cornelia Vogel, from the Mathematics Institutes at Eberhard Karls University Tübingen and Ludwig Maximilians University, investigate the emergence of the grand-canonical density matrix , a crucial concept for describing systems where particle numbers can fluctuate. Their research explores whether typical wave functions within a defined energy range and particle count closely resemble the predictions of grand-canonical statistics, extending previous work on canonical typicality to encompass chemical reactions and systems with permeable boundaries. This work is significant because it provides a rigorous justification for using the grand-canonical ensemble, not just as a practical tool, but as a genuinely accurate description of the state of a large system. By examining the density matrix and wave function distributions, the authors offer new insights into the foundations of statistical mechanics and its application to complex physical scenarios.

By examining the density matrix and wave function distributions, the authors offer new insights into the foundations of statistical mechanics and its application to complex physical scenarios.

Grand Canonical Emergence via Fock Space Analysis

The study investigates the emergence of the grand-canonical density matrix in macroscopic systems, extending canonical typicality to scenarios involving chemical reactions and systems with permeable spatial boundaries. Researchers meticulously examined how the reduced density matrix, derived from a generalized micro-canonical Hilbert subspace defined by energy and particle number intervals, approximates the grand-canonical density matrix, building upon existing theorems by Popescu et al. and Goldstein et al.

To explore chemical equilibrium, the team constructed a framework utilizing Fock spaces, infinite-dimensional spaces representing the number of molecules of each substance. The Hilbert space was defined as the tensor product of these Fock spaces, accounting for both fermionic and bosonic properties dependent on molecular composition. Crucially, the researchers developed a Hamiltonian operator, ˆH0, representing the combined kinetic and internal energies of individual molecules, alongside their interactions with external fields.

This Hamiltonian incorporates the ground state energy, E0i, of each molecule, linked to reaction energies, δEl, through a system of equations ensuring physical consistency. The methodology further involved defining a micro-canonical interval [E −∆E, E] to represent the initial energy of the system, then determining the equilibrium numbers of each substance, neq,i. Scientists then demonstrated that the conditional wave function exhibits a probability distribution consistent with the grand-canonical ensemble when projected onto a typical orthonormal basis.

A key innovation lies in the mathematical theorem, Proposition 1, which generalizes previous results concerning the canonical density matrix, providing a rigorous foundation for these grand-canonical extensions. This approach enables the study to address previously unexplored territory concerning the approach to thermal equilibrium, both in the canonical and grand-canonical cases.

Grand-Canonical Density from Micro-Canonical Wave Functions

Scientists have established a robust connection between the grand-canonical density matrix and the behaviour of macroscopic systems, extending the principle of canonical typicality to encompass scenarios involving variable particle numbers. The research demonstrates that for a typical wave function within a micro-canonical energy shell, the reduced density matrix closely approximates the grand-canonical density matrix when considering systems where molecules of a specific type can be added or removed.

Experiments, conducted through theoretical modelling, reveal that the density matrix of a generalized micro-canonical Hilbert subspace yields the grand-canonical density matrix after tracing out the environment. The team measured the reduced density matrix of typical states, confirming its proximity to the grand-canonical form, and further established that the conditional wave function exhibits a probability distribution consistent with grand-canonical expectations when projected onto a typical orthonormal basis. Results demonstrate the validity of both the density matrix and wave function distribution within the grand-canonical framework, extending these considerations to generalized Gibbs ensembles applicable to systems with conserved macroscopic observables.

Further investigations focused on the generalized Gibbs ensemble, represented by the equation ˆρgG = Z−1 gG exp K X k=1 λk ˆQk , where ˆQk are conserved macroscopic observables. Scientists formulated the General Gibbs Principle, stating that under specific conditions , namely, commuting ˆQk and a large dimensionality of the generalized micro-canonical subspace , the resulting density matrix ˆρgmc is equivalent to the generalized Gibbs ensemble ˆρgG with appropriately chosen parameters λk.

Measurements confirm this equivalence holds true, particularly when the observables ˆQk are approximately extensive, meaning their values scale with the size of the spatial region under consideration. The breakthrough delivers a locally equivalent relationship between ˆρgG and ˆρgmc, meaning that for any sufficiently small spatial region, the two density matrices are indistinguishable. This locally equivalent behaviour was demonstrated through derivations applicable when each ˆQk is approximately extensive, solidifying the understanding of thermal equilibrium in systems governed by conserved macroscopic quantities, and providing a physical basis for the approximate equality between the grand-canonical and micro-canonical descriptions.

Grand Canonical Emergence From Microcanonical Subspaces

This work establishes a framework for understanding how the grand-canonical density matrix emerges in macroscopic systems, extending the concept of canonical typicality to scenarios involving variable particle numbers. The authors demonstrate that, under typical conditions within a generalized micro-canonical subspace defined by energy and particle number constraints, the reduced density matrix closely approximates the grand-canonical form.

Crucially, they also show that the conditional wave function exhibits a probability distribution consistent with the grand-canonical ensemble when projected onto a typical orthonormal basis. The significance of these findings lies in providing a justification for the widespread use of the grand-canonical ensemble in describing systems open to particle exchange, such as those undergoing chemical reactions or defined by permeable boundaries. By rigorously connecting microscopic dynamics to macroscopic statistical descriptions, this research strengthens the foundations of statistical mechanics.

The authors acknowledge a limitation in that their analysis relies on the assumption of rationally independent eigenvalues for the Hamiltonian, though they note this condition holds generically. Future research directions include extending these considerations to generalized Gibbs ensembles, applicable to systems with conserved macroscopic observables, and further exploring the implications for non-equilibrium dynamics.