On quantum statistical mechanics of non-Hamiltonian systems

Román Ingarden

Kui Cao, Su-Peng Kou

We present a systematic study of statistical mechanics for non-Hermitian quantum systems. Our work reveals that the stability of a non-Hermitian system necessitates the existence of a single path-dependent conserved quantity, which, in conjunction with the system's Hamiltonian, dictates the equilibrium state. By elucidating the relationship between the Hamiltonian and the supported conserved quantity, we propose criteria for discerning equilibrium states with finite relaxation times. Although our findings indicate that only non-Hermitian systems with real energy spectrum precisely possess such conserved quantities, we also demonstrate that an effective conserved quantity can manifest in certain systems with complex energy spectra. The effective conserved quantity, alongside the effective transitions within their associated subspace, collectively determines the system's equilibrium state. Our results provide valuable insights into non-Hermitian systems across more realistic contexts and hold potential for applications in a diverse range of physical systems.

Ola Bratteli, Derek W. Robinson

Ajay Patwardhan

The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint components in phase space. Partition functions including the invariants, Kolmogorov Entropy and Euler number are introduced. The ergodic hypothesis for partial ergodicity is discussed. In the context of Quantum Mechanics the presence of symmetry groups with irreducible representations gives rise to degenerate and non degenerate spectrum for the Hamiltonian. Quantum Statistical Mechanics is formulated including these two cases ; by including the multiplicity dimension of the group representation and the Casimir invariants into the Partition function. The possibility of new kinds of phase transitions is discussed. The occurence of systems with non simply connected configuration spaces and Quantum Mechanics for them, also requires a possible generalisation of Statistical Mechanics. The developments of Quantum pure, mixed, and entangled states has made it neccessary to understand the Statistical Mechanics of the multipartite N particle system. And to obtain in terms of the density matrices, written in energy basis, the Trace of the Gibbs operator as the Partition function and use it to get statistical averages of operators. There are some issues of definition, interpretation and application that are discussed.

Derek W. Robinson

William L. Harper, C. A. Hooker

The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.