Beyond quantum microcanonical statistics

TL;DR

This paper proposes a new quantum statistical framework for isolated systems that derives thermodynamic properties from wavefunction analysis, bridging the gap between pure state dynamics and traditional mixed state thermodynamics.

Contribution

It introduces a formalism based on wavefunction probability distributions that naturally yields thermodynamic behavior in the macroscopic limit, without relying on mixed states.

Findings

Emergence of typical thermodynamic values from wavefunction analysis

Recovery of canonical statistics for subsystems

Thermodynamic properties derived without specific physical constraints

Abstract

Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e. the wavefunction, of an isolated system which is determined to calculate molecular properties and to consider the time evolution according to the unitary Schr\"odinger equation. On the other hand a mixed state, i.e. a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.