^{1,a)}and Giorgio J. Moro

^{1,b)}

### 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 is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger 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.

I. INTRODUCTION

II. PURE STATE DISTRIBUTION

III. RANDOM PURE STATE ENSEMBLE

IV. THERMODYNAMICAL BEHAVIOR

V. CONCLUDING REMARKS

### Key Topics

- Entropy
- 42.0
- Statistical properties
- 31.0
- Wave functions
- 25.0
- Thermodynamic properties
- 23.0
- Hilbert space
- 17.0

## Figures

Time evolution and distribution of the first diagonal element μ_{0, 0} of the reduced density matrix for an oscillator in the randomly perturbed Einstein oscillator model. The following parameters have been employed: *n* = 5, *E* _{max }/ℏω_{0} = 5.1 (corresponding to a dimension *N* = 252 of the active Hilbert space), λ/ℏω_{0} = 10^{−3}. Panels A and B display the evolution μ_{0, 0}(*t*) for two different random choices of the initial wavefunction, with the corresponding statistical distributions of μ_{0, 0} reported in panels C and D. The asymptotic time average is indicated by the red dotted line.

Time evolution and distribution of the first diagonal element μ_{0, 0} of the reduced density matrix for an oscillator in the randomly perturbed Einstein oscillator model. The following parameters have been employed: *n* = 5, *E* _{max }/ℏω_{0} = 5.1 (corresponding to a dimension *N* = 252 of the active Hilbert space), λ/ℏω_{0} = 10^{−3}. Panels A and B display the evolution μ_{0, 0}(*t*) for two different random choices of the initial wavefunction, with the corresponding statistical distributions of μ_{0, 0} reported in panels C and D. The asymptotic time average is indicated by the red dotted line.

Distribution within RPSE of first element, , of the equilibrium reduced density matrix of a single oscillator of the randomly perturbed Einstein oscillator model as obtained from the sampling of population sets. The distributions refer to systems composed of different number of oscillators (*n* = 5, 6, 7, 8), with *E* _{max }/*n*ℏω_{0} = 2 and λ/ℏω_{0} = 10^{−3}. In the inset we have reported the numerically determined variance (in a logarithmic scale) as a function of the number *n* of oscillators.

Distribution within RPSE of first element, , of the equilibrium reduced density matrix of a single oscillator of the randomly perturbed Einstein oscillator model as obtained from the sampling of population sets. The distributions refer to systems composed of different number of oscillators (*n* = 5, 6, 7, 8), with *E* _{max }/*n*ℏω_{0} = 2 and λ/ℏω_{0} = 10^{−3}. In the inset we have reported the numerically determined variance (in a logarithmic scale) as a function of the number *n* of oscillators.

Scaled internal energy per component *U*/*n*ℏω_{0} (panel A) and entropy per component *S*/*nk* _{ B } (panel B) as functions of the scaled cutoff energy *e* _{max }/ℏω_{0} per component, for systems of *n* = 5 (red points), *n* = 10 (black points), and *n* = 50 (blue points) oscillators. The asymptotic *n* → ∞ profiles are represented with black continuous lines.

Scaled internal energy per component *U*/*n*ℏω_{0} (panel A) and entropy per component *S*/*nk* _{ B } (panel B) as functions of the scaled cutoff energy *e* _{max }/ℏω_{0} per component, for systems of *n* = 5 (red points), *n* = 10 (black points), and *n* = 50 (blue points) oscillators. The asymptotic *n* → ∞ profiles are represented with black continuous lines.

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