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

^{1,b)}

### Abstract

A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the “global temperature,” derived from the entropyequation of state, with the “local temperature” determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.

The authors acknowledge the support by Univesità degli Studi di Padova through 60% grants.

I. INTRODUCTION

II. THE MODEL

III. THE FIXED EXPECTATION ENERGY ENSEMBLE (FEEE)

IV. THE RANDOM PURE STATE ENSEMBLE (RPSE)

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Entropy
- 40.0
- Electron densities of states
- 17.0
- Canonical forms
- 13.0
- Hilbert space
- 13.0
- Thermodynamic properties
- 13.0

## Figures

Domain for the independent occupation numbers represented as the gray area.

Domain for the independent occupation numbers represented as the gray area.

Gaussian density of states of ten identical spins (solid line). The histogram shows the degeneracy calculated according to Eq. (7).

Gaussian density of states of ten identical spins (solid line). The histogram shows the degeneracy calculated according to Eq. (7).

Average FEEE entropy per spin as a function of the internal energy per spin for a system composed of three levels systems. The continuous line represents the asymptotic behavior described by Eq. (23), while the other lines are the results of calculation according to Eq. (20) for finite systems: (dashed line), (dashed-dotted line), and (dotted line).

Average FEEE entropy per spin as a function of the internal energy per spin for a system composed of three levels systems. The continuous line represents the asymptotic behavior described by Eq. (23), while the other lines are the results of calculation according to Eq. (20) for finite systems: (dashed line), (dashed-dotted line), and (dotted line).

Parameter [Eq. (27)] which quantifies the deviations from the canonical form of the reduced density matrix as a function of the internal energy. The dashed and the dotted lines refer to a system with and with spins, while the continuous line represents the asymptotic result Eq. (29) for a large number of spins.

Parameter [Eq. (27)] which quantifies the deviations from the canonical form of the reduced density matrix as a function of the internal energy. The dashed and the dotted lines refer to a system with and with spins, while the continuous line represents the asymptotic result Eq. (29) for a large number of spins.

Ratio between the dimensions of RPSE active space and of the overall Hilbert space, in a logarithmic scale, as function of the scaled cutoff energy. Dotted lines: Eq. (32) resulting from the Gaussian density of states; continuous lines: exact calculations for a system of and of spins, respectively.

Ratio between the dimensions of RPSE active space and of the overall Hilbert space, in a logarithmic scale, as function of the scaled cutoff energy. Dotted lines: Eq. (32) resulting from the Gaussian density of states; continuous lines: exact calculations for a system of and of spins, respectively.

Logarithm of the dimensions of RPSE active space, divided by the number of spin, as a function of the energy cut off. Continuous line: asymptotic dependence [Eqs. (49) and (45)]; dotted lines: exact results for finite systems approaching the asymptotic profile for an increasing number of spins: , 30, and 100.

Logarithm of the dimensions of RPSE active space, divided by the number of spin, as a function of the energy cut off. Continuous line: asymptotic dependence [Eqs. (49) and (45)]; dotted lines: exact results for finite systems approaching the asymptotic profile for an increasing number of spins: , 30, and 100.

RPSE internal energy a function of the energy cutoff. Continuous line: asymptotic behavior described by Eqs. (56) and (58). The dotted lines represent the result for finite system approaching the asymptotic profile for an increasing number of spins: , 50, and 150.

RPSE internal energy a function of the energy cutoff. Continuous line: asymptotic behavior described by Eqs. (56) and (58). The dotted lines represent the result for finite system approaching the asymptotic profile for an increasing number of spins: , 50, and 150.

Inverse temperature as function of the internal energy from RPSE.

Inverse temperature as function of the internal energy from RPSE.

Dependence of the RPSE average of the reduced density matrix elements on the scaled cutoff energy. The asymptotic function which describes the dependence of the typical values on the energy Eq. (71) is reported as continuous lines. The other lines represent the result for finite system approaching the asymptotic profile for an increasing number of spins: , 30, and 100.

Dependence of the RPSE average of the reduced density matrix elements on the scaled cutoff energy. The asymptotic function which describes the dependence of the typical values on the energy Eq. (71) is reported as continuous lines. The other lines represent the result for finite system approaching the asymptotic profile for an increasing number of spins: , 30, and 100.

Plot of the location of maximum with respect to (continuous line) within the existence domain of the function (gray area).

Plot of the location of maximum with respect to (continuous line) within the existence domain of the function (gray area).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content